Daily Papers

Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such constraints can be softly imposed via loss function penalties, recent advancements in differentiable physics and optimization improve performance by incorporating PDE-constrained optimization as individual layers in neural networks. This enables a stricter adherence to physical constraints. However, imposing hard constraints significantly increases computational and memory costs, especially for complex dynamical systems. This is because it requires solving an optimization problem over a large number of points in a mesh, representing spatial and temporal discretizations, which greatly increases the complexity of the constraint. To address this challenge, we develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE), which can be used with any neural network architecture. Our approach imposes the constraint over smaller decomposed domains, each of which is solved by an "expert" through differentiable optimization. During training, each expert independently performs a localized backpropagation step by leveraging the implicit function theorem; the independence of each expert allows for parallelization across multiple GPUs. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting for predicting the dynamics of challenging non-linear systems. We also improve training stability and require significantly less computation time during both training and inference stages.

Unsupervised contrastive learning has achieved outstanding success, while the mechanism of contrastive loss has been less studied. In this paper, we concentrate on the understanding of the behaviours of unsupervised contrastive loss. We will show that the contrastive loss is a hardness-aware loss function, and the temperature {\tau} controls the strength of penalties on hard negative samples. The previous study has shown that uniformity is a key property of contrastive learning. We build relations between the uniformity and the temperature {\tau} . We will show that uniformity helps the contrastive learning to learn separable features, however excessive pursuit to the uniformity makes the contrastive loss not tolerant to semantically similar samples, which may break the underlying semantic structure and be harmful to the formation of features useful for downstream tasks. This is caused by the inherent defect of the instance discrimination objective. Specifically, instance discrimination objective tries to push all different instances apart, ignoring the underlying relations between samples. Pushing semantically consistent samples apart has no positive effect for acquiring a prior informative to general downstream tasks. A well-designed contrastive loss should have some extents of tolerance to the closeness of semantically similar samples. Therefore, we find that the contrastive loss meets a uniformity-tolerance dilemma, and a good choice of temperature can compromise these two properties properly to both learn separable features and tolerant to semantically similar samples, improving the feature qualities and the downstream performances.

Nesterov's accelerated gradient (AG) is a popular technique to optimize objective functions comprising two components: a convex loss and a penalty function. While AG methods perform well for convex penalties, such as the LASSO, convergence issues may arise when it is applied to nonconvex penalties, such as SCAD. A recent proposal generalizes Nesterov's AG method to the nonconvex setting. The proposed algorithm requires specification of several hyperparameters for its practical application. Aside from some general conditions, there is no explicit rule for selecting the hyperparameters, and how different selection can affect convergence of the algorithm. In this article, we propose a hyperparameter setting based on the complexity upper bound to accelerate convergence, and consider the application of this nonconvex AG algorithm to high-dimensional linear and logistic sparse learning problems. We further establish the rate of convergence and present a simple and useful bound to characterize our proposed optimal damping sequence. Simulation studies show that convergence can be made, on average, considerably faster than that of the conventional proximal gradient algorithm. Our experiments also show that the proposed method generally outperforms the current state-of-the-art methods in terms of signal recovery.

When training or evaluating deep learning models, two essential parts are picking the proper loss function and deciding on performance metrics. In this paper, we provide a comprehensive overview of the most common loss functions and metrics used across many different types of deep learning tasks, from general tasks such as regression and classification to more specific tasks in Computer Vision and Natural Language Processing. We introduce the formula for each loss and metric, discuss their strengths and limitations, and describe how these methods can be applied to various problems within deep learning. This work can serve as a reference for researchers and practitioners in the field, helping them make informed decisions when selecting the most appropriate loss function and performance metrics for their deep learning projects.

We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change arbitrarily over K episodes and the state space can be arbitrarily large. We assume that the Q-function of any policy is linear in some known features, that is, a linear function approximation exists. The best existing regret upper bound for this setting (Luo et al., 2021) is of order mathcal O(K^{2/3}) (omitting all other dependencies), given access to a simulator. This paper provides two algorithms that improve the regret to mathcal O(sqrt K) in the same setting. Our first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-barrier regularizer. This analysis allows the loss estimators to be arbitrarily negative and might be of independent interest. Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to logarithmic terms and dependency on the horizon). Moreover, we also extend the first algorithm to simulator-free linear MDPs, which achieves mathcal O(K^{8/9}) regret and greatly improves over the best existing bound mathcal O(K^{14/15}). This algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu & Olkhovskaya (2020), which could again be of independent interest.

Cross-entropy is a widely used loss function in applications. It coincides with the logistic loss applied to the outputs of a neural network, when the softmax is used. But, what guarantees can we rely on when using cross-entropy as a surrogate loss? We present a theoretical analysis of a broad family of loss functions, comp-sum losses, that includes cross-entropy (or logistic loss), generalized cross-entropy, the mean absolute error and other cross-entropy-like loss functions. We give the first H-consistency bounds for these loss functions. These are non-asymptotic guarantees that upper bound the zero-one loss estimation error in terms of the estimation error of a surrogate loss, for the specific hypothesis set H used. We further show that our bounds are tight. These bounds depend on quantities called minimizability gaps. To make them more explicit, we give a specific analysis of these gaps for comp-sum losses. We also introduce a new family of loss functions, smooth adversarial comp-sum losses, that are derived from their comp-sum counterparts by adding in a related smooth term. We show that these loss functions are beneficial in the adversarial setting by proving that they admit H-consistency bounds. This leads to new adversarial robustness algorithms that consist of minimizing a regularized smooth adversarial comp-sum loss. While our main purpose is a theoretical analysis, we also present an extensive empirical analysis comparing comp-sum losses. We further report the results of a series of experiments demonstrating that our adversarial robustness algorithms outperform the current state-of-the-art, while also achieving a superior non-adversarial accuracy.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as properness, which asserts that Bayes' rule is optimal. Recent works have sought to learn losses and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps R to [0,1] to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between R^{C-1} and the projected probability simplex Delta^{C-1} by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns proper canonical losses and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a t-test at 99% significance on all datasets with greater than 10 classes.

Time Series Forecasting has been an active area of research due to its many applications ranging from network usage prediction, resource allocation, anomaly detection, and predictive maintenance. Numerous publications published in the last five years have proposed diverse sets of objective loss functions to address cases such as biased data, long-term forecasting, multicollinear features, etc. In this paper, we have summarized 14 well-known regression loss functions commonly used for time series forecasting and listed out the circumstances where their application can aid in faster and better model convergence. We have also demonstrated how certain categories of loss functions perform well across all data sets and can be considered as a baseline objective function in circumstances where the distribution of the data is unknown. Our code is available at GitHub: https://github.com/aryan-jadon/Regression-Loss-Functions-in-Time-Series-Forecasting-Tensorflow.

We design learning rate schedules that minimize regret for SGD-based online learning in the presence of a changing data distribution. We fully characterize the optimal learning rate schedule for online linear regression via a novel analysis with stochastic differential equations. For general convex loss functions, we propose new learning rate schedules that are robust to distribution shift, and we give upper and lower bounds for the regret that only differ by constants. For non-convex loss functions, we define a notion of regret based on the gradient norm of the estimated models and propose a learning schedule that minimizes an upper bound on the total expected regret. Intuitively, one expects changing loss landscapes to require more exploration, and we confirm that optimal learning rate schedules typically increase in the presence of distribution shift. Finally, we provide experiments for high-dimensional regression models and neural networks to illustrate these learning rate schedules and their cumulative regret.

A novel gradient boosting framework is proposed where shallow neural networks are employed as ``weak learners''. General loss functions are considered under this unified framework with specific examples presented for classification, regression, and learning to rank. A fully corrective step is incorporated to remedy the pitfall of greedy function approximation of classic gradient boosting decision tree. The proposed model rendered outperforming results against state-of-the-art boosting methods in all three tasks on multiple datasets. An ablation study is performed to shed light on the effect of each model components and model hyperparameters.

Most value function learning algorithms in reinforcement learning are based on the mean squared (projected) Bellman error. However, squared errors are known to be sensitive to outliers, both skewing the solution of the objective and resulting in high-magnitude and high-variance gradients. To control these high-magnitude updates, typical strategies in RL involve clipping gradients, clipping rewards, rescaling rewards, or clipping errors. While these strategies appear to be related to robust losses -- like the Huber loss -- they are built on semi-gradient update rules which do not minimize a known loss. In this work, we build on recent insights reformulating squared Bellman errors as a saddlepoint optimization problem and propose a saddlepoint reformulation for a Huber Bellman error and Absolute Bellman error. We start from a formalization of robust losses, then derive sound gradient-based approaches to minimize these losses in both the online off-policy prediction and control settings. We characterize the solutions of the robust losses, providing insight into the problem settings where the robust losses define notably better solutions than the mean squared Bellman error. Finally, we show that the resulting gradient-based algorithms are more stable, for both prediction and control, with less sensitivity to meta-parameters.

L_2 regularization and weight decay regularization are equivalent for standard stochastic gradient descent (when rescaled by the learning rate), but as we demonstrate this is not the case for adaptive gradient algorithms, such as Adam. While common implementations of these algorithms employ L_2 regularization (often calling it "weight decay" in what may be misleading due to the inequivalence we expose), we propose a simple modification to recover the original formulation of weight decay regularization by decoupling the weight decay from the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam and (ii) substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter). Our proposed decoupled weight decay has already been adopted by many researchers, and the community has implemented it in TensorFlow and PyTorch; the complete source code for our experiments is available at https://github.com/loshchil/AdamW-and-SGDW

Cross-entropy loss and focal loss are the most common choices when training deep neural networks for classification problems. Generally speaking, however, a good loss function can take on much more flexible forms, and should be tailored for different tasks and datasets. Motivated by how functions can be approximated via Taylor expansion, we propose a simple framework, named PolyLoss, to view and design loss functions as a linear combination of polynomial functions. Our PolyLoss allows the importance of different polynomial bases to be easily adjusted depending on the targeting tasks and datasets, while naturally subsuming the aforementioned cross-entropy loss and focal loss as special cases. Extensive experimental results show that the optimal choice within the PolyLoss is indeed dependent on the task and dataset. Simply by introducing one extra hyperparameter and adding one line of code, our Poly-1 formulation outperforms the cross-entropy loss and focal loss on 2D image classification, instance segmentation, object detection, and 3D object detection tasks, sometimes by a large margin.

Empirical risk minimization (ERM) with a computationally feasible surrogate loss is a widely accepted approach for classification. Notably, the convexity and calibration (CC) properties of a loss function ensure consistency of ERM in maximizing accuracy, thereby offering a wide range of options for surrogate losses. In this article, we propose a novel ensemble method, namely EnsLoss, which extends the ensemble learning concept to combine loss functions within the ERM framework. A key feature of our method is the consideration on preserving the "legitimacy" of the combined losses, i.e., ensuring the CC properties. Specifically, we first transform the CC conditions of losses into loss-derivatives, thereby bypassing the need for explicit loss functions and directly generating calibrated loss-derivatives. Therefore, inspired by Dropout, EnsLoss enables loss ensembles through one training process with doubly stochastic gradient descent (i.e., random batch samples and random calibrated loss-derivatives). We theoretically establish the statistical consistency of our approach and provide insights into its benefits. The numerical effectiveness of EnsLoss compared to fixed loss methods is demonstrated through experiments on a broad range of 14 OpenML tabular datasets and 46 image datasets with various deep learning architectures. Python repository and source code are available on GitHub at https://github.com/statmlben/ensloss.

We study the problem of classification with a reject option for a fixed predictor, applicable in natural language processing. We introduce a new problem formulation for this scenario, and an algorithm minimizing a new surrogate loss function. We provide a complete theoretical analysis of the surrogate loss function with a strong H-consistency guarantee. For evaluation, we choose the decontextualization task, and provide a manually-labelled dataset of 2mathord,000 examples. Our algorithm significantly outperforms the baselines considered, with a sim!!25% improvement in coverage when halving the error rate, which is only sim!! 3 % away from the theoretical limit.

We study a family of loss functions named label-distributionally robust (LDR) losses for multi-class classification that are formulated from distributionally robust optimization (DRO) perspective, where the uncertainty in the given label information are modeled and captured by taking the worse case of distributional weights. The benefits of this perspective are several fold: (i) it provides a unified framework to explain the classical cross-entropy (CE) loss and SVM loss and their variants, (ii) it includes a special family corresponding to the temperature-scaled CE loss, which is widely adopted but poorly understood; (iii) it allows us to achieve adaptivity to the uncertainty degree of label information at an instance level. Our contributions include: (1) we study both consistency and robustness by establishing top-k (forall kgeq 1) consistency of LDR losses for multi-class classification, and a negative result that a top-1 consistent and symmetric robust loss cannot achieve top-k consistency simultaneously for all kgeq 2; (2) we propose a new adaptive LDR loss that automatically adapts the individualized temperature parameter to the noise degree of class label of each instance; (3) we demonstrate stable and competitive performance for the proposed adaptive LDR loss on 7 benchmark datasets under 6 noisy label and 1 clean settings against 13 loss functions, and on one real-world noisy dataset. The code is open-sourced at https://github.com/Optimization-AI/ICML2023_LDR.

Collaborative filtering (CF) is a widely studied research topic in recommender systems. The learning of a CF model generally depends on three major components, namely interaction encoder, loss function, and negative sampling. While many existing studies focus on the design of more powerful interaction encoders, the impacts of loss functions and negative sampling ratios have not yet been well explored. In this work, we show that the choice of loss function as well as negative sampling ratio is equivalently important. More specifically, we propose the cosine contrastive loss (CCL) and further incorporate it to a simple unified CF model, dubbed SimpleX. Extensive experiments have been conducted on 11 benchmark datasets and compared with 29 existing CF models in total. Surprisingly, the results show that, under our CCL loss and a large negative sampling ratio, SimpleX can surpass most sophisticated state-of-the-art models by a large margin (e.g., max 48.5% improvement in NDCG@20 over LightGCN). We believe that SimpleX could not only serve as a simple strong baseline to foster future research on CF, but also shed light on the potential research direction towards improving loss function and negative sampling. Our source code will be available at https://reczoo.github.io/SimpleX.

Policy optimization methods are popular reinforcement learning algorithms in practice. Recent works have built theoretical foundation for them by proving T regret bounds even when the losses are adversarial. Such bounds are tight in the worst case but often overly pessimistic. In this work, we show that in tabular Markov decision processes (MDPs), by properly designing the regularizer, the exploration bonus and the learning rates, one can achieve a more favorable polylog(T) regret when the losses are stochastic, without sacrificing the worst-case guarantee in the adversarial regime. To our knowledge, this is also the first time a gap-dependent polylog(T) regret bound is shown for policy optimization. Specifically, we achieve this by leveraging a Tsallis entropy or a Shannon entropy regularizer in the policy update. Then we show that under known transitions, we can further obtain a first-order regret bound in the adversarial regime by leveraging the log-barrier regularizer.

Due to the rise of privacy concerns, in many practical applications the training data is aggregated before being shared with the learner, in order to protect privacy of users' sensitive responses. In an aggregate learning framework, the dataset is grouped into bags of samples, where each bag is available only with an aggregate response, providing a summary of individuals' responses in that bag. In this paper, we study two natural loss functions for learning from aggregate responses: bag-level loss and the instance-level loss. In the former, the model is learnt by minimizing a loss between aggregate responses and aggregate model predictions, while in the latter the model aims to fit individual predictions to the aggregate responses. In this work, we show that the instance-level loss can be perceived as a regularized form of the bag-level loss. This observation lets us compare the two approaches with respect to bias and variance of the resulting estimators, and introduce a novel interpolating estimator which combines the two approaches. For linear regression tasks, we provide a precise characterization of the risk of the interpolating estimator in an asymptotic regime where the size of the training set grows in proportion to the features dimension. Our analysis allows us to theoretically understand the effect of different factors, such as bag size on the model prediction risk. In addition, we propose a mechanism for differentially private learning from aggregate responses and derive the optimal bag size in terms of prediction risk-privacy trade-off. We also carry out thorough experiments to corroborate our theory and show the efficacy of the interpolating estimator.

Aligning generative models with human preference via RLHF typically suffers from overoptimization, where an imperfectly learned reward model can misguide the generative model to output undesired responses. We investigate this problem in a principled manner by identifying the source of the misalignment as a form of distributional shift and uncertainty in learning human preferences. To mitigate overoptimization, we first propose a theoretical algorithm that chooses the best policy for an adversarially chosen reward model; one that simultaneously minimizes the maximum likelihood estimation of the loss and a reward penalty term. Here, the reward penalty term is introduced to prevent the policy from choosing actions with spurious high proxy rewards, resulting in provable sample efficiency of the algorithm under a partial coverage style condition. Moving from theory to practice, the proposed algorithm further enjoys an equivalent but surprisingly easy-to-implement reformulation. Using the equivalence between reward models and the corresponding optimal policy, the algorithm features a simple objective that combines: (i) a preference optimization loss that directly aligns the policy with human preference, and (ii) a supervised learning loss that explicitly imitates the policy with a (suitable) baseline distribution. In the context of aligning large language models (LLM), this objective fuses the direct preference optimization (DPO) loss with the supervised fune-tuning (SFT) loss to help mitigate the overoptimization towards undesired responses, for which we name the algorithm Regularized Preference Optimization (RPO). Experiments of aligning LLMs demonstrate the improved performance of RPO compared with DPO baselines. Our work sheds light on the interplay between preference optimization and SFT in tuning LLMs with both theoretical guarantees and empirical evidence.

It is well known that accurate probabilistic predictors can be trained through empirical risk minimisation with proper scoring rules as loss functions. While such learners capture so-called aleatoric uncertainty of predictions, various machine learning methods have recently been developed with the goal to let the learner also represent its epistemic uncertainty, i.e., the uncertainty caused by a lack of knowledge and data. An emerging branch of the literature proposes the use of a second-order learner that provides predictions in terms of distributions on probability distributions. However, recent work has revealed serious theoretical shortcomings for second-order predictors based on loss minimisation. In this paper, we generalise these findings and prove a more fundamental result: There seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its epistemic uncertainty in the same manner as proper scoring rules do for standard (first-order) learners. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.

The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of O(L(f)) and O(d_{eff}(mu)T) at a computational complexity in O(ln^2{T}). If the eigenvalues decay polynomially with degree pgeq 1, then our algorithms keep the same regret bounds at a computational complexity in o(T) in the case of p>4 and pgeq 10, respectively. L(f) is the cumulative losses of f and d_{eff}(mu) is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.

Generalization bounds which assess the difference between the true risk and the empirical risk, have been studied extensively. However, to obtain bounds, current techniques use strict assumptions such as a uniformly bounded or a Lipschitz loss function. To avoid these assumptions, in this paper, we follow an alternative approach: we relax uniform bounds assumptions by using on-average bounded loss and on-average bounded gradient norm assumptions. Following this relaxation, we propose a new generalization bound that exploits the contractivity of the log-Sobolev inequalities. These inequalities add an additional loss-gradient norm term to the generalization bound, which is intuitively a surrogate of the model complexity. We apply the proposed bound on Bayesian deep nets and empirically analyze the effect of this new loss-gradient norm term on different neural architectures.

Training a modern machine learning architecture on a new task requires extensive learning-rate tuning, which comes at a high computational cost. Here we develop new adaptive learning rates that can be used with any momentum method, and require less tuning to perform well. We first develop MoMo, a Momentum Model based adaptive learning rate for SGD-M (Stochastic gradient descent with momentum). MoMo uses momentum estimates of the batch losses and gradients sampled at each iteration to build a model of the loss function. Our model also makes use of any known lower bound of the loss function by using truncation, e.g. most losses are lower-bounded by zero. We then approximately minimize this model at each iteration to compute the next step. We show how MoMo can be used in combination with any momentum-based method, and showcase this by developing MoMo-Adam - which is Adam with our new model-based adaptive learning rate. Additionally, for losses with unknown lower bounds, we develop on-the-fly estimates of a lower bound, that are incorporated in our model. Through extensive numerical experiments, we demonstrate that MoMo and MoMo-Adam improve over SGD-M and Adam in terms of accuracy and robustness to hyperparameter tuning for training image classifiers on MNIST, CIFAR10, CIFAR100, Imagenet, recommender systems on the Criteo dataset, and a transformer model on the translation task IWSLT14.

Levin Tree Search (LTS) is a search algorithm that makes use of a policy (a probability distribution over actions) and comes with a theoretical guarantee on the number of expansions before reaching a goal node, depending on the quality of the policy. This guarantee can be used as a loss function, which we call the LTS loss, to optimize neural networks representing the policy (LTS+NN). In this work we show that the neural network can be substituted with parameterized context models originating from the online compression literature (LTS+CM). We show that the LTS loss is convex under this new model, which allows for using standard convex optimization tools, and obtain convergence guarantees to the optimal parameters in an online setting for a given set of solution trajectories -- guarantees that cannot be provided for neural networks. The new LTS+CM algorithm compares favorably against LTS+NN on several benchmarks: Sokoban (Boxoban), The Witness, and the 24-Sliding Tile puzzle (STP). The difference is particularly large on STP, where LTS+NN fails to solve most of the test instances while LTS+CM solves each test instance in a fraction of a second. Furthermore, we show that LTS+CM is able to learn a policy that solves the Rubik's cube in only a few hundred expansions, which considerably improves upon previous machine learning techniques.

Harmful fine-tuning issue qi2023fine poses serious safety concerns for Large language models' fine-tuning-as-a-service. While existing defenses huang2024vaccine,rosati2024representation have been proposed to mitigate the issue, their performances are still far away from satisfactory, and the root cause of the problem has not been fully recovered. For the first time in the literature, we in this paper show that harmful perturbation over the model weights should be the root cause of alignment-broken of harmful fine-tuning. In order to attenuate the negative impact of harmful perturbation, we propose an alignment-stage solution, dubbed Booster. Technically, along with the original alignment loss, we append a loss regularizer in the alignment stage's optimization. The regularizer ensures that the model's harmful loss reduction before/after simulated harmful perturbation is attenuated, thereby mitigating the subsequent fine-tuning risk. Empirical results show that Booster can effectively reduce the harmful score of the fine-tuned models while maintaining the performance of downstream tasks. Our code is available at https://github.com/git-disl/Booster.

We extend conformal prediction to control the expected value of any monotone loss function. The algorithm generalizes split conformal prediction together with its coverage guarantee. Like conformal prediction, the conformal risk control procedure is tight up to an O(1/n) factor. We also introduce extensions of the idea to distribution shift, quantile risk control, multiple and adversarial risk control, and expectations of U-statistics. Worked examples from computer vision and natural language processing demonstrate the usage of our algorithm to bound the false negative rate, graph distance, and token-level F1-score.

Despite being the standard loss function to train multi-class neural networks, the log-softmax has two potential limitations. First, it involves computations that scale linearly with the number of output classes, which can restrict the size of problems we are able to tackle with current hardware. Second, it remains unclear how close it matches the task loss such as the top-k error rate or other non-differentiable evaluation metrics which we aim to optimize ultimately. In this paper, we introduce an alternative classification loss function, the Z-loss, which is designed to address these two issues. Unlike the log-softmax, it has the desirable property of belonging to the spherical loss family (Vincent et al., 2015), a class of loss functions for which training can be performed very efficiently with a complexity independent of the number of output classes. We show experimentally that it significantly outperforms the other spherical loss functions previously investigated. Furthermore, we show on a word language modeling task that it also outperforms the log-softmax with respect to certain ranking scores, such as top-k scores, suggesting that the Z-loss has the flexibility to better match the task loss. These qualities thus makes the Z-loss an appealing candidate to train very efficiently large output networks such as word-language models or other extreme classification problems. On the One Billion Word (Chelba et al., 2014) dataset, we are able to train a model with the Z-loss 40 times faster than the log-softmax and more than 4 times faster than the hierarchical softmax.

The notion of omnipredictors (Gopalan, Kalai, Reingold, Sharan and Wieder ITCS 2021), suggested a new paradigm for loss minimization. Rather than learning a predictor based on a known loss function, omnipredictors can easily be post-processed to minimize any one of a rich family of loss functions compared with the loss of hypotheses in a class mathcal C. It has been shown that such omnipredictors exist and are implied (for all convex and Lipschitz loss functions) by the notion of multicalibration from the algorithmic fairness literature. In this paper, we introduce omnipredictors for constrained optimization and study their complexity and implications. The notion that we introduce allows the learner to be unaware of the loss function that will be later assigned as well as the constraints that will be later imposed, as long as the subpopulations that are used to define these constraints are known. We show how to obtain omnipredictors for constrained optimization problems, relying on appropriate variants of multicalibration. We also investigate the implications of this notion when the constraints used are so-called group fairness notions.

The loss functions of many learning problems contain multiple additive terms that can disagree and yield conflicting update directions. For Physics-Informed Neural Networks (PINNs), loss terms on initial/boundary conditions and physics equations are particularly interesting as they are well-established as highly difficult tasks. To improve learning the challenging multi-objective task posed by PINNs, we propose the ConFIG method, which provides conflict-free updates by ensuring a positive dot product between the final update and each loss-specific gradient. It also maintains consistent optimization rates for all loss terms and dynamically adjusts gradient magnitudes based on conflict levels. We additionally leverage momentum to accelerate optimizations by alternating the back-propagation of different loss terms. We provide a mathematical proof showing the convergence of the ConFIG method, and it is evaluated across a range of challenging PINN scenarios. ConFIG consistently shows superior performance and runtime compared to baseline methods. We also test the proposed method in a classic multi-task benchmark, where the ConFIG method likewise exhibits a highly promising performance. Source code is available at https://tum-pbs.github.io/ConFIG

We propose a game-based formulation for learning dimensionality-reducing representations of feature vectors, when only a prior knowledge on future prediction tasks is available. In this game, the first player chooses a representation, and then the second player adversarially chooses a prediction task from a given class, representing the prior knowledge. The first player aims is to minimize, and the second player to maximize, the regret: The minimal prediction loss using the representation, compared to the same loss using the original features. For the canonical setting in which the representation, the response to predict and the predictors are all linear functions, and under the mean squared error loss function, we derive the theoretically optimal representation in pure strategies, which shows the effectiveness of the prior knowledge, and the optimal regret in mixed strategies, which shows the usefulness of randomizing the representation. For general representations and loss functions, we propose an efficient algorithm to optimize a randomized representation. The algorithm only requires the gradients of the loss function, and is based on incrementally adding a representation rule to a mixture of such rules.

In the field of object detection, domain generalisation (DG) aims to ensure robust performance across diverse and unseen target domains by learning the robust domain-invariant features corresponding to the objects of interest across multiple source domains. While there are many approaches established for performing DG for the task of classification, there has been a very little focus on object detection. In this paper, we propose a domain penalisation (DP) framework for the task of object detection, where the data is assumed to be sampled from multiple source domains and tested on completely unseen test domains. We assign penalisation weights to each domain, with the values updated based on the detection networks performance on the respective source domains. By prioritising the domains that needs more attention, our approach effectively balances the training process. We evaluate our solution on the GWHD 2021 dataset, a component of the WiLDS benchmark and we compare against ERM and GroupDRO as these are primarily loss function based. Our extensive experimental results reveals that the proposed approach improves the accuracy by 0.3 percent and 0.5 percent on validation and test out-of-distribution (OOD) sets, respectively for FasterRCNN. We also compare the performance of our approach on FCOS detector and show that our approach improves the baseline OOD performance over the existing approaches by 1.3 percent and 1.4 percent on validation and test sets, respectively. This study underscores the potential of performance based domain penalisation in enhancing the generalisation ability of object detection models across diverse environments.

Neural network training relies on our ability to find "good" minimizers of highly non-convex loss functions. It is well-known that certain network architecture designs (e.g., skip connections) produce loss functions that train easier, and well-chosen training parameters (batch size, learning rate, optimizer) produce minimizers that generalize better. However, the reasons for these differences, and their effects on the underlying loss landscape, are not well understood. In this paper, we explore the structure of neural loss functions, and the effect of loss landscapes on generalization, using a range of visualization methods. First, we introduce a simple "filter normalization" method that helps us visualize loss function curvature and make meaningful side-by-side comparisons between loss functions. Then, using a variety of visualizations, we explore how network architecture affects the loss landscape, and how training parameters affect the shape of minimizers.

In this paper, we introduce a novel predict-and-optimize method for profit-driven churn prevention. We frame the task of targeting customers for a retention campaign as a regret minimization problem. The main objective is to leverage individual customer lifetime values (CLVs) to ensure that only the most valuable customers are targeted. In contrast, many profit-driven strategies focus on churn probabilities while considering average CLVs. This often results in significant information loss due to data aggregation. Our proposed model aligns with the guidelines of Predict-and-Optimize (PnO) frameworks and can be efficiently solved using stochastic gradient descent methods. Results from 12 churn prediction datasets underscore the effectiveness of our approach, which achieves the best average performance compared to other well-established strategies in terms of average profit.

In this paper, we study the predict-then-optimize problem where the output of a machine learning prediction task is used as the input of some downstream optimization problem, say, the objective coefficient vector of a linear program. The problem is also known as predictive analytics or contextual linear programming. The existing approaches largely suffer from either (i) optimization intractability (a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii) requiring strong condition(s) such as no constraint or loss calibration. We develop a new approach to the problem called maximum optimality margin which designs the machine learning loss function by the optimality condition of the downstream optimization. The max-margin formulation enjoys both computational efficiency and good theoretical properties for the learning procedure. More importantly, our new approach only needs the observations of the optimal solution in the training data rather than the objective function, which makes it a new and natural approach to the inverse linear programming problem under both contextual and context-free settings; we also analyze the proposed method under both offline and online settings, and demonstrate its performance using numerical experiments.

This paper investigates the issue of an adequate loss function in the optimization of machine learning models used in the forecasting of financial time series for the purpose of algorithmic investment strategies (AIS) construction. We propose the Mean Absolute Directional Loss (MADL) function, solving important problems of classical forecast error functions in extracting information from forecasts to create efficient buy/sell signals in algorithmic investment strategies. Finally, based on the data from two different asset classes (cryptocurrencies: Bitcoin and commodities: Crude Oil), we show that the new loss function enables us to select better hyperparameters for the LSTM model and obtain more efficient investment strategies, with regard to risk-adjusted return metrics on the out-of-sample data.

Depth estimation is a cornerstone for autonomous driving, yet acquiring per-pixel depth ground truth for supervised learning is challenging. Self-Supervised Surround Depth Estimation (SSSDE) from consecutive images offers an economical alternative. While previous SSSDE methods have proposed different mechanisms to fuse information across images, few of them explicitly consider the cross-view constraints, leading to inferior performance, particularly in overlapping regions. This paper proposes an efficient and consistent pose estimation design and two loss functions to enhance cross-view consistency for SSSDE. For pose estimation, we propose to use only front-view images to reduce training memory and sustain pose estimation consistency. The first loss function is the dense depth consistency loss, which penalizes the difference between predicted depths in overlapping regions. The second one is the multi-view reconstruction consistency loss, which aims to maintain consistency between reconstruction from spatial and spatial-temporal contexts. Additionally, we introduce a novel flipping augmentation to improve the performance further. Our techniques enable a simple neural model to achieve state-of-the-art performance on the DDAD and nuScenes datasets. Last but not least, our proposed techniques can be easily applied to other methods. The code will be made public.

Mixture of experts (MoE) has a well-principled finite mixture model construction for prediction, allowing the gating network (mixture weights) to learn from the predictors (explanatory variables) together with the experts' network (mixture component densities). We investigate the estimation properties of MoEs in a high-dimensional setting, where the number of predictors is much larger than the sample size, for which the literature lacks computational and especially theoretical results. We consider the class of finite MoE models with softmax gating functions and Gaussian regression experts, and focus on the theoretical properties of their l_1-regularized estimation via the Lasso. We provide a lower bound on the regularization parameter of the Lasso penalty that ensures an l_1-oracle inequality is satisfied by the Lasso estimator according to the Kullback--Leibler loss. We further state an l_1-ball oracle inequality for the l_1-penalized maximum likelihood estimator from the model selection.

Likelihood training and maximization-based decoding result in dull and repetitive generated texts even when using powerful language models (Holtzman et al., 2019). Adding a loss function for regularization was shown to improve text generation output by helping avoid unwanted properties, such as contradiction or repetition (Li at al., 2020). In this work, we propose fine-tuning a language model by using policy gradient reinforcement learning, directly optimizing for better generation. We apply this approach to minimizing repetition in generated text, and show that, when combined with unlikelihood training (Welleck et al., 2020), our method further reduces repetition without impacting the language model quality. We also evaluate other methods for improving generation at training and decoding time, and compare them using various metrics aimed at control for better text generation output.

In order to train networks for verified adversarial robustness, it is common to over-approximate the worst-case loss over perturbation regions, resulting in networks that attain verifiability at the expense of standard performance. As shown in recent work, better trade-offs between accuracy and robustness can be obtained by carefully coupling adversarial training with over-approximations. We hypothesize that the expressivity of a loss function, which we formalize as the ability to span a range of trade-offs between lower and upper bounds to the worst-case loss through a single parameter (the over-approximation coefficient), is key to attaining state-of-the-art performance. To support our hypothesis, we show that trivial expressive losses, obtained via convex combinations between adversarial attacks and IBP bounds, yield state-of-the-art results across a variety of settings in spite of their conceptual simplicity. We provide a detailed analysis of the relationship between the over-approximation coefficient and performance profiles across different expressive losses, showing that, while expressivity is essential, better approximations of the worst-case loss are not necessarily linked to superior robustness-accuracy trade-offs.

This paper proposes a new loss function for adversarial training. Since adversarial training has difficulties, e.g., necessity of high model capacity, focusing on important data points by weighting cross-entropy loss has attracted much attention. However, they are vulnerable to sophisticated attacks, e.g., Auto-Attack. This paper experimentally reveals that the cause of their vulnerability is their small margins between logits for the true label and the other labels. Since neural networks classify the data points based on the logits, logit margins should be large enough to avoid flipping the largest logit by the attacks. Importance-aware methods do not increase logit margins of important samples but decrease those of less-important samples compared with cross-entropy loss. To increase logit margins of important samples, we propose switching one-vs-the-rest loss (SOVR), which switches from cross-entropy to one-vs-the-rest loss for important samples that have small logit margins. We prove that one-vs-the-rest loss increases logit margins two times larger than the weighted cross-entropy loss for a simple problem. We experimentally confirm that SOVR increases logit margins of important samples unlike existing methods and achieves better robustness against Auto-Attack than importance-aware methods.

Robust loss functions are essential for training accurate deep neural networks (DNNs) in the presence of noisy (incorrect) labels. It has been shown that the commonly used Cross Entropy (CE) loss is not robust to noisy labels. Whilst new loss functions have been designed, they are only partially robust. In this paper, we theoretically show by applying a simple normalization that: any loss can be made robust to noisy labels. However, in practice, simply being robust is not sufficient for a loss function to train accurate DNNs. By investigating several robust loss functions, we find that they suffer from a problem of underfitting. To address this, we propose a framework to build robust loss functions called Active Passive Loss (APL). APL combines two robust loss functions that mutually boost each other. Experiments on benchmark datasets demonstrate that the family of new loss functions created by our APL framework can consistently outperform state-of-the-art methods by large margins, especially under large noise rates such as 60% or 80% incorrect labels.

This paper offers a new perspective to ease the challenge of domain generalization, which involves maintaining robust results even in unseen environments. Our design focuses on the decision-making process in the final classifier layer. Specifically, we propose treating the element-wise contributions to the final results as the rationale for making a decision and representing the rationale for each sample as a matrix. For a well-generalized model, we suggest the rationale matrices for samples belonging to the same category should be similar, indicating the model relies on domain-invariant clues to make decisions, thereby ensuring robust results. To implement this idea, we introduce a rationale invariance loss as a simple regularization technique, requiring only a few lines of code. Our experiments demonstrate that the proposed approach achieves competitive results across various datasets, despite its simplicity. Code is available at https://github.com/liangchen527/RIDG.

We propose a new regularization method based on virtual adversarial loss: a new measure of local smoothness of the conditional label distribution given input. Virtual adversarial loss is defined as the robustness of the conditional label distribution around each input data point against local perturbation. Unlike adversarial training, our method defines the adversarial direction without label information and is hence applicable to semi-supervised learning. Because the directions in which we smooth the model are only "virtually" adversarial, we call our method virtual adversarial training (VAT). The computational cost of VAT is relatively low. For neural networks, the approximated gradient of virtual adversarial loss can be computed with no more than two pairs of forward- and back-propagations. In our experiments, we applied VAT to supervised and semi-supervised learning tasks on multiple benchmark datasets. With a simple enhancement of the algorithm based on the entropy minimization principle, our VAT achieves state-of-the-art performance for semi-supervised learning tasks on SVHN and CIFAR-10.

We propose sparsemax, a new activation function similar to the traditional softmax, but able to output sparse probabilities. After deriving its properties, we show how its Jacobian can be efficiently computed, enabling its use in a network trained with backpropagation. Then, we propose a new smooth and convex loss function which is the sparsemax analogue of the logistic loss. We reveal an unexpected connection between this new loss and the Huber classification loss. We obtain promising empirical results in multi-label classification problems and in attention-based neural networks for natural language inference. For the latter, we achieve a similar performance as the traditional softmax, but with a selective, more compact, attention focus.

While language models have exceptional capabilities at text generation, they lack a natural inductive bias for emitting numbers and thus struggle in tasks involving reasoning over quantities, especially arithmetics. This has particular relevance in scientific datasets where combinations of text and numerical data are abundant. One fundamental limitation is the nature of the CE loss, which assumes a nominal (categorical) scale and thus cannot convey proximity between generated number tokens. As a remedy, we here present two versions of a number token loss. The first is based on an L_p loss between the ground truth token value and the weighted sum of the predicted class probabilities. The second loss minimizes the Wasserstein-1 distance between the distribution of the predicted output probabilities and the ground truth distribution. These regression-like losses can easily be added to any language model and extend the CE objective during training. We compare the proposed schemes on a mathematics dataset against existing tokenization, encoding, and decoding schemes for improving number representation in language models. Our results reveal a significant improvement in numerical accuracy when equipping a standard T5 model with the proposed loss schemes.

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter sigma > 0. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be O(sigma)-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as O(sigma)-approximation of the original BO, we propose first-order algorithms that find an epsilon-stationary solution by optimizing the penalty formulation with sigma = O(epsilon). When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an epsilon-stationary point of the penalty function, using in total O(epsilon^{-3}) and O(epsilon^{-7}) accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, i.e., with O(1) samples per iteration, and achieves the improved oracle-complexity of O(epsilon^{-3}) and O(epsilon^{-5}), respectively.

The Invariant Risk Minimization (IRM) principle was first proposed by Arjovsky et al. [2019] to address the domain generalization problem by leveraging data heterogeneity from differing experimental conditions. Specifically, IRM seeks to find a data representation under which an optimal classifier remains invariant across all domains. Despite the conceptual appeal of IRM, the effectiveness of the originally proposed invariance penalty has recently been brought into question. In particular, there exists counterexamples for which that invariance penalty can be arbitrarily small for non-invariant data representations. We propose an alternative invariance penalty by revisiting the Gramian matrix of the data representation. We discuss the role of its eigenvalues in the relationship between the risk and the invariance penalty, and demonstrate that it is ill-conditioned for said counterexamples. The proposed approach is guaranteed to recover an invariant representation for linear settings under mild non-degeneracy conditions. Its effectiveness is substantiated by experiments on DomainBed and InvarianceUnitTest, two extensive test beds for domain generalization.

In today's heavily overparameterized models, the value of the training loss provides few guarantees on model generalization ability. Indeed, optimizing only the training loss value, as is commonly done, can easily lead to suboptimal model quality. Motivated by prior work connecting the geometry of the loss landscape and generalization, we introduce a novel, effective procedure for instead simultaneously minimizing loss value and loss sharpness. In particular, our procedure, Sharpness-Aware Minimization (SAM), seeks parameters that lie in neighborhoods having uniformly low loss; this formulation results in a min-max optimization problem on which gradient descent can be performed efficiently. We present empirical results showing that SAM improves model generalization across a variety of benchmark datasets (e.g., CIFAR-10, CIFAR-100, ImageNet, finetuning tasks) and models, yielding novel state-of-the-art performance for several. Additionally, we find that SAM natively provides robustness to label noise on par with that provided by state-of-the-art procedures that specifically target learning with noisy labels. We open source our code at https://github.com/google-research/sam.

Diffusion models have emerged as a powerful tool rivaling GANs in generating high-quality samples with improved fidelity, flexibility, and robustness. A key component of these models is to learn the score function through score matching. Despite empirical success on various tasks, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. As a first step toward answering this question, this paper establishes a mathematical framework for analyzing score estimation using neural networks trained by gradient descent. Our analysis covers both the optimization and the generalization aspects of the learning procedure. In particular, we propose a parametric form to formulate the denoising score-matching problem as a regression with noisy labels. Compared to the standard supervised learning setup, the score-matching problem introduces distinct challenges, including unbounded input, vector-valued output, and an additional time variable, preventing existing techniques from being applied directly. In this paper, we show that with proper designs, the evolution of neural networks during training can be accurately modeled by a series of kernel regression tasks. Furthermore, by applying an early-stopping rule for gradient descent and leveraging recent developments in neural tangent kernels, we establish the first generalization error (sample complexity) bounds for learning the score function with neural networks, despite the presence of noise in the observations. Our analysis is grounded in a novel parametric form of the neural network and an innovative connection between score matching and regression analysis, facilitating the application of advanced statistical and optimization techniques.

Recent works have shown a reduction from contextual bandits to online regression under a realizability assumption [Foster and Rakhlin, 2020, Foster and Krishnamurthy, 2021]. In this work, we investigate the use of neural networks for such online regression and associated Neural Contextual Bandits (NeuCBs). Using existing results for wide networks, one can readily show a {O}(T) regret for online regression with square loss, which via the reduction implies a {O}(K T^{3/4}) regret for NeuCBs. Departing from this standard approach, we first show a O(log T) regret for online regression with almost convex losses that satisfy QG (Quadratic Growth) condition, a generalization of the PL (Polyak-\L ojasiewicz) condition, and that have a unique minima. Although not directly applicable to wide networks since they do not have unique minima, we show that adding a suitable small random perturbation to the network predictions surprisingly makes the loss satisfy QG with unique minima. Based on such a perturbed prediction, we show a {O}(log T) regret for online regression with both squared loss and KL loss, and subsequently convert these respectively to mathcal{O}(KT) and mathcal{O}(KL^* + K) regret for NeuCB, where L^* is the loss of the best policy. Separately, we also show that existing regret bounds for NeuCBs are Omega(T) or assume i.i.d. contexts, unlike this work. Finally, our experimental results on various datasets demonstrate that our algorithms, especially the one based on KL loss, persistently outperform existing algorithms.

Reinforcement Learning with Human Feedback (RLHF) has achieved great success in aligning large language models (LLMs) with human preferences. Prevalent RLHF approaches are reward-based, following the Bradley-Terry (BT) model assumption, which may not fully capture the complexity of human preferences. In this paper, we explore RLHF under a general preference framework and approach it from a game-theoretic perspective. Specifically, we formulate the problem as a two-player game and propose a novel algorithm, iterative Nash policy optimization (INPO). The key idea is to let the policy play against itself via no-regret learning, thereby approximating the Nash policy. Unlike previous methods, INPO bypasses the need for estimating the expected win rate for individual responses, which typically incurs high computational or annotation costs. Instead, we introduce a new loss objective that is directly minimized over a preference dataset. We provide theoretical analysis for our approach and demonstrate its effectiveness through experiments on various representative benchmarks. With an LLaMA-3-8B-based SFT model, INPO achieves a 41.5% length-controlled win rate on AlpacaEval 2.0 and a 38.3% win rate on Arena-Hard, showing substantial improvement over the state-of-the-art iterative algorithm [Dong et al., 2024] under the BT model assumption. Additionally, our ablation study highlights the benefits of incorporating KL regularization for response length control.

We study online reinforcement learning in linear Markov decision processes with adversarial losses and bandit feedback, without prior knowledge on transitions or access to simulators. We introduce two algorithms that achieve improved regret performance compared to existing approaches. The first algorithm, although computationally inefficient, ensures a regret of mathcal{O}left(Kright), where K is the number of episodes. This is the first result with the optimal K dependence in the considered setting. The second algorithm, which is based on the policy optimization framework, guarantees a regret of mathcal{O}left(K^{3{4}} right) and is computationally efficient. Both our results significantly improve over the state-of-the-art: a computationally inefficient algorithm by Kong et al. [2023] with mathcal{O}left(K^{4{5}}+polyleft(1{lambda_{min}}right) right) regret, for some problem-dependent constant lambda_{min} that can be arbitrarily close to zero, and a computationally efficient algorithm by Sherman et al. [2023b] with mathcal{O}left(K^{6{7}} right) regret.

We seek to understand fundamental tradeoffs between the accuracy of prior information that a learner has on a given problem and its learning performance. We introduce the notion of prioritized risk, which differs from traditional notions of minimax and Bayes risk by allowing us to study such fundamental tradeoffs in settings where reality does not necessarily conform to the learner's prior. We present a general reduction-based approach for extending classical minimax lower-bound techniques in order to lower bound the prioritized risk for statistical estimation problems. We also introduce a novel generalization of Fano's inequality (which may be of independent interest) for lower bounding the prioritized risk in more general settings involving unbounded losses. We illustrate the ability of our framework to provide insights into tradeoffs between prior information and learning performance for problems in estimation, regression, and reinforcement learning.

We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating C^s smooth functions with s >0 and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests, for nonparametric quantile regression.

Recently, contrastive learning has found impressive success in advancing the state of the art in solving various machine learning tasks. However, the existing generalization analysis is very limited or even not meaningful. In particular, the existing generalization error bounds depend linearly on the number k of negative examples while it was widely shown in practice that choosing a large k is necessary to guarantee good generalization of contrastive learning in downstream tasks. In this paper, we establish novel generalization bounds for contrastive learning which do not depend on k, up to logarithmic terms. Our analysis uses structural results on empirical covering numbers and Rademacher complexities to exploit the Lipschitz continuity of loss functions. For self-bounding Lipschitz loss functions, we further improve our results by developing optimistic bounds which imply fast rates in a low noise condition. We apply our results to learning with both linear representation and nonlinear representation by deep neural networks, for both of which we derive Rademacher complexity bounds to get improved generalization bounds.

We introduce Gravity, another algorithm for gradient-based optimization. In this paper, we explain how our novel idea change parameters to reduce the deep learning model's loss. It has three intuitive hyper-parameters that the best values for them are proposed. Also, we propose an alternative to moving average. To compare the performance of the Gravity optimizer with two common optimizers, Adam and RMSProp, five standard datasets were trained on two VGGNet models with a batch size of 128 for 100 epochs. Gravity hyper-parameters did not need to be tuned for different models. As will be explained more in the paper, to investigate the direct impact of the optimizer itself on loss reduction no overfitting prevention technique was used. The obtained results show that the Gravity optimizer has more stable performance than Adam and RMSProp and gives greater values of validation accuracy for datasets with more output classes like CIFAR-100 (Fine).

Macro-AUC is the arithmetic mean of the class-wise AUCs in multi-label learning and is commonly used in practice. However, its theoretical understanding is far lacking. Toward solving it, we characterize the generalization properties of various learning algorithms based on the corresponding surrogate losses w.r.t. Macro-AUC. We theoretically identify a critical factor of the dataset affecting the generalization bounds: the label-wise class imbalance. Our results on the imbalance-aware error bounds show that the widely-used univariate loss-based algorithm is more sensitive to the label-wise class imbalance than the proposed pairwise and reweighted loss-based ones, which probably implies its worse performance. Moreover, empirical results on various datasets corroborate our theory findings. To establish it, technically, we propose a new (and more general) McDiarmid-type concentration inequality, which may be of independent interest.

We study the problem of constrained efficient global optimization, where both the objective and constraints are expensive black-box functions that can be learned with Gaussian processes. We propose CONFIG (CONstrained efFIcient Global Optimization), a simple and effective algorithm to solve it. Under certain regularity assumptions, we show that our algorithm enjoys the same cumulative regret bound as that in the unconstrained case and similar cumulative constraint violation upper bounds. For commonly used Matern and Squared Exponential kernels, our bounds are sublinear and allow us to derive a convergence rate to the optimal solution of the original constrained problem. In addition, our method naturally provides a scheme to declare infeasibility when the original black-box optimization problem is infeasible. Numerical experiments on sampled instances from the Gaussian process, artificial numerical problems, and a black-box building controller tuning problem all demonstrate the competitive performance of our algorithm. Compared to the other state-of-the-art methods, our algorithm significantly improves the theoretical guarantees, while achieving competitive empirical performance.

We study the problem of learning a single neuron with respect to the L_2^2-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal L_2^2-error within a constant factor. Our algorithm applies under much milder distributional assumptions compared to prior work. The key ingredient enabling our results is a novel connection to local error bounds from optimization theory.

We study the design of loss functions for click-through rates (CTR) to optimize (social) welfare in advertising auctions. Existing works either only focus on CTR predictions without consideration of business objectives (e.g., welfare) in auctions or assume that the distribution over the participants' expected cost-per-impression (eCPM) is known a priori, then use various additional assumptions on the parametric form of the distribution to derive loss functions for predicting CTRs. In this work, we bring back the welfare objectives of ad auctions into CTR predictions and propose a novel weighted rankloss to train the CTR model. Compared to existing literature, our approach provides a provable guarantee on welfare but without assumptions on the eCPMs' distribution while also avoiding the intractability of naively applying existing learning-to-rank methods. Further, we propose a theoretically justifiable technique for calibrating the losses using labels generated from a teacher network, only assuming that the teacher network has bounded ell_2 generalization error. Finally, we demonstrate the advantages of the proposed loss on synthetic and real-world data.

We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

We propose a new class of online learning algorithms, generalized implicit Follow-The-Regularized-Leader (FTRL), that expands the scope of FTRL framework. Generalized implicit FTRL can recover known algorithms, as FTRL with linearized losses and implicit FTRL, and it allows the design of new update rules, as extensions of aProx and Mirror-Prox to FTRL. Our theory is constructive in the sense that it provides a simple unifying framework to design updates that directly improve the worst-case upper bound on the regret. The key idea is substituting the linearization of the losses with a Fenchel-Young inequality. We show the flexibility of the framework by proving that some known algorithms, like the Mirror-Prox updates, are instantiations of the generalized implicit FTRL. Finally, the new framework allows us to recover the temporal variation bound of implicit OMD, with the same computational complexity.

We present the OMG-CMDP! algorithm for regret minimization in adversarial Contextual MDPs. The algorithm operates under the minimal assumptions of realizable function class and access to online least squares and log loss regression oracles. Our algorithm is efficient (assuming efficient online regression oracles), simple and robust to approximation errors. It enjoys an O(H^{2.5} T|S||A| ( mathcal{R(O) + H log(delta^{-1}) )}) regret guarantee, with T being the number of episodes, S the state space, A the action space, H the horizon and R(O) = R(O_{sq}^F) + R(O_{log}^P) is the sum of the regression oracles' regret, used to approximate the context-dependent rewards and dynamics, respectively. To the best of our knowledge, our algorithm is the first efficient rate optimal regret minimization algorithm for adversarial CMDPs that operates under the minimal standard assumption of online function approximation.

We consider algorithms for learning reward functions from human preferences over pairs of trajectory segments, as used in reinforcement learning from human feedback (RLHF). Most recent work assumes that human preferences are generated based only upon the reward accrued within those segments, or their partial return. Recent work casts doubt on the validity of this assumption, proposing an alternative preference model based upon regret. We investigate the consequences of assuming preferences are based upon partial return when they actually arise from regret. We argue that the learned function is an approximation of the optimal advantage function, A^*_r, not a reward function. We find that if a specific pitfall is addressed, this incorrect assumption is not particularly harmful, resulting in a highly shaped reward function. Nonetheless, this incorrect usage of A^*_r is less desirable than the appropriate and simpler approach of greedy maximization of A^*_r. From the perspective of the regret preference model, we also provide a clearer interpretation of fine tuning contemporary large language models with RLHF. This paper overall provides insight regarding why learning under the partial return preference model tends to work so well in practice, despite it conforming poorly to how humans give preferences.

Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al. [2022] as an interpolation between stochastic and adversarial online convex optimization. Under the smoothness condition, they demonstrate that the expected regret of optimistic follow-the-regularized-leader (FTRL) depends on the cumulative stochastic variance sigma_{1:T}^2 and the cumulative adversarial variation Sigma_{1:T}^2 for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance sigma_{max}^2 and the maximal adversarial variation Sigma_{max}^2 for strongly convex functions. Inspired by their work, we investigate the theoretical guarantees of optimistic online mirror descent (OMD) for the SEA model. For convex and smooth functions, we obtain the same O(sigma_{1:T^2}+Sigma_{1:T^2}) regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an O(min{log (sigma_{1:T}^2+Sigma_{1:T}^2), (sigma_{max}^2 + Sigma_{max}^2) log T}) bound, better than their O((sigma_{max}^2 + Sigma_{max}^2) log T) bound. For exp-concave and smooth functions, we achieve a new O(dlog(sigma_{1:T}^2+Sigma_{1:T}^2)) bound. Owing to the OMD framework, we can further extend our result to obtain dynamic regret guarantees, which are more favorable in non-stationary online scenarios. The attained results allow us to recover excess risk bounds of the stochastic setting and regret bounds of the adversarial setting, and derive new guarantees for many intermediate scenarios.

We consider the distributionally robust optimization (DRO) problem with spectral risk-based uncertainty set and f-divergence penalty. This formulation includes common risk-sensitive learning objectives such as regularized condition value-at-risk (CVaR) and average top-k loss. We present Prospect, a stochastic gradient-based algorithm that only requires tuning a single learning rate hyperparameter, and prove that it enjoys linear convergence for smooth regularized losses. This contrasts with previous algorithms that either require tuning multiple hyperparameters or potentially fail to converge due to biased gradient estimates or inadequate regularization. Empirically, we show that Prospect can converge 2-3times faster than baselines such as stochastic gradient and stochastic saddle-point methods on distribution shift and fairness benchmarks spanning tabular, vision, and language domains.

LOBSTER (LOss-Based SensiTivity rEgulaRization) is a method for training neural networks having a sparse topology. Let the sensitivity of a network parameter be the variation of the loss function with respect to the variation of the parameter. Parameters with low sensitivity, i.e. having little impact on the loss when perturbed, are shrunk and then pruned to sparsify the network. Our method allows to train a network from scratch, i.e. without preliminary learning or rewinding. Experiments on multiple architectures and datasets show competitive compression ratios with minimal computational overhead.

Representation learning methods have revolutionized machine learning on networks by converting discrete network structures into continuous domains. However, dynamic networks that evolve over time pose new challenges. To address this, dynamic representation learning methods have gained attention, offering benefits like reduced learning time and improved accuracy by utilizing temporal information. T-batching is a valuable technique for training dynamic network models that reduces training time while preserving vital conditions for accurate modeling. However, we have identified a limitation in the training loss function used with t-batching. Through mathematical analysis, we propose two alternative loss functions that overcome these issues, resulting in enhanced training performance. We extensively evaluate the proposed loss functions on synthetic and real-world dynamic networks. The results consistently demonstrate superior performance compared to the original loss function. Notably, in a real-world network characterized by diverse user interaction histories, the proposed loss functions achieved more than 26.9% enhancement in Mean Reciprocal Rank (MRR) and more than 11.8% improvement in Recall@10. These findings underscore the efficacy of the proposed loss functions in dynamic network modeling.

Few-shot learning is a challenging area of research that aims to learn new concepts with only a few labeled samples of data. Recent works based on metric-learning approaches leverage the meta-learning approach, which is encompassed by episodic tasks that make use a support (training) and query set (test) with the objective of learning a similarity comparison metric between those sets. Due to the lack of data, the learning process of the embedding network becomes an important part of the few-shot task. Previous works have addressed this problem using metric learning approaches, but the properties of the underlying latent space and the separability of the difference classes on it was not entirely enforced. In this work, we propose two different loss functions which consider the importance of the embedding vectors by looking at the intra-class and inter-class distance between the few data. The first loss function is the Proto-Triplet Loss, which is based on the original triplet loss with the modifications needed to better work on few-shot scenarios. The second loss function, which we dub ICNN loss is based on an inter and intra class nearest neighbors score, which help us to assess the quality of embeddings obtained from the trained network. Our results, obtained from a extensive experimental setup show a significant improvement in accuracy in the miniImagenNet benchmark compared to other metric-based few-shot learning methods by a margin of 2%, demonstrating the capability of these loss functions to allow the network to generalize better to previously unseen classes. In our experiments, we demonstrate competitive generalization capabilities to other domains, such as the Caltech CUB, Dogs and Cars datasets compared with the state of the art.

The trade-off between robustness and accuracy has been widely studied in the adversarial literature. Although still controversial, the prevailing view is that this trade-off is inherent, either empirically or theoretically. Thus, we dig for the origin of this trade-off in adversarial training and find that it may stem from the improperly defined robust error, which imposes an inductive bias of local invariance -- an overcorrection towards smoothness. Given this, we advocate employing local equivariance to describe the ideal behavior of a robust model, leading to a self-consistent robust error named SCORE. By definition, SCORE facilitates the reconciliation between robustness and accuracy, while still handling the worst-case uncertainty via robust optimization. By simply substituting KL divergence with variants of distance metrics, SCORE can be efficiently minimized. Empirically, our models achieve top-rank performance on RobustBench under AutoAttack. Besides, SCORE provides instructive insights for explaining the overfitting phenomenon and semantic input gradients observed on robust models. Code is available at https://github.com/P2333/SCORE.

Loss of plasticity is a phenomenon where neural networks become more difficult to train during the course of learning. Continual learning algorithms seek to mitigate this effect by sustaining good predictive performance while maintaining network trainability. We develop new techniques for improving continual learning by first reconsidering how initialization can ensure trainability during early phases of learning. From this perspective, we derive new regularization strategies for continual learning that ensure beneficial initialization properties are better maintained throughout training. In particular, we investigate two new regularization techniques for continual learning: (i) Wasserstein regularization toward the initial weight distribution, which is less restrictive than regularizing toward initial weights; and (ii) regularizing weight matrix singular values, which directly ensures gradient diversity is maintained throughout training. We present an experimental analysis that shows these alternative regularizers can improve continual learning performance across a range of supervised learning tasks and model architectures. The alternative regularizers prove to be less sensitive to hyperparameters while demonstrating better training in individual tasks, sustaining trainability as new tasks arrive, and achieving better generalization performance.

Offline preference optimization is a key method for enhancing and controlling the quality of Large Language Model (LLM) outputs. Typically, preference optimization is approached as an offline supervised learning task using manually-crafted convex loss functions. While these methods are based on theoretical insights, they are inherently constrained by human creativity, so the large search space of possible loss functions remains under explored. We address this by performing LLM-driven objective discovery to automatically discover new state-of-the-art preference optimization algorithms without (expert) human intervention. Specifically, we iteratively prompt an LLM to propose and implement new preference optimization loss functions based on previously-evaluated performance metrics. This process leads to the discovery of previously-unknown and performant preference optimization algorithms. The best performing of these we call Discovered Preference Optimization (DiscoPOP), a novel algorithm that adaptively blends logistic and exponential losses. Experiments demonstrate the state-of-the-art performance of DiscoPOP and its successful transfer to held-out tasks.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

Nowadays, state-of-the-art learning-to-rank (LTR) methods are based on gradient-boosted decision trees (GBDT). The most well-known algorithm is LambdaMART that was proposed more than a decade ago. Recently, several other GBDT-based ranking algorithms were proposed. In this paper, we conduct a thorough analysis of these methods in a unified setup. In particular, we address the following questions. Is direct optimization of a smoothed ranking loss preferable over optimizing a convex surrogate? How to properly construct and smooth surrogate ranking losses? To address these questions, we compare LambdaMART with YetiRank and StochasticRank methods and their modifications. We also improve the YetiRank approach to allow for optimizing specific ranking loss functions. As a result, we gain insights into learning-to-rank approaches and obtain a new state-of-the-art algorithm.

We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret {O} big( varepsilon^{-1} log^{1.5}{d} big) where d is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are {O} big( varepsilon^{-1} minbig{d, T^{1/3}log dbig} big). We also develop an adaptive algorithm for the small-loss setting with regret O(L^starlog d + varepsilon^{-1} log^{1.5}{d}) where L^star is the total loss of the best expert. Additionally, we consider DP online convex optimization in the realizable setting and propose an algorithm with near-optimal regret O big(varepsilon^{-1} d^{1.5} big), as well as an algorithm for the smooth case with regret O big( varepsilon^{-2/3} (dT)^{1/3} big), both significantly improving over existing bounds in the non-realizable regime.

Overfitting widely exists in adversarial robust training of deep networks. An effective remedy is adversarial weight perturbation, which injects the worst-case weight perturbation during network training by maximizing the classification loss on adversarial examples. Adversarial weight perturbation helps reduce the robust generalization gap; however, it also undermines the robustness improvement. A criterion that regulates the weight perturbation is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation. With LSC, we find that it is essential to conduct weight perturbation on adversarial data with small classification loss to eliminate robust overfitting. Weight perturbation on adversarial data with large classification loss is not necessary and may even lead to poor robustness. Based on these observations, we propose a robust perturbation strategy to constrain the extent of weight perturbation. The perturbation strategy prevents deep networks from overfitting while avoiding the side effect of excessive weight perturbation, significantly improving the robustness of adversarial training. Extensive experiments demonstrate the superiority of the proposed method over the state-of-the-art adversarial training methods.

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made important advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another interesting type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. For low-rank matrices the Hessian of this loss can theoretically blow up, which creates challenges to analyze convergence of optimizaton methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss and convergence results for finite step size gradient descent under certain assumptions on the initial weights.

We present SURE-Score: an approach for learning score-based generative models using training samples corrupted by additive Gaussian noise. When a large training set of clean samples is available, solving inverse problems via score-based (diffusion) generative models trained on the underlying fully-sampled data distribution has recently been shown to outperform end-to-end supervised deep learning. In practice, such a large collection of training data may be prohibitively expensive to acquire in the first place. In this work, we present an approach for approximately learning a score-based generative model of the clean distribution, from noisy training data. We formulate and justify a novel loss function that leverages Stein's unbiased risk estimate to jointly denoise the data and learn the score function via denoising score matching, while using only the noisy samples. We demonstrate the generality of SURE-Score by learning priors and applying posterior sampling to ill-posed inverse problems in two practical applications from different domains: compressive wireless multiple-input multiple-output channel estimation and accelerated 2D multi-coil magnetic resonance imaging reconstruction, where we demonstrate competitive reconstruction performance when learning at signal-to-noise ratio values of 0 and 10 dB, respectively.

Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimization challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https://github.com/duntrain/topo.

Bounding box regression is the crucial step in object detection. In existing methods, while ell_n-norm loss is widely adopted for bounding box regression, it is not tailored to the evaluation metric, i.e., Intersection over Union (IoU). Recently, IoU loss and generalized IoU (GIoU) loss have been proposed to benefit the IoU metric, but still suffer from the problems of slow convergence and inaccurate regression. In this paper, we propose a Distance-IoU (DIoU) loss by incorporating the normalized distance between the predicted box and the target box, which converges much faster in training than IoU and GIoU losses. Furthermore, this paper summarizes three geometric factors in bounding box regression, \ie, overlap area, central point distance and aspect ratio, based on which a Complete IoU (CIoU) loss is proposed, thereby leading to faster convergence and better performance. By incorporating DIoU and CIoU losses into state-of-the-art object detection algorithms, e.g., YOLO v3, SSD and Faster RCNN, we achieve notable performance gains in terms of not only IoU metric but also GIoU metric. Moreover, DIoU can be easily adopted into non-maximum suppression (NMS) to act as the criterion, further boosting performance improvement. The source code and trained models are available at https://github.com/Zzh-tju/DIoU.

Traditional knowledge distillation focuses on aligning the student's predicted probabilities with both ground-truth labels and the teacher's predicted probabilities. However, the transition to predicted probabilities from logits would obscure certain indispensable information. To address this issue, it is intuitive to additionally introduce a logit-level loss function as a supplement to the widely used probability-level loss function, for exploiting the latent information of logits. Unfortunately, we empirically find that the amalgamation of the newly introduced logit-level loss and the previous probability-level loss will lead to performance degeneration, even trailing behind the performance of employing either loss in isolation. We attribute this phenomenon to the collapse of the classification head, which is verified by our theoretical analysis based on the neural collapse theory. Specifically, the gradients of the two loss functions exhibit contradictions in the linear classifier yet display no such conflict within the backbone. Drawing from the theoretical analysis, we propose a novel method called dual-head knowledge distillation, which partitions the linear classifier into two classification heads responsible for different losses, thereby preserving the beneficial effects of both losses on the backbone while eliminating adverse influences on the classification head. Extensive experiments validate that our method can effectively exploit the information inside the logits and achieve superior performance against state-of-the-art counterparts.

Dynamic model pruning is a recent direction that allows for the inference of a different sub-network for each input sample during deployment. However, current dynamic methods rely on learning a continuous channel gating through regularization by inducing sparsity loss. This formulation introduces complexity in balancing different losses (e.g task loss, regularization loss). In addition, regularization based methods lack transparent tradeoff hyperparameter selection to realize a computational budget. Our contribution is two-fold: 1) decoupled task and pruning losses. 2) Simple hyperparameter selection that enables FLOPs reduction estimation before training. Inspired by the Hebbian theory in Neuroscience: "neurons that fire together wire together", we propose to predict a mask to process k filters in a layer based on the activation of its previous layer. We pose the problem as a self-supervised binary classification problem. Each mask predictor module is trained to predict if the log-likelihood for each filter in the current layer belongs to the top-k activated filters. The value k is dynamically estimated for each input based on a novel criterion using the mass of heatmaps. We show experiments on several neural architectures, such as VGG, ResNet and MobileNet on CIFAR and ImageNet datasets. On CIFAR, we reach similar accuracy to SOTA methods with 15% and 24% higher FLOPs reduction. Similarly in ImageNet, we achieve lower drop in accuracy with up to 13% improvement in FLOPs reduction.

We initiate the study of proper losses for evaluating generative models in the discrete setting. Unlike traditional proper losses, we treat both the generative model and the target distribution as black-boxes, only assuming ability to draw i.i.d. samples. We define a loss to be black-box proper if the generative distribution that minimizes expected loss is equal to the target distribution. Using techniques from statistical estimation theory, we give a general construction and characterization of black-box proper losses: they must take a polynomial form, and the number of draws from the model and target distribution must exceed the degree of the polynomial. The characterization rules out a loss whose expectation is the cross-entropy between the target distribution and the model. By extending the construction to arbitrary sampling schemes such as Poisson sampling, however, we show that one can construct such a loss.

Large language models (LLMs) have been increasingly employed for (interactive) decision-making, via the development of LLM-based autonomous agents. Despite their emerging successes, the performance of LLM agents in decision-making has not been fully investigated through quantitative metrics, especially in the multi-agent setting when they interact with each other, a typical scenario in real-world LLM-agent applications. To better understand the limits of LLM agents in these interactive environments, we propose to study their interactions in benchmark decision-making settings in online learning and game theory, through the performance metric of regret. We first empirically study the {no-regret} behaviors of LLMs in canonical (non-stationary) online learning problems, as well as the emergence of equilibria when LLM agents interact through playing repeated games. We then provide some theoretical insights into the no-regret behaviors of LLM agents, under certain assumptions on the supervised pre-training and the rationality model of human decision-makers who generate the data. Notably, we also identify (simple) cases where advanced LLMs such as GPT-4 fail to be no-regret. To promote the no-regret behaviors, we propose a novel unsupervised training loss of regret-loss, which, in contrast to the supervised pre-training loss, does not require the labels of (optimal) actions. We then establish the statistical guarantee of generalization bound for regret-loss minimization, followed by the optimization guarantee that minimizing such a loss may automatically lead to known no-regret learning algorithms. Our further experiments demonstrate the effectiveness of our regret-loss, especially in addressing the above ``regrettable'' cases.

Happiness computing based on large-scale online web data and machine learning methods is an emerging research topic that underpins a range of issues, from personal growth to social stability. Many advanced Machine Learning (ML) models with explanations are used to compute the happiness online assessment while maintaining high accuracy of results. However, domain knowledge constraints, such as the primary and secondary relations of happiness factors, are absent from these models, which limits the association between computing results and the right reasons for why they occurred. This article attempts to provide new insights into the explanation consistency from an empirical study perspective. Then we study how to represent and introduce domain knowledge constraints to make ML models more trustworthy. We achieve this through: (1) proving that multiple prediction models with additive factor attributions will have the desirable property of primary and secondary relations consistency, and (2) showing that factor relations with quantity can be represented as an importance distribution for encoding domain knowledge. Factor explanation difference is penalized by the Kullback-Leibler divergence-based loss among computing models. Experimental results using two online web datasets show that domain knowledge of stable factor relations exists. Using this knowledge not only improves happiness computing accuracy but also reveals more significative happiness factors for assisting decisions well.

While scaling laws provide a reliable methodology for predicting train loss across compute scales for a single data distribution, less is known about how these predictions should change as we change the distribution. In this paper, we derive a strategy for predicting one loss from another and apply it to predict across different pre-training datasets and from pre-training data to downstream task data. Our predictions extrapolate well even at 20x the largest FLOP budget used to fit the curves. More precisely, we find that there are simple shifted power law relationships between (1) the train losses of two models trained on two separate datasets when the models are paired by training compute (train-to-train), (2) the train loss and the test loss on any downstream distribution for a single model (train-to-test), and (3) the test losses of two models trained on two separate train datasets (test-to-test). The results hold up for pre-training datasets that differ substantially (some are entirely code and others have no code at all) and across a variety of downstream tasks. Finally, we find that in some settings these shifted power law relationships can yield more accurate predictions than extrapolating single-dataset scaling laws.

Transfer learning is widely used for training deep neural networks (DNN) for building a powerful representation. Even after the pre-trained model is adapted for the target task, the representation performance of the feature extractor is retained to some extent. As the performance of the pre-trained model can be considered the private property of the owner, it is natural to seek the exclusive right of the generalized performance of the pre-trained weight. To address this issue, we suggest a new paradigm of transfer learning called disposable transfer learning (DTL), which disposes of only the source task without degrading the performance of the target task. To achieve knowledge disposal, we propose a novel loss named Gradient Collision loss (GC loss). GC loss selectively unlearns the source knowledge by leading the gradient vectors of mini-batches in different directions. Whether the model successfully unlearns the source task is measured by piggyback learning accuracy (PL accuracy). PL accuracy estimates the vulnerability of knowledge leakage by retraining the scrubbed model on a subset of source data or new downstream data. We demonstrate that GC loss is an effective approach to the DTL problem by showing that the model trained with GC loss retains the performance on the target task with a significantly reduced PL accuracy.

Catastrophic overfitting (CO) in single-step adversarial training (AT) results in abrupt drops in the adversarial test accuracy (even down to 0%). For models trained with multi-step AT, it has been observed that the loss function behaves locally linearly with respect to the input, this is however lost in single-step AT. To address CO in single-step AT, several methods have been proposed to enforce local linearity of the loss via regularization. However, these regularization terms considerably slow down training due to Double Backpropagation. Instead, in this work, we introduce a regularization term, called ELLE, to mitigate CO effectively and efficiently in classical AT evaluations, as well as some more difficult regimes, e.g., large adversarial perturbations and long training schedules. Our regularization term can be theoretically linked to curvature of the loss function and is computationally cheaper than previous methods by avoiding Double Backpropagation. Our thorough experimental validation demonstrates that our work does not suffer from CO, even in challenging settings where previous works suffer from it. We also notice that adapting our regularization parameter during training (ELLE-A) greatly improves the performance, specially in large epsilon setups. Our implementation is available in https://github.com/LIONS-EPFL/ELLE .

Training classifiers is difficult with severe class imbalance, but many rare events are the culmination of a sequence with much more common intermediate outcomes. For example, in online marketing a user first sees an ad, then may click on it, and finally may make a purchase; estimating the probability of purchases is difficult because of their rarity. We show both theoretically and through data experiments that the more abundant data in earlier steps may be leveraged to improve estimation of probabilities of rare events. We present PRESTO, a relaxation of the proportional odds model for ordinal regression. Instead of estimating weights for one separating hyperplane that is shifted by separate intercepts for each of the estimated Bayes decision boundaries between adjacent pairs of categorical responses, we estimate separate weights for each of these transitions. We impose an L1 penalty on the differences between weights for the same feature in adjacent weight vectors in order to shrink towards the proportional odds model. We prove that PRESTO consistently estimates the decision boundary weights under a sparsity assumption. Synthetic and real data experiments show that our method can estimate rare probabilities in this setting better than both logistic regression on the rare category, which fails to borrow strength from more abundant categories, and the proportional odds model, which is too inflexible.

For observational studies, we study the sensitivity of causal inference when treatment assignments may depend on unobserved confounders. We develop a loss minimization approach for estimating bounds on the conditional average treatment effect (CATE) when unobserved confounders have a bounded effect on the odds ratio of treatment selection. Our approach is scalable and allows flexible use of model classes in estimation, including nonparametric and black-box machine learning methods. Based on these bounds for the CATE, we propose a sensitivity analysis for the average treatment effect (ATE). Our semi-parametric estimator extends/bounds the augmented inverse propensity weighted (AIPW) estimator for the ATE under bounded unobserved confounding. By constructing a Neyman orthogonal score, our estimator of the bound for the ATE is a regular root-n estimator so long as the nuisance parameters are estimated at the o_p(n^{-1/4}) rate. We complement our methodology with optimality results showing that our proposed bounds are tight in certain cases. We demonstrate our method on simulated and real data examples, and show accurate coverage of our confidence intervals in practical finite sample regimes with rich covariate information.

This work examines the challenges of training neural networks using vector quantization using straight-through estimation. We find that a primary cause of training instability is the discrepancy between the model embedding and the code-vector distribution. We identify the factors that contribute to this issue, including the codebook gradient sparsity and the asymmetric nature of the commitment loss, which leads to misaligned code-vector assignments. We propose to address this issue via affine re-parameterization of the code vectors. Additionally, we introduce an alternating optimization to reduce the gradient error introduced by the straight-through estimation. Moreover, we propose an improvement to the commitment loss to ensure better alignment between the codebook representation and the model embedding. These optimization methods improve the mathematical approximation of the straight-through estimation and, ultimately, the model performance. We demonstrate the effectiveness of our methods on several common model architectures, such as AlexNet, ResNet, and ViT, across various tasks, including image classification and generative modeling.

The S{\o}rensen--Dice Coefficient has recently seen rising popularity as a loss function (also known as Dice loss) due to its robustness in tasks where the number of negative samples significantly exceeds that of positive samples, such as semantic segmentation, natural language processing, and sound event detection. Conventional training of polyphonic sound event detection systems with binary cross-entropy loss often results in suboptimal detection performance as the training is often overwhelmed by updates from negative samples. In this paper, we investigated the effect of the Dice loss, intra- and inter-modal transfer learning, data augmentation, and recording formats, on the performance of polyphonic sound event detection systems with multichannel inputs. Our analysis showed that polyphonic sound event detection systems trained with Dice loss consistently outperformed those trained with cross-entropy loss across different training settings and recording formats in terms of F1 score and error rate. We achieved further performance gains via the use of transfer learning and an appropriate combination of different data augmentation techniques.

Overparameterized neural networks can be highly accurate on average on an i.i.d. test set yet consistently fail on atypical groups of the data (e.g., by learning spurious correlations that hold on average but not in such groups). Distributionally robust optimization (DRO) allows us to learn models that instead minimize the worst-case training loss over a set of pre-defined groups. However, we find that naively applying group DRO to overparameterized neural networks fails: these models can perfectly fit the training data, and any model with vanishing average training loss also already has vanishing worst-case training loss. Instead, the poor worst-case performance arises from poor generalization on some groups. By coupling group DRO models with increased regularization---a stronger-than-typical L2 penalty or early stopping---we achieve substantially higher worst-group accuracies, with 10-40 percentage point improvements on a natural language inference task and two image tasks, while maintaining high average accuracies. Our results suggest that regularization is important for worst-group generalization in the overparameterized regime, even if it is not needed for average generalization. Finally, we introduce a stochastic optimization algorithm, with convergence guarantees, to efficiently train group DRO models.

Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generalization performance. We theoretically analyze a solvable model, a diagonal linear network, and clarify that GR has a desirable implicit bias to so-called rich regime and finite-difference computation strengthens this bias. Furthermore, finite-difference GR is closely related to some other algorithms based on iterative ascent and descent steps for exploring flat minima. In particular, we reveal that the flooding method can perform finite-difference GR in an implicit way. Thus, this work broadens our understanding of GR for both practice and theory.

We introduce a new mechanism for stochastic convex optimization (SCO) with user-level differential privacy guarantees. The convergence rates of this mechanism are similar to those in the prior work of Levy et al. (2021); Narayanan et al. (2022), but with two important improvements. Our mechanism does not require any smoothness assumptions on the loss. Furthermore, our bounds are also the first where the minimum number of users needed for user-level privacy has no dependence on the dimension and only a logarithmic dependence on the desired excess error. The main idea underlying the new mechanism is to show that the optimizers of strongly convex losses have low local deletion sensitivity, along with an output perturbation method for functions with low local deletion sensitivity, which could be of independent interest.

This paper investigates the distinctions between gradient methods applied to non-differentiable functions (NGDMs) and classical gradient descents (GDs) designed for differentiable functions. First, we demonstrate significant differences in the convergence properties of NGDMs compared to GDs, challenging the applicability of the extensive neural network convergence literature based on L-smoothness to non-smooth neural networks. Next, we demonstrate the paradoxical nature of NGDM solutions for L_{1}-regularized problems, showing that increasing the regularization penalty leads to an increase in the L_{1} norm of optimal solutions in NGDMs. Consequently, we show that widely adopted L_{1} penalization-based techniques for network pruning do not yield expected results. Finally, we explore the Edge of Stability phenomenon, indicating its inapplicability even to Lipschitz continuous convex differentiable functions, leaving its relevance to non-convex non-differentiable neural networks inconclusive. Our analysis exposes misguided interpretations of NGDMs in widely referenced papers and texts due to an overreliance on strong smoothness assumptions, emphasizing the necessity for a nuanced understanding of foundational assumptions in the analysis of these systems.

Reinforcement learning from human feedback (RLHF) has emerged as a reliable approach to aligning large language models (LLMs) to human preferences. Among the plethora of RLHF techniques, proximal policy optimization (PPO) is of the most widely used methods. Despite its popularity, however, PPO may suffer from mode collapse, instability, and poor sample efficiency. We show that these issues can be alleviated by a novel algorithm that we refer to as Advantage-Induced Policy Alignment (APA), which leverages a squared error loss function based on the estimated advantages. We demonstrate empirically that APA consistently outperforms PPO in language tasks by a large margin, when a separate reward model is employed as the evaluator. In addition, compared with PPO, APA offers a more stable form of control over the deviation from the model's initial policy, ensuring that the model improves its performance without collapsing to deterministic output. In addition to empirical results, we also provide a theoretical justification supporting the design of our loss function.

Algorithms for online learning typically require one or more boundedness assumptions: that the domain is bounded, that the losses are Lipschitz, or both. In this paper, we develop a new setting for online learning with unbounded domains and non-Lipschitz losses. For this setting we provide an algorithm which guarantees R_{T}(u)le tilde O(G|u|T+L|u|^{2}T) regret on any problem where the subgradients satisfy |g_{t}|le G+L|w_{t}|, and show that this bound is unimprovable without further assumptions. We leverage this algorithm to develop new saddle-point optimization algorithms that converge in duality gap in unbounded domains, even in the absence of meaningful curvature. Finally, we provide the first algorithm achieving non-trivial dynamic regret in an unbounded domain for non-Lipschitz losses, as well as a matching lower bound. The regret of our dynamic regret algorithm automatically improves to a novel L^{*} bound when the losses are smooth.

Effective regularization techniques are highly desired in deep learning for alleviating overfitting and improving generalization. This work proposes a new regularization scheme, based on the understanding that the flat local minima of the empirical risk cause the model to generalize better. This scheme is referred to as adversarial model perturbation (AMP), where instead of directly minimizing the empirical risk, an alternative "AMP loss" is minimized via SGD. Specifically, the AMP loss is obtained from the empirical risk by applying the "worst" norm-bounded perturbation on each point in the parameter space. Comparing with most existing regularization schemes, AMP has strong theoretical justifications, in that minimizing the AMP loss can be shown theoretically to favour flat local minima of the empirical risk. Extensive experiments on various modern deep architectures establish AMP as a new state of the art among regularization schemes. Our code is available at https://github.com/hiyouga/AMP-Regularizer.

Pretraining large language models (LLMs) on vast and heterogeneous datasets is crucial for achieving state-of-the-art performance across diverse downstream tasks. However, current training paradigms treat all samples equally, overlooking the importance or relevance of individual samples throughout the training process. Existing reweighting strategies, which primarily focus on group-level data importance, fail to leverage fine-grained instance-level information and do not adapt dynamically to individual sample importance as training progresses. In this paper, we introduce novel algorithms for dynamic, instance-level data reweighting aimed at improving both the efficiency and effectiveness of LLM pretraining. Our methods adjust the weight of each training sample based on its loss value in an online fashion, allowing the model to dynamically focus on more informative or important samples at the current training stage. In particular, our framework allows us to systematically devise reweighting strategies deprioritizing redundant or uninformative data, which we find tend to work best. Furthermore, we develop a new theoretical framework for analyzing the impact of loss-based reweighting on the convergence of gradient-based optimization, providing the first formal characterization of how these strategies affect convergence bounds. We empirically validate our approach across a spectrum of tasks, from pretraining 7B and 1.4B parameter LLMs to smaller-scale language models and linear regression problems, demonstrating that our loss-based reweighting approach can lead to faster convergence and significantly improved performance.

We consider minimizing functions for which it is expensive to compute the (possibly stochastic) gradient. Such functions are prevalent in reinforcement learning, imitation learning and adversarial training. Our target optimization framework uses the (expensive) gradient computation to construct surrogate functions in a target space (e.g. the logits output by a linear model for classification) that can be minimized efficiently. This allows for multiple parameter updates to the model, amortizing the cost of gradient computation. In the full-batch setting, we prove that our surrogate is a global upper-bound on the loss, and can be (locally) minimized using a black-box optimization algorithm. We prove that the resulting majorization-minimization algorithm ensures convergence to a stationary point of the loss. Next, we instantiate our framework in the stochastic setting and propose the SSO algorithm, which can be viewed as projected stochastic gradient descent in the target space. This connection enables us to prove theoretical guarantees for SSO when minimizing convex functions. Our framework allows the use of standard stochastic optimization algorithms to construct surrogates which can be minimized by any deterministic optimization method. To evaluate our framework, we consider a suite of supervised learning and imitation learning problems. Our experiments indicate the benefits of target optimization and the effectiveness of SSO.

Developers rely on code comments to document their work, track issues, and understand the source code. As such, comments provide valuable insights into developers' understanding of their code and describe their various intentions in writing the surrounding code. Recent research leverages natural language processing and deep learning to classify comments based on developers' intentions. However, such labelled data are often imbalanced, causing learning models to perform poorly. This work investigates the use of different weighting strategies of the loss function to mitigate the scarcity of certain classes in the dataset. In particular, various RoBERTa-based transformer models are fine-tuned by means of a hyperparameter search to identify their optimal parameter configurations. Additionally, we fine-tuned the transformers with different weighting strategies for the loss function to address class imbalances. Our approach outperforms the STACC baseline by 8.9 per cent on the NLBSE'25 Tool Competition dataset in terms of the average F1_c score, and exceeding the baseline approach in 17 out of 19 cases with a gain ranging from -5.0 to 38.2. The source code is publicly available at https://github.com/moritzmock/NLBSE2025.

A large-scale deep model pre-trained on massive labeled or unlabeled data transfers well to downstream tasks. Linear evaluation freezes parameters in the pre-trained model and trains a linear classifier separately, which is efficient and attractive for transfer. However, little work has investigated the classifier in linear evaluation except for the default logistic regression. Inspired by the statistical efficiency of naive Bayes, the paper revisits the classical topic on discriminative vs. generative classifiers. Theoretically, the paper considers the surrogate loss instead of the zero-one loss in analyses and generalizes the classical results from binary cases to multiclass ones. We show that, under mild assumptions, multiclass naive Bayes requires O(log n) samples to approach its asymptotic error while the corresponding multiclass logistic regression requires O(n) samples, where n is the feature dimension. To establish it, we present a multiclass H-consistency bound framework and an explicit bound for logistic loss, which are of independent interests. Simulation results on a mixture of Gaussian validate our theoretical findings. Experiments on various pre-trained deep vision models show that naive Bayes consistently converges faster as the number of data increases. Besides, naive Bayes shows promise in few-shot cases and we observe the "two regimes" phenomenon in pre-trained supervised models. Our code is available at https://github.com/ML-GSAI/Revisiting-Dis-vs-Gen-Classifiers.

We study robustness to test-time adversarial attacks in the regression setting with ell_p losses and arbitrary perturbation sets. We address the question of which function classes are PAC learnable in this setting. We show that classes of finite fat-shattering dimension are learnable in both realizable and agnostic settings. Moreover, for convex function classes, they are even properly learnable. In contrast, some non-convex function classes provably require improper learning algorithms. Our main technique is based on a construction of an adversarially robust sample compression scheme of a size determined by the fat-shattering dimension. Along the way, we introduce a novel agnostic sample compression scheme for real-valued functions, which may be of independent interest.

Efficiently and flexibly estimating treatment effect heterogeneity is an important task in a wide variety of settings ranging from medicine to marketing, and there are a considerable number of promising conditional average treatment effect estimators currently available. These, however, typically rely on the assumption that the measured covariates are enough to justify conditional exchangeability. We propose the P-learner, motivated by the R- and DR-learner, a tailored two-stage loss function for learning heterogeneous treatment effects in settings where exchangeability given observed covariates is an implausible assumption, and we wish to rely on proxy variables for causal inference. Our proposed estimator can be implemented by off-the-shelf loss-minimizing machine learning methods, which in the case of kernel regression satisfies an oracle bound on the estimated error as long as the nuisance components are estimated reasonably well.

Diffusion models have emerged as a pivotal advancement in generative models, setting new standards to the quality of the generated instances. In the current paper we aim to underscore a discrepancy between conventional training methods and the desired conditional sampling behavior of these models. While the prevalent classifier-free guidance technique works well, it's not without flaws. At higher values for the guidance scale parameter w, we often get out of distribution samples and mode collapse, whereas at lower values for w we may not get the desired specificity. To address these challenges, we introduce an updated loss function that better aligns training objectives with sampling behaviors. Experimental validation with FID scores on CIFAR-10 elucidates our method's ability to produce higher quality samples with fewer sampling timesteps, and be more robust to the choice of guidance scale w. We also experiment with fine-tuning Stable Diffusion on the proposed loss, to provide early evidence that large diffusion models may also benefit from this refined loss function.

Training a large and state-of-the-art machine learning model typically necessitates the use of large-scale datasets, which, in turn, makes the training and parameter-tuning process expensive and time-consuming. Some researchers opt to distil information from real-world datasets into tiny and compact synthetic datasets while maintaining their ability to train a well-performing model, hence proposing a data-efficient method known as Dataset Distillation (DD). Despite recent progress in this field, existing methods still underperform and cannot effectively replace large datasets. In this paper, unlike previous methods that focus solely on improving the efficacy of student distillation, we are the first to recognize the important interplay between expert and student. We argue the significant impact of expert smoothness when employing more potent expert trajectories in subsequent dataset distillation. Based on this, we introduce the integration of clipping loss and gradient penalty to regulate the rate of parameter changes in expert trajectories. Furthermore, in response to the sensitivity exhibited towards randomly initialized variables during distillation, we propose representative initialization for synthetic dataset and balanced inner-loop loss. Finally, we present two enhancement strategies, namely intermediate matching loss and weight perturbation, to mitigate the potential occurrence of cumulative errors. We conduct extensive experiments on datasets of different scales, sizes, and resolutions. The results demonstrate that the proposed method significantly outperforms prior methods.

The ability to learn different tasks sequentially is essential to the development of artificial intelligence. In general, neural networks lack this capability, the major obstacle being catastrophic forgetting. It occurs when the incrementally available information from non-stationary data distributions is continually acquired, disrupting what the model has already learned. Our approach remembers old tasks by projecting the representations of new tasks close to that of old tasks while keeping the decision boundaries unchanged. We employ the center loss as a regularization penalty that enforces new tasks' features to have the same class centers as old tasks and makes the features highly discriminative. This, in turn, leads to the least forgetting of already learned information. This method is easy to implement, requires minimal computational and memory overhead, and allows the neural network to maintain high performance across many sequentially encountered tasks. We also demonstrate that using the center loss in conjunction with the memory replay outperforms other replay-based strategies. Along with standard MNIST variants for continual learning, we apply our method to continual domain adaptation scenarios with the Digits and PACS datasets. We demonstrate that our approach is scalable, effective, and gives competitive performance compared to state-of-the-art continual learning methods.

Deep neural networks exploiting millions of parameters are nowadays the norm in deep learning applications. This is a potential issue because of the great amount of computational resources needed for training, and of the possible loss of generalization performance of overparametrized networks. We propose in this paper a method for learning sparse neural topologies via a regularization technique which identifies non relevant weights and selectively shrinks their norm, while performing a classic update for relevant ones. This technique, which is an improvement of classical weight decay, is based on the definition of a regularization term which can be added to any loss functional regardless of its form, resulting in a unified general framework exploitable in many different contexts. The actual elimination of parameters identified as irrelevant is handled by an iterative pruning algorithm. We tested the proposed technique on different image classification and Natural language generation tasks, obtaining results on par or better then competitors in terms of sparsity and metrics, while achieving strong models compression.

Self-distillation (SD) is the process of first training a teacher model and then using its predictions to train a student model with the same architecture. Specifically, the student's objective function is big(xi*ell(teacher's predictions, student's predictions) + (1-xi)*ell(given labels, student's predictions)big), where ell is some loss function and xi is some parameter in [0,1]. Empirically, SD has been observed to provide performance gains in several settings. In this paper, we theoretically characterize the effect of SD in two supervised learning problems with noisy labels. We first analyze SD for regularized linear regression and show that in the high label noise regime, the optimal value of xi that minimizes the expected error in estimating the ground truth parameter is surprisingly greater than 1. Empirically, we show that xi > 1 works better than xi leq 1 even with the cross-entropy loss for several classification datasets when 50\% or 30\% of the labels are corrupted. Further, we quantify when optimal SD is better than optimal regularization. Next, we analyze SD in the case of logistic regression for binary classification with random label corruption and quantify the range of label corruption in which the student outperforms the teacher in terms of accuracy. To our knowledge, this is the first result of its kind for the cross-entropy loss.

This paper provides a pair similarity optimization viewpoint on deep feature learning, aiming to maximize the within-class similarity s_p and minimize the between-class similarity s_n. We find a majority of loss functions, including the triplet loss and the softmax plus cross-entropy loss, embed s_n and s_p into similarity pairs and seek to reduce (s_n-s_p). Such an optimization manner is inflexible, because the penalty strength on every single similarity score is restricted to be equal. Our intuition is that if a similarity score deviates far from the optimum, it should be emphasized. To this end, we simply re-weight each similarity to highlight the less-optimized similarity scores. It results in a Circle loss, which is named due to its circular decision boundary. The Circle loss has a unified formula for two elemental deep feature learning approaches, i.e. learning with class-level labels and pair-wise labels. Analytically, we show that the Circle loss offers a more flexible optimization approach towards a more definite convergence target, compared with the loss functions optimizing (s_n-s_p). Experimentally, we demonstrate the superiority of the Circle loss on a variety of deep feature learning tasks. On face recognition, person re-identification, as well as several fine-grained image retrieval datasets, the achieved performance is on par with the state of the art.

Reinforcement Learning from Human Feedback (RLHF) has become the predominant approach for language model (LM) alignment. At its core, RLHF uses a margin-based loss for preference optimization, specifying ideal LM behavior only by the difference between preferred and dispreferred responses. In this paper, we identify a common pitfall of margin-based methods -- the under-specification of ideal LM behavior on preferred and dispreferred responses individually, which leads to two unintended consequences as the margin increases: (1) The probability of dispreferred (e.g., unsafe) responses may increase, resulting in potential safety alignment failures. (2) The probability of preferred responses may decrease, even when those responses are ideal. We demystify the reasons behind these problematic behaviors: margin-based losses couple the change in the preferred probability to the gradient of the dispreferred one, and vice versa, often preventing the preferred probability from increasing while the dispreferred one decreases, and thus causing a synchronized increase or decrease in both probabilities. We term this effect, inherent in margin-based objectives, gradient entanglement. Formally, we derive conditions for general margin-based alignment objectives under which gradient entanglement becomes concerning: the inner product of the gradients of preferred and dispreferred log-probabilities is large relative to the individual gradient norms. We theoretically investigate why such inner products can be large when aligning language models and empirically validate our findings. Empirical implications of our framework extend to explaining important differences in the training dynamics of various preference optimization algorithms, and suggesting potential algorithm designs to mitigate the under-specification issue of margin-based methods and thereby improving language model alignment.

Offline reinforcement learning (RL), where the agent aims to learn the optimal policy based on the data collected by a behavior policy, has attracted increasing attention in recent years. While offline RL with linear function approximation has been extensively studied with optimal results achieved under certain assumptions, many works shift their interest to offline RL with non-linear function approximation. However, limited works on offline RL with non-linear function approximation have instance-dependent regret guarantees. In this paper, we propose an oracle-efficient algorithm, dubbed Pessimistic Nonlinear Least-Square Value Iteration (PNLSVI), for offline RL with non-linear function approximation. Our algorithmic design comprises three innovative components: (1) a variance-based weighted regression scheme that can be applied to a wide range of function classes, (2) a subroutine for variance estimation, and (3) a planning phase that utilizes a pessimistic value iteration approach. Our algorithm enjoys a regret bound that has a tight dependency on the function class complexity and achieves minimax optimal instance-dependent regret when specialized to linear function approximation. Our work extends the previous instance-dependent results within simpler function classes, such as linear and differentiable function to a more general framework.

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e.g. performance on another dataset, robustness, agreement with a prior). Although the simplest approach to incorporating an auxiliary loss is to sum it with the training loss as a regularizer, recent works have shown that one can improve performance by blending the gradients beyond a simple sum; this is known as gradient surgery. We cast the problem as a constrained minimization problem where the auxiliary objective is minimized among the set of minimizers of the training loss. To solve this bilevel problem, we follow a parameter update direction that combines the training loss gradient and the orthogonal projection of the auxiliary gradient to the training gradient. In a setting where gradients come from mini-batches, we explain how, using a moving average of the training loss gradients, we can carefully maintain this critical orthogonality property. We demonstrate that our method, Bloop, can lead to much better performances on NLP and vision experiments than other gradient surgery methods without EMA.

In the presence of noisy labels, designing robust loss functions is critical for securing the generalization performance of deep neural networks. Cross Entropy (CE) loss has been shown to be not robust to noisy labels due to its unboundedness. To alleviate this issue, existing works typically design specialized robust losses with the symmetric condition, which usually lead to the underfitting issue. In this paper, our key idea is to induce a loss bound at the logit level, thus universally enhancing the noise robustness of existing losses. Specifically, we propose logit clipping (LogitClip), which clamps the norm of the logit vector to ensure that it is upper bounded by a constant. In this manner, CE loss equipped with our LogitClip method is effectively bounded, mitigating the overfitting to examples with noisy labels. Moreover, we present theoretical analyses to certify the noise-tolerant ability of LogitClip. Extensive experiments show that LogitClip not only significantly improves the noise robustness of CE loss, but also broadly enhances the generalization performance of popular robust losses.

The loss landscape of neural networks is a critical aspect of their training, and understanding its properties is essential for improving their performance. In this paper, we investigate how the loss surface changes when the sample size increases, a previously unexplored issue. We theoretically analyze the convergence of the loss landscape in a fully connected neural network and derive upper bounds for the difference in loss function values when adding a new object to the sample. Our empirical study confirms these results on various datasets, demonstrating the convergence of the loss function surface for image classification tasks. Our findings provide insights into the local geometry of neural loss landscapes and have implications for the development of sample size determination techniques.

In most machine learning tasks, we evaluate a model M on a given data population S by measuring a population-level metric F(S;M). Examples of such evaluation metric F include precision/recall for (binary) recognition, the F1 score for multi-class classification, and the BLEU metric for language generation. On the other hand, the model M is trained by optimizing a sample-level loss G(S_t;M) at each learning step t, where S_t is a subset of S (a.k.a. the mini-batch). Popular choices of G include cross-entropy loss, the Dice loss, and sentence-level BLEU scores. A fundamental assumption behind this paradigm is that the mean value of the sample-level loss G, if averaged over all possible samples, should effectively represent the population-level metric F of the task, such as, that E[ G(S_t;M) ] approx F(S;M). In this paper, we systematically investigate the above assumption in several NLP tasks. We show, both theoretically and experimentally, that some popular designs of the sample-level loss G may be inconsistent with the true population-level metric F of the task, so that models trained to optimize the former can be substantially sub-optimal to the latter, a phenomenon we call it, Simpson's bias, due to its deep connections with the classic paradox known as Simpson's reversal paradox in statistics and social sciences.

In our era of enormous neural networks, empirical progress has been driven by the philosophy that more is better. Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained. Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and the number of samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is obligatory: near-optimal performance can only be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural tangent kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization, overfitting, and more data in random feature models.

In this paper, we introduce refined Direct Preference Optimization (rDPO), a method for improving the behavioral alignment of Large Language Models (LLMs) without the need for human-annotated data. The method involves creating synthetic data using self-critique prompting by a teacher LLM and then utilising a generalized DPO loss function to distil to a student LLM. The loss function incorporates an additional external reward model to improve the quality of synthetic data, making rDPO robust to potential noise in the synthetic dataset. rDPO is shown to be effective in a diverse set of behavioural alignment tasks, such as improved safety, robustness against role-playing, and reduced sycophancy. Code to be released at https://github.com/vicgalle/refined-dpo.

In order to solve a task using reinforcement learning, it is necessary to first formalise the goal of that task as a reward function. However, for many real-world tasks, it is very difficult to manually specify a reward function that never incentivises undesirable behaviour. As a result, it is increasingly popular to use reward learning algorithms, which attempt to learn a reward function from data. However, the theoretical foundations of reward learning are not yet well-developed. In particular, it is typically not known when a given reward learning algorithm with high probability will learn a reward function that is safe to optimise. This means that reward learning algorithms generally must be evaluated empirically, which is expensive, and that their failure modes are difficult to anticipate in advance. One of the roadblocks to deriving better theoretical guarantees is the lack of good methods for quantifying the difference between reward functions. In this paper we provide a solution to this problem, in the form of a class of pseudometrics on the space of all reward functions that we call STARC (STAndardised Reward Comparison) metrics. We show that STARC metrics induce both an upper and a lower bound on worst-case regret, which implies that our metrics are tight, and that any metric with the same properties must be bilipschitz equivalent to ours. Moreover, we also identify a number of issues with reward metrics proposed by earlier works. Finally, we evaluate our metrics empirically, to demonstrate their practical efficacy. STARC metrics can be used to make both theoretical and empirical analysis of reward learning algorithms both easier and more principled.

Bilevel optimization enjoys a wide range of applications in hyper-parameter optimization, meta-learning and reinforcement learning. However, bilevel optimization problems are difficult to solve. Recent progress on scalable bilevel algorithms mainly focuses on bilevel optimization problems where the lower-level objective is either strongly convex or unconstrained. In this work, we tackle the bilevel problem through the lens of the penalty method. We show that under certain conditions, the penalty reformulation recovers the solutions of the original bilevel problem. Further, we propose the penalty-based bilevel gradient descent (PBGD) algorithm and establish its finite-time convergence for the constrained bilevel problem without lower-level strong convexity. Experiments showcase the efficiency of the proposed PBGD algorithm.

We introduce a novel and general loss function, called Symmetric Contrastive (Sy-CON) loss, for effective continual self-supervised learning (CSSL). We first argue that the conventional loss form of continual learning which consists of single task-specific loss (for plasticity) and a regularizer (for stability) may not be ideal for contrastive loss based CSSL that focus on representation learning. Our reasoning is that, in contrastive learning based methods, the task-specific loss would suffer from decreasing diversity of negative samples and the regularizer may hinder learning new distinctive representations. To that end, we propose Sy-CON that consists of two losses (one for plasticity and the other for stability) with symmetric dependence on current and past models' negative sample embeddings. We argue our model can naturally find good trade-off between the plasticity and stability without any explicit hyperparameter tuning. We validate the effectiveness of our approach through extensive experiments, demonstrating that MoCo-based implementation of Sy-CON loss achieves superior performance compared to other state-of-the-art CSSL methods.

Nearly all practical neural models for classification are trained using cross-entropy loss. Yet this ubiquitous choice is supported by little theoretical or empirical evidence. Recent work (Hui & Belkin, 2020) suggests that training using the (rescaled) square loss is often superior in terms of the classification accuracy. In this paper we propose the "squentropy" loss, which is the sum of two terms: the cross-entropy loss and the average square loss over the incorrect classes. We provide an extensive set of experiments on multi-class classification problems showing that the squentropy loss outperforms both the pure cross entropy and rescaled square losses in terms of the classification accuracy. We also demonstrate that it provides significantly better model calibration than either of these alternative losses and, furthermore, has less variance with respect to the random initialization. Additionally, in contrast to the square loss, squentropy loss can typically be trained using exactly the same optimization parameters, including the learning rate, as the standard cross-entropy loss, making it a true "plug-and-play" replacement. Finally, unlike the rescaled square loss, multiclass squentropy contains no parameters that need to be adjusted.

We introduce Gradient Descent with Adaptive Momentum Scaling (Grams), a novel optimization algorithm that decouples the direction and magnitude of parameter updates in deep learning. Unlike traditional optimizers that directly integrate momentum into updates, Grams separates the update direction, derived from current gradients, from momentum, which is used solely for adaptive magnitude scaling. This approach enables Grams to achieve improved loss descent compared to state-of-the-art cautious and momentum-based optimizers. We establish a global convergence guarantee for Grams and validate its effectiveness through extensive empirical evaluations. The results demonstrate Grams' superior performance, including faster convergence and better generalization, compared to widely-used optimizers such as Adam, Lion, and their cautious variants. Our results highlight Grams' potential as a transformative approach for efficient optimization in large-scale machine learning.

In this work, we study the evolution of the loss Hessian across many classification tasks in order to understand the effect the curvature of the loss has on the training dynamics. Whereas prior work has focused on how different learning rates affect the loss Hessian observed during training, we also analyze the effects of model initialization, architectural choices, and common training heuristics such as gradient clipping and learning rate warmup. Our results demonstrate that successful model and hyperparameter choices allow the early optimization trajectory to either avoid -- or navigate out of -- regions of high curvature and into flatter regions that tolerate a higher learning rate. Our results suggest a unifying perspective on how disparate mitigation strategies for training instability ultimately address the same underlying failure mode of neural network optimization, namely poor conditioning. Inspired by the conditioning perspective, we show that learning rate warmup can improve training stability just as much as batch normalization, layer normalization, MetaInit, GradInit, and Fixup initialization.

Real-world datasets exhibit imbalances of varying types and degrees. Several techniques based on re-weighting and margin adjustment of loss are often used to enhance the performance of neural networks, particularly on minority classes. In this work, we analyze the class-imbalanced learning problem by examining the loss landscape of neural networks trained with re-weighting and margin-based techniques. Specifically, we examine the spectral density of Hessian of class-wise loss, through which we observe that the network weights converge to a saddle point in the loss landscapes of minority classes. Following this observation, we also find that optimization methods designed to escape from saddle points can be effectively used to improve generalization on minority classes. We further theoretically and empirically demonstrate that Sharpness-Aware Minimization (SAM), a recent technique that encourages convergence to a flat minima, can be effectively used to escape saddle points for minority classes. Using SAM results in a 6.2\% increase in accuracy on the minority classes over the state-of-the-art Vector Scaling Loss, leading to an overall average increase of 4\% across imbalanced datasets. The code is available at: https://github.com/val-iisc/Saddle-LongTail.

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

Deep neural networks are highly performant, but might base their decision on spurious or background features that co-occur with certain classes, which can hurt generalization. To mitigate this issue, the usage of 'model guidance' has gained popularity recently: for this, models are guided to be "right for the right reasons" by regularizing the models' explanations to highlight the right features. Experimental validation of these approaches has thus far however been limited to relatively simple and / or synthetic datasets. To gain a better understanding of which model-guiding approaches actually transfer to more challenging real-world datasets, in this work we conduct an in-depth evaluation across various loss functions, attribution methods, models, and 'guidance depths' on the PASCAL VOC 2007 and MS COCO 2014 datasets, and show that model guidance can sometimes even improve model performance. In this context, we further propose a novel energy loss, show its effectiveness in directing the model to focus on object features. We also show that these gains can be achieved even with a small fraction (e.g. 1%) of bounding box annotations, highlighting the cost effectiveness of this approach. Lastly, we show that this approach can also improve generalization under distribution shifts. Code will be made available.

With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness requirements. These requirements can be imposed (with generalization guarantees) by formulating constrained learning problems that can then be tackled by dual ascent algorithms. Yet, though these algorithms converge in objective value, even in non-convex settings, they cannot guarantee that their outcome is feasible. Doing so requires randomizing over all iterates, which is impractical in virtually any modern applications. Still, final iterates have been observed to perform well in practice. In this work, we address this gap between theory and practice by characterizing the constraint violation of Lagrangian minimizers associated with optimal dual variables, despite lack of convexity. To do this, we leverage the fact that non-convex, finite-dimensional constrained learning problems can be seen as parametrizations of convex, functional problems. Our results show that rich parametrizations effectively mitigate the issue of feasibility in dual methods, shedding light on prior empirical successes of dual learning. We illustrate our findings in fair learning tasks.

Practitioners commonly align large language models using pairwise preferences, i.e., given labels of the type response A is preferred to response B for a given input. Perhaps less commonly, methods have also been developed for binary feedback, i.e. training models given labels of type response A is good or bad. We show how an existing performant binary feedback method, the Cringe Loss (Adolphs et al., 2022), can be generalized to the pairwise preference setting using a simple soft margin extension. Pairwise Cringe Loss is straightforward to implement and efficient to train, and we find it outperforms state-of-the-art preference optimization algorithms such as PPO and DPO on the AlpacaFarm benchmark.

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

We focus on the classification problem with a separable dataset, one of the most important and classical problems from machine learning. The standard approach to this task is logistic regression with gradient descent (LR+GD). Recent studies have observed that LR+GD can find a solution with arbitrarily large step sizes, defying conventional optimization theory. Our work investigates this phenomenon and makes three interconnected key observations about LR+GD with large step sizes. First, we find a remarkably simple explanation of why LR+GD with large step sizes solves the classification problem: LR+GD reduces to a batch version of the celebrated perceptron algorithm when the step size gamma to infty. Second, we observe that larger step sizes lead LR+GD to higher logistic losses when it tends to the perceptron algorithm, but larger step sizes also lead to faster convergence to a solution for the classification problem, meaning that logistic loss is an unreliable metric of the proximity to a solution. Surprisingly, high loss values can actually indicate faster convergence. Third, since the convergence rate in terms of loss function values of LR+GD is unreliable, we examine the iteration complexity required by LR+GD with large step sizes to solve the classification problem and prove that this complexity is suboptimal. To address this, we propose a new method, Normalized LR+GD - based on the connection between LR+GD and the perceptron algorithm - with much better theoretical guarantees.

Label Smoothing (LS) is widely adopted to curb overconfidence in neural network predictions and enhance generalization. However, previous research shows that LS can force feature representations into excessively tight clusters, eroding intra-class distinctions. More recent findings suggest that LS also induces overconfidence in misclassifications, yet the precise mechanism remained unclear. In this work, we decompose the loss term introduced by LS, revealing two key components: (i) a regularization term that functions only when the prediction is correct, and (ii) an error-enhancement term that emerges under misclassifications. This latter term compels the model to reinforce incorrect predictions with exaggerated certainty, further collapsing the feature space. To address these issues, we propose Max Suppression (MaxSup), which uniformly applies the intended regularization to both correct and incorrect predictions by penalizing the top-1 logit instead of the ground-truth logit. Through feature analyses, we show that MaxSup restores intra-class variation and sharpens inter-class boundaries. Extensive experiments on image classification and downstream tasks confirm that MaxSup is a more robust alternative to LS. Code is available at: https://github.com/ZhouYuxuanYX/Maximum-Suppression-Regularization.

The recent integration of deep learning and pairwise similarity annotation-based constrained clustering -- i.e., deep constrained clustering (DCC) -- has proven effective for incorporating weak supervision into massive data clustering: Less than 1% of pair similarity annotations can often substantially enhance the clustering accuracy. However, beyond empirical successes, there is a lack of understanding of DCC. In addition, many DCC paradigms are sensitive to annotation noise, but performance-guaranteed noisy DCC methods have been largely elusive. This work first takes a deep look into a recently emerged logistic loss function of DCC, and characterizes its theoretical properties. Our result shows that the logistic DCC loss ensures the identifiability of data membership under reasonable conditions, which may shed light on its effectiveness in practice. Building upon this understanding, a new loss function based on geometric factor analysis is proposed to fend against noisy annotations. It is shown that even under unknown annotation confusions, the data membership can still be provably identified under our proposed learning criterion. The proposed approach is tested over multiple datasets to validate our claims.

Reinforcement learning from human feedback (RLHF) is a standard approach for fine-tuning large language models to follow instructions. As part of this process, learned reward models are used to approximately model human preferences. However, as imperfect representations of the "true" reward, these learned reward models are susceptible to overoptimization. Gao et al. (2023) studied this phenomenon in a synthetic human feedback setup with a significantly larger "gold" reward model acting as the true reward (instead of humans) and showed that overoptimization remains a persistent problem regardless of the size of the proxy reward model and training data used. Using a similar setup, we conduct a systematic study to evaluate the efficacy of using ensemble-based conservative optimization objectives, specifically worst-case optimization (WCO) and uncertainty-weighted optimization (UWO), for mitigating reward model overoptimization when using two optimization methods: (a) best-of-n sampling (BoN) (b) proximal policy optimization (PPO). We additionally extend the setup of Gao et al. (2023) to include 25% label noise to better mirror real-world conditions. Both with and without label noise, we find that conservative optimization practically eliminates overoptimization and improves performance by up to 70% for BoN sampling. For PPO, ensemble-based conservative optimization always reduces overoptimization and outperforms single reward model optimization. Moreover, combining it with a small KL penalty successfully prevents overoptimization at no performance cost. Overall, our results demonstrate that ensemble-based conservative optimization can effectively counter overoptimization.

Although deep learning has produced dazzling successes for applications of image, speech, and video processing in the past few years, most trainings are with suboptimal hyper-parameters, requiring unnecessarily long training times. Setting the hyper-parameters remains a black art that requires years of experience to acquire. This report proposes several efficient ways to set the hyper-parameters that significantly reduce training time and improves performance. Specifically, this report shows how to examine the training validation/test loss function for subtle clues of underfitting and overfitting and suggests guidelines for moving toward the optimal balance point. Then it discusses how to increase/decrease the learning rate/momentum to speed up training. Our experiments show that it is crucial to balance every manner of regularization for each dataset and architecture. Weight decay is used as a sample regularizer to show how its optimal value is tightly coupled with the learning rates and momentums. Files to help replicate the results reported here are available.

The study on improving the robustness of deep neural networks against adversarial examples grows rapidly in recent years. Among them, adversarial training is the most promising one, which flattens the input loss landscape (loss change with respect to input) via training on adversarially perturbed examples. However, how the widely used weight loss landscape (loss change with respect to weight) performs in adversarial training is rarely explored. In this paper, we investigate the weight loss landscape from a new perspective, and identify a clear correlation between the flatness of weight loss landscape and robust generalization gap. Several well-recognized adversarial training improvements, such as early stopping, designing new objective functions, or leveraging unlabeled data, all implicitly flatten the weight loss landscape. Based on these observations, we propose a simple yet effective Adversarial Weight Perturbation (AWP) to explicitly regularize the flatness of weight loss landscape, forming a double-perturbation mechanism in the adversarial training framework that adversarially perturbs both inputs and weights. Extensive experiments demonstrate that AWP indeed brings flatter weight loss landscape and can be easily incorporated into various existing adversarial training methods to further boost their adversarial robustness.

Objective functions that optimize deep neural networks play a vital role in creating an enhanced feature representation of the input data. Although cross-entropy-based loss formulations have been extensively used in a variety of supervised deep-learning applications, these methods tend to be less adequate when there is large intra-class variance and low inter-class variance in input data distribution. Deep Metric Learning seeks to develop methods that aim to measure the similarity between data samples by learning a representation function that maps these data samples into a representative embedding space. It leverages carefully designed sampling strategies and loss functions that aid in optimizing the generation of a discriminative embedding space even for distributions having low inter-class and high intra-class variances. In this chapter, we will provide an overview of recent progress in this area and discuss state-of-the-art Deep Metric Learning approaches.

We study the notion of a generalization bound being uniformly tight, meaning that the difference between the bound and the population loss is small for all learning algorithms and all population distributions. Numerous generalization bounds have been proposed in the literature as potential explanations for the ability of neural networks to generalize in the overparameterized setting. However, in their paper ``Fantastic Generalization Measures and Where to Find Them,'' Jiang et al. (2020) examine more than a dozen generalization bounds, and show empirically that none of them are uniformly tight. This raises the question of whether uniformly-tight generalization bounds are at all possible in the overparameterized setting. We consider two types of generalization bounds: (1) bounds that may depend on the training set and the learned hypothesis (e.g., margin bounds). We prove mathematically that no such bound can be uniformly tight in the overparameterized setting; (2) bounds that may in addition also depend on the learning algorithm (e.g., stability bounds). For these bounds, we show a trade-off between the algorithm's performance and the bound's tightness. Namely, if the algorithm achieves good accuracy on certain distributions, then no generalization bound can be uniformly tight for it in the overparameterized setting. We explain how these formal results can, in our view, inform research on generalization bounds for neural networks, while stressing that other interpretations of these results are also possible.

With the increase in the number of parameters in large language models, the process of pre-training and fine-tuning increasingly demands larger volumes of GPU memory. A significant portion of this memory is typically consumed by the optimizer state. To overcome this challenge, recent approaches such as low-rank adaptation (LoRA (Hu et al., 2021)), low-rank gradient projection (GaLore (Zhao et al., 2024)), and blockwise optimization (BAdam (Luo et al., 2024)) have been proposed. However, in all these algorithms, the effective rank of the weight updates remains low-rank, which can lead to a substantial loss of information from the gradient. This loss can be critically important, especially during the pre-training stage. In this paper, we introduce FRUGAL (Full-Rank Updates with GrAdient spLitting), a new memory-efficient optimization framework. FRUGAL leverages gradient splitting to perform low-dimensional updates using advanced algorithms (such as Adam), while updates along the remaining directions are executed via state-free methods like SGD or signSGD (Bernstein et al., 2018). Our framework can be integrated with various low-rank update selection techniques, including GaLore and BAdam. We provide theoretical convergence guarantees for our framework when using SGDM for low-dimensional updates and SGD for state-free updates. Additionally, our method consistently outperforms concurrent approaches across various fixed memory budgets, achieving state-of-the-art results in pre-training and fine-tuning tasks while balancing memory efficiency and performance metrics.

We deal with the combinatorial problem of learning directed acyclic graph (DAG) structure from observational data adhering to a linear structural equation model (SEM). Leveraging advances in differentiable, nonconvex characterizations of acyclicity, recent efforts have advocated a continuous constrained optimization paradigm to efficiently explore the space of DAGs. Most existing methods employ lasso-type score functions to guide this search, which (i) require expensive penalty parameter retuning when the unknown SEM noise variances change across problem instances; and (ii) implicitly rely on limiting homoscedasticity assumptions. In this work, we propose a new convex score function for sparsity-aware learning of linear DAGs, which incorporates concomitant estimation of scale and thus effectively decouples the sparsity parameter from the exogenous noise levels. Regularization via a smooth, nonconvex acyclicity penalty term yields CoLiDE (Concomitant Linear DAG Estimation), a regression-based criterion amenable to efficient gradient computation and closed-form estimation of noise variances in heteroscedastic scenarios. Our algorithm outperforms state-of-the-art methods without incurring added complexity, especially when the DAGs are larger and the noise level profile is heterogeneous. We also find CoLiDE exhibits enhanced stability manifested via reduced standard deviations in several domain-specific metrics, underscoring the robustness of our novel linear DAG estimator.

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.

In reinforcement learning from human feedback, it is common to optimize against a reward model trained to predict human preferences. Because the reward model is an imperfect proxy, optimizing its value too much can hinder ground truth performance, in accordance with Goodhart's law. This effect has been frequently observed, but not carefully measured due to the expense of collecting human preference data. In this work, we use a synthetic setup in which a fixed "gold-standard" reward model plays the role of humans, providing labels used to train a proxy reward model. We study how the gold reward model score changes as we optimize against the proxy reward model using either reinforcement learning or best-of-n sampling. We find that this relationship follows a different functional form depending on the method of optimization, and that in both cases its coefficients scale smoothly with the number of reward model parameters. We also study the effect on this relationship of the size of the reward model dataset, the number of reward model and policy parameters, and the coefficient of the KL penalty added to the reward in the reinforcement learning setup. We explore the implications of these empirical results for theoretical considerations in AI alignment.

We propose a class of greedy algorithms for weighted sparse recovery by considering new loss function-based generalizations of Orthogonal Matching Pursuit (OMP). Given a (regularized) loss function, the proposed algorithms alternate the iterative construction of the signal support via greedy index selection and a signal update based on solving a local data-fitting problem restricted to the current support. We show that greedy selection rules associated with popular weighted sparsity-promoting loss functions admit explicitly computable and simple formulas. Specifically, we consider ell^0 - and ell^1 -based versions of the weighted LASSO (Least Absolute Shrinkage and Selection Operator), the Square-Root LASSO (SR-LASSO) and the Least Absolute Deviations LASSO (LAD-LASSO). Through numerical experiments on Gaussian compressive sensing and high-dimensional function approximation, we demonstrate the effectiveness of the proposed algorithms and empirically show that they inherit desirable characteristics from the corresponding loss functions, such as SR-LASSO's noise-blind optimal parameter tuning and LAD-LASSO's fault tolerance. In doing so, our study sheds new light on the connection between greedy sparse recovery and convex relaxation.

In multi-task learning (MTL), gradient balancing has recently attracted more research interest than loss balancing since it often leads to better performance. However, loss balancing is much more efficient than gradient balancing, and thus it is still worth further exploration in MTL. Note that prior studies typically ignore that there exist varying improvable gaps across multiple tasks, where the improvable gap per task is defined as the distance between the current training progress and desired final training progress. Therefore, after loss balancing, the performance imbalance still arises in many cases. In this paper, following the loss balancing framework, we propose two novel improvable gap balancing (IGB) algorithms for MTL: one takes a simple heuristic, and the other (for the first time) deploys deep reinforcement learning for MTL. Particularly, instead of directly balancing the losses in MTL, both algorithms choose to dynamically assign task weights for improvable gap balancing. Moreover, we combine IGB and gradient balancing to show the complementarity between the two types of algorithms. Extensive experiments on two benchmark datasets demonstrate that our IGB algorithms lead to the best results in MTL via loss balancing and achieve further improvements when combined with gradient balancing. Code is available at https://github.com/YanqiDai/IGB4MTL.

Over-parameterized deep networks trained using gradient-based optimizers are a popular choice for solving classification and ranking problems. Without appropriately tuned ell_2 regularization or weight decay, such networks have the tendency to make output scores (logits) and network weights large, causing training loss to become too small and the network to lose its adaptivity (ability to move around) in the parameter space. Although regularization is typically understood from an overfitting perspective, we highlight its role in making the network more adaptive and enabling it to escape more easily from weights that generalize poorly. To provide such a capability, we propose a method called Logit Attenuating Weight Normalization (LAWN), that can be stacked onto any gradient-based optimizer. LAWN controls the logits by constraining the weight norms of layers in the final homogeneous sub-network. Empirically, we show that the resulting LAWN variant of the optimizer makes a deep network more adaptive to finding minimas with superior generalization performance on large-scale image classification and recommender systems. While LAWN is particularly impressive in improving Adam, it greatly improves all optimizers when used with large batch sizes

Contrastive learning has been applied successfully to learn vector representations of text. Previous research demonstrated that learning high-quality representations benefits from batch-wise contrastive loss with a large number of negatives. In practice, the technique of in-batch negative is used, where for each example in a batch, other batch examples' positives will be taken as its negatives, avoiding encoding extra negatives. This, however, still conditions each example's loss on all batch examples and requires fitting the entire large batch into GPU memory. This paper introduces a gradient caching technique that decouples backpropagation between contrastive loss and the encoder, removing encoder backward pass data dependency along the batch dimension. As a result, gradients can be computed for one subset of the batch at a time, leading to almost constant memory usage.

Metric learning involves learning a discriminative representation such that embeddings of similar classes are encouraged to be close, while embeddings of dissimilar classes are pushed far apart. State-of-the-art methods focus mostly on sophisticated loss functions or mining strategies. On the one hand, metric learning losses consider two or more examples at a time. On the other hand, modern data augmentation methods for classification consider two or more examples at a time. The combination of the two ideas is under-studied. In this work, we aim to bridge this gap and improve representations using mixup, which is a powerful data augmentation approach interpolating two or more examples and corresponding target labels at a time. This task is challenging because unlike classification, the loss functions used in metric learning are not additive over examples, so the idea of interpolating target labels is not straightforward. To the best of our knowledge, we are the first to investigate mixing both examples and target labels for deep metric learning. We develop a generalized formulation that encompasses existing metric learning loss functions and modify it to accommodate for mixup, introducing Metric Mix, or Metrix. We also introduce a new metric - utilization, to demonstrate that by mixing examples during training, we are exploring areas of the embedding space beyond the training classes, thereby improving representations. To validate the effect of improved representations, we show that mixing inputs, intermediate representations or embeddings along with target labels significantly outperforms state-of-the-art metric learning methods on four benchmark deep metric learning datasets.

The use of deep neural networks in real-world applications require well-calibrated networks with confidence scores that accurately reflect the actual probability. However, it has been found that these networks often provide over-confident predictions, which leads to poor calibration. Recent efforts have sought to address this issue by focal loss to reduce over-confidence, but this approach can also lead to under-confident predictions. While different variants of focal loss have been explored, it is difficult to find a balance between over-confidence and under-confidence. In our work, we propose a new loss function by focusing on dual logits. Our method not only considers the ground truth logit, but also take into account the highest logit ranked after the ground truth logit. By maximizing the gap between these two logits, our proposed dual focal loss can achieve a better balance between over-confidence and under-confidence. We provide theoretical evidence to support our approach and demonstrate its effectiveness through evaluations on multiple models and datasets, where it achieves state-of-the-art performance. Code is available at https://github.com/Linwei94/DualFocalLoss

Denoising Score Matching estimates the score of a noised version of a target distribution by minimizing a regression loss and is widely used to train the popular class of Denoising Diffusion Models. A well known limitation of Denoising Score Matching, however, is that it yields poor estimates of the score at low noise levels. This issue is particularly unfavourable for problems in the physical sciences and for Monte Carlo sampling tasks for which the score of the clean original target is known. Intuitively, estimating the score of a slightly noised version of the target should be a simple task in such cases. In this paper, we address this shortcoming and show that it is indeed possible to leverage knowledge of the target score. We present a Target Score Identity and corresponding Target Score Matching regression loss which allows us to obtain score estimates admitting favourable properties at low noise levels.

Recent research shows that when Gradient Descent (GD) is applied to neural networks, the loss almost never decreases monotonically. Instead, the loss oscillates as gradient descent converges to its ''Edge of Stability'' (EoS). Here, we find a quantity that does decrease monotonically throughout GD training: the sharpness attained by the gradient flow solution (GFS)-the solution that would be obtained if, from now until convergence, we train with an infinitesimal step size. Theoretically, we analyze scalar neural networks with the squared loss, perhaps the simplest setting where the EoS phenomena still occur. In this model, we prove that the GFS sharpness decreases monotonically. Using this result, we characterize settings where GD provably converges to the EoS in scalar networks. Empirically, we show that GD monotonically decreases the GFS sharpness in a squared regression model as well as practical neural network architectures.

Process Reward Modeling (PRM) is critical for complex reasoning and decision-making tasks where the accuracy of intermediate steps significantly influences the overall outcome. Existing PRM approaches, primarily framed as classification problems, employ cross-entropy loss to independently evaluate each step's correctness. This method can lead to suboptimal reward distribution and does not adequately address the interdependencies among steps. To address these limitations, we introduce the Process Q-value Model (PQM), a novel framework that redefines PRM in the context of a Markov Decision Process. PQM optimizes Q-value rankings based on a novel comparative loss function, enhancing the model's ability to capture the intricate dynamics among sequential decisions. This approach provides a more granular and theoretically grounded methodology for process rewards. Our extensive empirical evaluations across various sampling policies, language model backbones, and multi-step reasoning benchmarks show that PQM outperforms classification-based PRMs. The effectiveness of the comparative loss function is highlighted in our comprehensive ablation studies, confirming PQM's practical efficacy and theoretical advantage.

Reinforcement learning from human feedback (RLHF) is one of the key techniques that helps large language models (LLMs) to follow instructions and provide helpful and harmless responses. While direct policy optimization methods exist, state-of-the-art LLMs adopt RL-based methods (usually PPO) in RLHF to train the policy to generate good responses guided by a reward model learned from preference data. The main challenge of these methods is the inaccuracy of the intermediate reward model, especially in code generation tasks that require long and complex reasoning to score a response. We find that the reliability of the reward model varies across responses assigned with different rewards. This motivates us to filter the samples whose rewards may be unreliable to improve signal-to-noise ratio during policy learning, resulting in Policy Filtration for Proximal Policy Optimization (PF-PPO). To choose a proper policy filtration strategy for a given reward model, the coefficient of determination (R^2) between rewards and actual scores on filtered samples serves as a good metrics and helps us find several promising strategies. We provide extensive experiments to validate the effectiveness of PF-PPO in code generation tasks, and find that some variants of PF-PPO are highly effective and achieve new state-of-the-art performance across 7-billion-parameter models on HumanEval, MBPP, and a new and more challenging LeetCode Contest benchmark.

Existing Score-Based Models (SBMs) can be categorized into constrained SBMs (CSBMs) or unconstrained SBMs (USBMs) according to their parameterization approaches. CSBMs model probability density functions as Boltzmann distributions, and assign their predictions as the negative gradients of some scalar-valued energy functions. On the other hand, USBMs employ flexible architectures capable of directly estimating scores without the need to explicitly model energy functions. In this paper, we demonstrate that the architectural constraints of CSBMs may limit their modeling ability. In addition, we show that USBMs' inability to preserve the property of conservativeness may lead to degraded performance in practice. To address the above issues, we propose Quasi-Conservative Score-Based Models (QCSBMs) for keeping the advantages of both CSBMs and USBMs. Our theoretical derivations demonstrate that the training objective of QCSBMs can be efficiently integrated into the training processes by leveraging the Hutchinson's trace estimator. In addition, our experimental results on the CIFAR-10, CIFAR-100, ImageNet, and SVHN datasets validate the effectiveness of QCSBMs. Finally, we justify the advantage of QCSBMs using an example of a one-layered autoencoder.

In a typical multi-label setting, a picture contains on average few positive labels, and many negative ones. This positive-negative imbalance dominates the optimization process, and can lead to under-emphasizing gradients from positive labels during training, resulting in poor accuracy. In this paper, we introduce a novel asymmetric loss ("ASL"), which operates differently on positive and negative samples. The loss enables to dynamically down-weights and hard-thresholds easy negative samples, while also discarding possibly mislabeled samples. We demonstrate how ASL can balance the probabilities of different samples, and how this balancing is translated to better mAP scores. With ASL, we reach state-of-the-art results on multiple popular multi-label datasets: MS-COCO, Pascal-VOC, NUS-WIDE and Open Images. We also demonstrate ASL applicability for other tasks, such as single-label classification and object detection. ASL is effective, easy to implement, and does not increase the training time or complexity. Implementation is available at: https://github.com/Alibaba-MIIL/ASL.

Current model quantization methods have shown their promising capability in reducing storage space and computation complexity. However, due to the diversity of quantization forms supported by different hardware, one limitation of existing solutions is that usually require repeated optimization for different scenarios. How to construct a model with flexible quantization forms has been less studied. In this paper, we explore a one-shot network quantization regime, named Elastic Quantization Neural Networks (EQ-Net), which aims to train a robust weight-sharing quantization supernet. First of all, we propose an elastic quantization space (including elastic bit-width, granularity, and symmetry) to adapt to various mainstream quantitative forms. Secondly, we propose the Weight Distribution Regularization Loss (WDR-Loss) and Group Progressive Guidance Loss (GPG-Loss) to bridge the inconsistency of the distribution for weights and output logits in the elastic quantization space gap. Lastly, we incorporate genetic algorithms and the proposed Conditional Quantization-Aware Accuracy Predictor (CQAP) as an estimator to quickly search mixed-precision quantized neural networks in supernet. Extensive experiments demonstrate that our EQ-Net is close to or even better than its static counterparts as well as state-of-the-art robust bit-width methods. Code can be available at https://github.com/xuke225/EQ-Net.git{https://github.com/xuke225/EQ-Net}.

Knowledge distillation transfers knowledge from a large model to a small one via task and distillation losses. In this paper, we observe a trade-off between task and distillation losses, i.e., introducing distillation loss limits the convergence of task loss. We believe that the trade-off results from the insufficient optimization of distillation loss. The reason is: The teacher has a lower task loss than the student, and a lower distillation loss drives the student more similar to the teacher, then a better-converged task loss could be obtained. To break the trade-off, we propose the Distillation-Oriented Trainer (DOT). DOT separately considers gradients of task and distillation losses, then applies a larger momentum to distillation loss to accelerate its optimization. We empirically prove that DOT breaks the trade-off, i.e., both losses are sufficiently optimized. Extensive experiments validate the superiority of DOT. Notably, DOT achieves a +2.59% accuracy improvement on ImageNet-1k for the ResNet50-MobileNetV1 pair. Conclusively, DOT greatly benefits the student's optimization properties in terms of loss convergence and model generalization. Code will be made publicly available.

Text generation models are notoriously vulnerable to errors in the training data. With the wide-spread availability of massive amounts of web-crawled data becoming more commonplace, how can we enhance the robustness of models trained on a massive amount of noisy web-crawled text? In our work, we propose Error Norm Truncation (ENT), a robust enhancement method to the standard training objective that truncates noisy data. Compared to methods that only uses the negative log-likelihood loss to estimate data quality, our method provides a more accurate estimation by considering the distribution of non-target tokens, which is often overlooked by previous work. Through comprehensive experiments across language modeling, machine translation, and text summarization, we show that equipping text generation models with ENT improves generation quality over standard training and previous soft and hard truncation methods. Furthermore, we show that our method improves the robustness of models against two of the most detrimental types of noise in machine translation, resulting in an increase of more than 2 BLEU points over the MLE baseline when up to 50% of noise is added to the data.

Recently, there has been extensive research interest in training deep networks to denoise images without clean reference. However, the representative approaches such as Noise2Noise, Noise2Void, Stein's unbiased risk estimator (SURE), etc. seem to differ from one another and it is difficult to find the coherent mathematical structure. To address this, here we present a novel approach, called Noise2Score, which reveals a missing link in order to unite these seemingly different approaches. Specifically, we show that image denoising problems without clean images can be addressed by finding the mode of the posterior distribution and that the Tweedie's formula offers an explicit solution through the score function (i.e. the gradient of log likelihood). Our method then uses the recent finding that the score function can be stably estimated from the noisy images using the amortized residual denoising autoencoder, the method of which is closely related to Noise2Noise or Nose2Void. Our Noise2Score approach is so universal that the same network training can be used to remove noises from images that are corrupted by any exponential family distributions and noise parameters. Using extensive experiments with Gaussian, Poisson, and Gamma noises, we show that Noise2Score significantly outperforms the state-of-the-art self-supervised denoising methods in the benchmark data set such as (C)BSD68, Set12, and Kodak, etc.

Our motivation is to shed light the performance of the widely popular "R-Learner." Like many other methods for estimating conditional average treatment effects (CATEs), R-Learning can be expressed as a weighted pseudo-outcome regression (POR). Previous comparisons of POR techniques have paid careful attention to the choice of pseudo-outcome transformation. However, we argue that the dominant driver of performance is actually the choice of weights. Specifically, we argue that R-Learning implicitly performs an inverse-variance weighted form of POR. These weights stabilize the regression and allow for convenient simplifications of bias terms.

Heavy-tail phenomena in stochastic gradient descent (SGD) have been reported in several empirical studies. Experimental evidence in previous works suggests a strong interplay between the heaviness of the tails and generalization behavior of SGD. To address this empirical phenomena theoretically, several works have made strong topological and statistical assumptions to link the generalization error to heavy tails. Very recently, new generalization bounds have been proven, indicating a non-monotonic relationship between the generalization error and heavy tails, which is more pertinent to the reported empirical observations. While these bounds do not require additional topological assumptions given that SGD can be modeled using a heavy-tailed stochastic differential equation (SDE), they can only apply to simple quadratic problems. In this paper, we build on this line of research and develop generalization bounds for a more general class of objective functions, which includes non-convex functions as well. Our approach is based on developing Wasserstein stability bounds for heavy-tailed SDEs and their discretizations, which we then convert to generalization bounds. Our results do not require any nontrivial assumptions; yet, they shed more light to the empirical observations, thanks to the generality of the loss functions.

Uncertainty quantification has received increasing attention in machine learning in the recent past. In particular, a distinction between aleatoric and epistemic uncertainty has been found useful in this regard. The latter refers to the learner's (lack of) knowledge and appears to be especially difficult to measure and quantify. In this paper, we analyse a recent proposal based on the idea of a second-order learner, which yields predictions in the form of distributions over probability distributions. While standard (first-order) learners can be trained to predict accurate probabilities, namely by minimising suitable loss functions on sample data, we show that loss minimisation does not work for second-order predictors: The loss functions proposed for inducing such predictors do not incentivise the learner to represent its epistemic uncertainty in a faithful way.

Recent work has shown that it is possible to train deep neural networks that are provably robust to norm-bounded adversarial perturbations. Most of these methods are based on minimizing an upper bound on the worst-case loss over all possible adversarial perturbations. While these techniques show promise, they often result in difficult optimization procedures that remain hard to scale to larger networks. Through a comprehensive analysis, we show how a simple bounding technique, interval bound propagation (IBP), can be exploited to train large provably robust neural networks that beat the state-of-the-art in verified accuracy. While the upper bound computed by IBP can be quite weak for general networks, we demonstrate that an appropriate loss and clever hyper-parameter schedule allow the network to adapt such that the IBP bound is tight. This results in a fast and stable learning algorithm that outperforms more sophisticated methods and achieves state-of-the-art results on MNIST, CIFAR-10 and SVHN. It also allows us to train the largest model to be verified beyond vacuous bounds on a downscaled version of ImageNet.

Mixture of experts (MoE) are a popular class of statistical and machine learning models that have gained attention over the years due to their flexibility and efficiency. In this work, we consider Gaussian-gated localized MoE (GLoME) and block-diagonal covariance localized MoE (BLoME) regression models to present nonlinear relationships in heterogeneous data with potential hidden graph-structured interactions between high-dimensional predictors. These models pose difficult statistical estimation and model selection questions, both from a computational and theoretical perspective. This paper is devoted to the study of the problem of model selection among a collection of GLoME or BLoME models characterized by the number of mixture components, the complexity of Gaussian mean experts, and the hidden block-diagonal structures of the covariance matrices, in a penalized maximum likelihood estimation framework. In particular, we establish non-asymptotic risk bounds that take the form of weak oracle inequalities, provided that lower bounds for the penalties hold. The good empirical behavior of our models is then demonstrated on synthetic and real datasets.

Many optimization problems are inherently multi-objective. To address them, we formalize Jacobian descent (JD), a direct generalization of gradient descent for vector-valued functions. Each step of this algorithm relies on a Jacobian matrix consisting of one gradient per objective. The aggregator, responsible for reducing this matrix into an update vector, characterizes JD. While the multi-task learning literature already contains a variety of aggregators, they often lack some natural properties. In particular, the update should not conflict with any objective and should scale proportionally to the norm of each gradient. We propose a new aggregator specifically designed to satisfy this. Emphasizing conflict between objectives, we then highlight direct applications for our methods. Most notably, we introduce instance-wise risk minimization (IWRM), a learning paradigm in which the loss of each training example is considered a separate objective. On simple image classification tasks, IWRM exhibits promising results compared to the direct minimization of the average loss. The performance of our aggregator in those experiments also corroborates our theoretical findings. Lastly, as speed is the main limitation of JD, we provide a path towards a more efficient implementation.

We study the data selection problem, whose aim is to select a small representative subset of data that can be used to efficiently train a machine learning model. We present a new data selection approach based on k-means clustering and sensitivity sampling. Assuming access to an embedding representation of the data with respect to which the model loss is H\"older continuous, our approach provably allows selecting a set of ``typical'' k + 1/varepsilon^2 elements whose average loss corresponds to the average loss of the whole dataset, up to a multiplicative (1pmvarepsilon) factor and an additive varepsilon lambda Phi_k, where Phi_k represents the k-means cost for the input embeddings and lambda is the H\"older constant. We furthermore demonstrate the performance and scalability of our approach on fine-tuning foundation models and show that it outperforms state-of-the-art methods. We also show how it can be applied on linear regression, leading to a new sampling strategy that surprisingly matches the performances of leverage score sampling, while being conceptually simpler and more scalable.

Loss spikes often occur during pre-training of large language models. The spikes degrade the performance of large language models and sometimes ruin the pre-training. Since the pre-training needs a vast computational budget, we should avoid such spikes. To investigate the cause of loss spikes, we focus on gradients of internal layers. Through theoretical analyses, we reveal two causes of the exploding gradients, and provide requirements to prevent the explosion. In addition, we propose a method to satisfy the requirements by combining the initialization method and a simple modification to embeddings. We conduct various experiments to verify our theoretical analyses empirically. Experimental results indicate that the combination is effective in preventing spikes during pre-training.

We consider online learning in multi-player smooth monotone games. Existing algorithms have limitations such as (1) being only applicable to strongly monotone games; (2) lacking the no-regret guarantee; (3) having only asymptotic or slow O(1{T}) last-iterate convergence rate to a Nash equilibrium. While the O(1{T}) rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms. We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no-regret learning algorithm for smooth monotone games. Namely, our algorithm achieves both (i) the optimal O(T) regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal O(1{T}) last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games. As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an O(log T) individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art O(T) bound.

In this paper, we focus on the task of conditional image generation, where an image is synthesized according to user instructions. The critical challenge underpinning this task is ensuring both the fidelity of the generated images and their semantic alignment with the provided conditions. To tackle this issue, previous studies have employed supervised perceptual losses derived from pre-trained models, i.e., reward models, to enforce alignment between the condition and the generated result. However, we observe one inherent shortcoming: considering the diversity of synthesized images, the reward model usually provides inaccurate feedback when encountering newly generated data, which can undermine the training process. To address this limitation, we propose an uncertainty-aware reward modeling, called Ctrl-U, including uncertainty estimation and uncertainty-aware regularization, designed to reduce the adverse effects of imprecise feedback from the reward model. Given the inherent cognitive uncertainty within reward models, even images generated under identical conditions often result in a relatively large discrepancy in reward loss. Inspired by the observation, we explicitly leverage such prediction variance as an uncertainty indicator. Based on the uncertainty estimation, we regularize the model training by adaptively rectifying the reward. In particular, rewards with lower uncertainty receive higher loss weights, while those with higher uncertainty are given reduced weights to allow for larger variability. The proposed uncertainty regularization facilitates reward fine-tuning through consistency construction. Extensive experiments validate the effectiveness of our methodology in improving the controllability and generation quality, as well as its scalability across diverse conditional scenarios. Code will soon be available at https://grenoble-zhang.github.io/Ctrl-U-Page/.

Learning from preference-based feedback has recently gained traction as a promising approach to align language models with human interests. While these aligned generative models have demonstrated impressive capabilities across various tasks, their dependence on high-quality human preference data poses a bottleneck in practical applications. Specifically, noisy (incorrect and ambiguous) preference pairs in the dataset might restrict the language models from capturing human intent accurately. While practitioners have recently proposed heuristics to mitigate the effect of noisy preferences, a complete theoretical understanding of their workings remain elusive. In this work, we aim to bridge this gap by by introducing a general framework for policy optimization in the presence of random preference flips. We focus on the direct preference optimization (DPO) algorithm in particular since it assumes that preferences adhere to the Bradley-Terry-Luce (BTL) model, raising concerns about the impact of noisy data on the learned policy. We design a novel loss function, which de-bias the effect of noise on average, making a policy trained by minimizing that loss robust to the noise. Under log-linear parameterization of the policy class and assuming good feature coverage of the SFT policy, we prove that the sub-optimality gap of the proposed robust DPO (rDPO) policy compared to the optimal policy is of the order O(1{1-2epsilon}frac{d{n}}), where epsilon < 1/2 is flip rate of labels, d is policy parameter dimension and n is size of dataset. Our experiments on IMDb sentiment generation and Anthropic's helpful-harmless dataset show that rDPO is robust to noise in preference labels compared to vanilla DPO and other heuristics proposed by practitioners.

Cinematic audio source separation is a relatively new subtask of audio source separation, with the aim of extracting the dialogue, music, and effects stems from their mixture. In this work, we developed a model generalizing the Bandsplit RNN for any complete or overcomplete partitions of the frequency axis. Psychoacoustically motivated frequency scales were used to inform the band definitions which are now defined with redundancy for more reliable feature extraction. A loss function motivated by the signal-to-noise ratio and the sparsity-promoting property of the 1-norm was proposed. We additionally exploit the information-sharing property of a common-encoder setup to reduce computational complexity during both training and inference, improve separation performance for hard-to-generalize classes of sounds, and allow flexibility during inference time with detachable decoders. Our best model sets the state of the art on the Divide and Remaster dataset with performance above the ideal ratio mask for the dialogue stem.

When quantizing neural networks, assigning each floating-point weight to its nearest fixed-point value is the predominant approach. We find that, perhaps surprisingly, this is not the best we can do. In this paper, we propose AdaRound, a better weight-rounding mechanism for post-training quantization that adapts to the data and the task loss. AdaRound is fast, does not require fine-tuning of the network, and only uses a small amount of unlabelled data. We start by theoretically analyzing the rounding problem for a pre-trained neural network. By approximating the task loss with a Taylor series expansion, the rounding task is posed as a quadratic unconstrained binary optimization problem. We simplify this to a layer-wise local loss and propose to optimize this loss with a soft relaxation. AdaRound not only outperforms rounding-to-nearest by a significant margin but also establishes a new state-of-the-art for post-training quantization on several networks and tasks. Without fine-tuning, we can quantize the weights of Resnet18 and Resnet50 to 4 bits while staying within an accuracy loss of 1%.

We develop a variant of the stochastic prox-linear method for minimizing the Conditional Value-at-Risk (CVaR) objective. CVaR is a risk measure focused on minimizing worst-case performance, defined as the average of the top quantile of the losses. In machine learning, such a risk measure is useful to train more robust models. Although the stochastic subgradient method (SGM) is a natural choice for minimizing the CVaR objective, we show that our stochastic prox-linear (SPL+) algorithm can better exploit the structure of the objective, while still providing a convenient closed form update. Our SPL+ method also adapts to the scaling of the loss function, which allows for easier tuning. We then specialize a general convergence theorem for SPL+ to our setting, and show that it allows for a wider selection of step sizes compared to SGM. We support this theoretical finding experimentally.

We introduce a novel neural network-based algorithm to compute optimal transport (OT) plans for general cost functionals. In contrast to common Euclidean costs, i.e., ell^1 or ell^2, such functionals provide more flexibility and allow using auxiliary information, such as class labels, to construct the required transport map. Existing methods for general costs are discrete and have limitations in practice, i.e. they do not provide an out-of-sample estimation. We address the challenge of designing a continuous OT approach for general costs that generalizes to new data points in high-dimensional spaces, such as images. Additionally, we provide the theoretical error analysis for our recovered transport plans. As an application, we construct a cost functional to map data distributions while preserving the class-wise structure.

One prominent approach toward resolving the adversarial vulnerability of deep neural networks is the two-player zero-sum paradigm of adversarial training, in which predictors are trained against adversarially chosen perturbations of data. Despite the promise of this approach, algorithms based on this paradigm have not engendered sufficient levels of robustness and suffer from pathological behavior like robust overfitting. To understand this shortcoming, we first show that the commonly used surrogate-based relaxation used in adversarial training algorithms voids all guarantees on the robustness of trained classifiers. The identification of this pitfall informs a novel non-zero-sum bilevel formulation of adversarial training, wherein each player optimizes a different objective function. Our formulation yields a simple algorithmic framework that matches and in some cases outperforms state-of-the-art attacks, attains comparable levels of robustness to standard adversarial training algorithms, and does not suffer from robust overfitting.

Autoencoders are a popular model in many branches of machine learning and lossy data compression. However, their fundamental limits, the performance of gradient methods and the features learnt during optimization remain poorly understood, even in the two-layer setting. In fact, earlier work has considered either linear autoencoders or specific training regimes (leading to vanishing or diverging compression rates). Our paper addresses this gap by focusing on non-linear two-layer autoencoders trained in the challenging proportional regime in which the input dimension scales linearly with the size of the representation. Our results characterize the minimizers of the population risk, and show that such minimizers are achieved by gradient methods; their structure is also unveiled, thus leading to a concise description of the features obtained via training. For the special case of a sign activation function, our analysis establishes the fundamental limits for the lossy compression of Gaussian sources via (shallow) autoencoders. Finally, while the results are proved for Gaussian data, numerical simulations on standard datasets display the universality of the theoretical predictions.

Contrastive learning has become an important tool in learning representations from unlabeled data mainly relying on the idea of minimizing distance between positive data pairs, e.g., views from the same images, and maximizing distance between negative data pairs, e.g., views from different images. This paper proposes a new variation of the contrastive learning objective, Group Ordering Constraints (GroCo), that leverages the idea of sorting the distances of positive and negative pairs and computing the respective loss based on how many positive pairs have a larger distance than the negative pairs, and thus are not ordered correctly. To this end, the GroCo loss is based on differentiable sorting networks, which enable training with sorting supervision by matching a differentiable permutation matrix, which is produced by sorting a given set of scores, to a respective ground truth permutation matrix. Applying this idea to groupwise pre-ordered inputs of multiple positive and negative pairs allows introducing the GroCo loss with implicit emphasis on strong positives and negatives, leading to better optimization of the local neighborhood. We evaluate the proposed formulation on various self-supervised learning benchmarks and show that it not only leads to improved results compared to vanilla contrastive learning but also shows competitive performance to comparable methods in linear probing and outperforms current methods in k-NN performance.

Off-policy learning (OPL) aims at finding improved policies from logged bandit data, often by minimizing the inverse propensity scoring (IPS) estimator of the risk. In this work, we investigate a smooth regularization for IPS, for which we derive a two-sided PAC-Bayes generalization bound. The bound is tractable, scalable, interpretable and provides learning certificates. In particular, it is also valid for standard IPS without making the assumption that the importance weights are bounded. We demonstrate the relevance of our approach and its favorable performance through a set of learning tasks. Since our bound holds for standard IPS, we are able to provide insight into when regularizing IPS is useful. Namely, we identify cases where regularization might not be needed. This goes against the belief that, in practice, clipped IPS often enjoys favorable performance than standard IPS in OPL.

Offline reinforcement learning faces a significant challenge of value over-estimation due to the distributional drift between the dataset and the current learned policy, leading to learning failure in practice. The common approach is to incorporate a penalty term to reward or value estimation in the Bellman iterations. Meanwhile, to avoid extrapolation on out-of-distribution (OOD) states and actions, existing methods focus on conservative Q-function estimation. In this paper, we propose Conservative State Value Estimation (CSVE), a new approach that learns conservative V-function via directly imposing penalty on OOD states. Compared to prior work, CSVE allows more effective in-data policy optimization with conservative value guarantees. Further, we apply CSVE and develop a practical actor-critic algorithm in which the critic does the conservative value estimation by additionally sampling and penalizing the states around the dataset, and the actor applies advantage weighted updates extended with state exploration to improve the policy. We evaluate in classic continual control tasks of D4RL, showing that our method performs better than the conservative Q-function learning methods and is strongly competitive among recent SOTA methods.

Adversarial robustness is a research area that has recently received a lot of attention in the quest for trustworthy artificial intelligence. However, recent works on adversarial robustness have focused on supervised learning where it is assumed that labeled data is plentiful. In this paper, we investigate semi-supervised adversarial training where labeled data is scarce. We derive two upper bounds for the robust risk and propose a regularization term for unlabeled data motivated by these two upper bounds. Then, we develop a semi-supervised adversarial training algorithm that combines the proposed regularization term with knowledge distillation using a semi-supervised teacher (i.e., a teacher model trained using a semi-supervised learning algorithm). Our experiments show that our proposed algorithm achieves state-of-the-art performance with significant margins compared to existing algorithms. In particular, compared to supervised learning algorithms, performance of our proposed algorithm is not much worse even when the amount of labeled data is very small. For example, our algorithm with only 8\% labeled data is comparable to supervised adversarial training algorithms that use all labeled data, both in terms of standard and robust accuracies on CIFAR-10.

In this paper, we propose a new loss function called generalized end-to-end (GE2E) loss, which makes the training of speaker verification models more efficient than our previous tuple-based end-to-end (TE2E) loss function. Unlike TE2E, the GE2E loss function updates the network in a way that emphasizes examples that are difficult to verify at each step of the training process. Additionally, the GE2E loss does not require an initial stage of example selection. With these properties, our model with the new loss function decreases speaker verification EER by more than 10%, while reducing the training time by 60% at the same time. We also introduce the MultiReader technique, which allows us to do domain adaptation - training a more accurate model that supports multiple keywords (i.e. "OK Google" and "Hey Google") as well as multiple dialects.

We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for sigma-sub-Gaussian label noise, our analysis provides a regret upper bound of O(sigma^2 d log T) + o(log T), where d is the dimension of the input vector, T is the total number of rounds. We also prove a Omega(sigma^2dlog(T/d)) lower bound for stochastic online linear regression, which indicates that our upper bound is nearly optimal. In addition, we extend our analysis to a more refined Bernstein noise condition. As an application, we study generalized linear bandits with heteroscedastic noise and propose an algorithm based on FTRL to achieve the first variance-aware regret bound.

Much research in machine learning involves finding appropriate inductive biases (e.g. convolutional neural networks, momentum-based optimizers, transformers) to promote generalization on tasks. However, quantification of the amount of inductive bias associated with these architectures and hyperparameters has been limited. We propose a novel method for efficiently computing the inductive bias required for generalization on a task with a fixed training data budget; formally, this corresponds to the amount of information required to specify well-generalizing models within a specific hypothesis space of models. Our approach involves modeling the loss distribution of random hypotheses drawn from a hypothesis space to estimate the required inductive bias for a task relative to these hypotheses. Unlike prior work, our method provides a direct estimate of inductive bias without using bounds and is applicable to diverse hypothesis spaces. Moreover, we derive approximation error bounds for our estimation approach in terms of the number of sampled hypotheses. Consistent with prior results, our empirical results demonstrate that higher dimensional tasks require greater inductive bias. We show that relative to other expressive model classes, neural networks as a model class encode large amounts of inductive bias. Furthermore, our measure quantifies the relative difference in inductive bias between different neural network architectures. Our proposed inductive bias metric provides an information-theoretic interpretation of the benefits of specific model architectures for certain tasks and provides a quantitative guide to developing tasks requiring greater inductive bias, thereby encouraging the development of more powerful inductive biases.

Simple regret is a natural and parameter-free performance criterion for pure exploration in multi-armed bandits yet is less popular than the probability of missing the best arm or an epsilon-good arm, perhaps due to lack of easy ways to characterize it. In this paper, we make significant progress on minimizing simple regret in both data-rich (Tge n) and data-poor regime (T le n) where n is the number of arms, and T is the number of samples. At its heart is our improved instance-dependent analysis of the well-known Sequential Halving (SH) algorithm, where we bound the probability of returning an arm whose mean reward is not within epsilon from the best (i.e., not epsilon-good) for any choice of epsilon>0, although epsilon is not an input to SH. Our bound not only leads to an optimal worst-case simple regret bound of n/T up to logarithmic factors but also essentially matches the instance-dependent lower bound for returning an epsilon-good arm reported by Katz-Samuels and Jamieson (2020). For the more challenging data-poor regime, we propose Bracketing SH (BSH) that enjoys the same improvement even without sampling each arm at least once. Our empirical study shows that BSH outperforms existing methods on real-world tasks.

We address the problem of network calibration adjusting miscalibrated confidences of deep neural networks. Many approaches to network calibration adopt a regularization-based method that exploits a regularization term to smooth the miscalibrated confidences. Although these approaches have shown the effectiveness on calibrating the networks, there is still a lack of understanding on the underlying principles of regularization in terms of network calibration. We present in this paper an in-depth analysis of existing regularization-based methods, providing a better understanding on how they affect to network calibration. Specifically, we have observed that 1) the regularization-based methods can be interpreted as variants of label smoothing, and 2) they do not always behave desirably. Based on the analysis, we introduce a novel loss function, dubbed ACLS, that unifies the merits of existing regularization methods, while avoiding the limitations. We show extensive experimental results for image classification and semantic segmentation on standard benchmarks, including CIFAR10, Tiny-ImageNet, ImageNet, and PASCAL VOC, demonstrating the effectiveness of our loss function.

Model calibration aims to align confidence with prediction correctness. The Cross-Entropy (CE) loss is widely used for calibrator training, which enforces the model to increase confidence on the ground truth class. However, we find the CE loss has intrinsic limitations. For example, for a narrow misclassification, a calibrator trained by the CE loss often produces high confidence on the wrongly predicted class (e.g., a test sample is wrongly classified and its softmax score on the ground truth class is around 0.4), which is undesirable. In this paper, we propose a new post-hoc calibration objective derived from the aim of calibration. Intuitively, the proposed objective function asks that the calibrator decrease model confidence on wrongly predicted samples and increase confidence on correctly predicted samples. Because a sample itself has insufficient ability to indicate correctness, we use its transformed versions (e.g., rotated, greyscaled and color-jittered) during calibrator training. Trained on an in-distribution validation set and tested with isolated, individual test samples, our method achieves competitive calibration performance on both in-distribution and out-of-distribution test sets compared with the state of the art. Further, our analysis points out the difference between our method and commonly used objectives such as CE loss and mean square error loss, where the latters sometimes deviates from the calibration aim.

Pretrained language models are commonly aligned with human preferences and downstream tasks via reinforcement finetuning (RFT), which entails maximizing a (possibly learned) reward function using policy gradient algorithms. This work highlights a fundamental optimization obstacle in RFT: we prove that the expected gradient for an input vanishes when its reward standard deviation under the model is small, even if the expected reward is far from optimal. Through experiments on an RFT benchmark and controlled environments, as well as a theoretical analysis, we then demonstrate that vanishing gradients due to small reward standard deviation are prevalent and detrimental, leading to extremely slow reward maximization. Lastly, we explore ways to overcome vanishing gradients in RFT. We find the common practice of an initial supervised finetuning (SFT) phase to be the most promising candidate, which sheds light on its importance in an RFT pipeline. Moreover, we show that a relatively small number of SFT optimization steps on as few as 1% of the input samples can suffice, indicating that the initial SFT phase need not be expensive in terms of compute and data labeling efforts. Overall, our results emphasize that being mindful for inputs whose expected gradient vanishes, as measured by the reward standard deviation, is crucial for successful execution of RFT.

We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in the paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. A direct computational implication of our results is that by solving either the barycenter problem and its dual, or the MOT problem and its dual, we can recover the optimal robust classification rule and the optimal adversarial strategy for the original adversarial problem. Examples with synthetic and real data illustrate our results.

Large language models can memorize and repeat their training data, causing privacy and copyright risks. To mitigate memorization, we introduce a subtle modification to the next-token training objective that we call the goldfish loss. During training, a randomly sampled subset of tokens are excluded from the loss computation. These dropped tokens are not memorized by the model, which prevents verbatim reproduction of a complete chain of tokens from the training set. We run extensive experiments training billion-scale Llama-2 models, both pre-trained and trained from scratch, and demonstrate significant reductions in extractable memorization with little to no impact on downstream benchmarks.

We examine the problem of regret minimization when the learner is involved in a continuous game with other optimizing agents: in this case, if all players follow a no-regret algorithm, it is possible to achieve significantly lower regret relative to fully adversarial environments. We study this problem in the context of variationally stable games (a class of continuous games which includes all convex-concave and monotone games), and when the players only have access to noisy estimates of their individual payoff gradients. If the noise is additive, the game-theoretic and purely adversarial settings enjoy similar regret guarantees; however, if the noise is multiplicative, we show that the learners can, in fact, achieve constant regret. We achieve this faster rate via an optimistic gradient scheme with learning rate separation -- that is, the method's extrapolation and update steps are tuned to different schedules, depending on the noise profile. Subsequently, to eliminate the need for delicate hyperparameter tuning, we propose a fully adaptive method that attains nearly the same guarantees as its non-adapted counterpart, while operating without knowledge of either the game or of the noise profile.

Learning to Optimize (L2O), a technique that utilizes machine learning to learn an optimization algorithm automatically from data, has gained arising attention in recent years. A generic L2O approach parameterizes the iterative update rule and learns the update direction as a black-box network. While the generic approach is widely applicable, the learned model can overfit and may not generalize well to out-of-distribution test sets. In this paper, we derive the basic mathematical conditions that successful update rules commonly satisfy. Consequently, we propose a novel L2O model with a mathematics-inspired structure that is broadly applicable and generalized well to out-of-distribution problems. Numerical simulations validate our theoretical findings and demonstrate the superior empirical performance of the proposed L2O model.

Assigning importance weights to adversarial data has achieved great success in training adversarially robust networks under limited model capacity. However, existing instance-reweighted adversarial training (AT) methods heavily depend on heuristics and/or geometric interpretations to determine those importance weights, making these algorithms lack rigorous theoretical justification/guarantee. Moreover, recent research has shown that adversarial training suffers from a severe non-uniform robust performance across the training distribution, e.g., data points belonging to some classes can be much more vulnerable to adversarial attacks than others. To address both issues, in this paper, we propose a novel doubly-robust instance reweighted AT framework, which allows to obtain the importance weights via exploring distributionally robust optimization (DRO) techniques, and at the same time boosts the robustness on the most vulnerable examples. In particular, our importance weights are obtained by optimizing the KL-divergence regularized loss function, which allows us to devise new algorithms with a theoretical convergence guarantee. Experiments on standard classification datasets demonstrate that our proposed approach outperforms related state-of-the-art baseline methods in terms of average robust performance, and at the same time improves the robustness against attacks on the weakest data points. Codes will be available soon.

Learning rate schedules used in practice bear little resemblance to those recommended by theory. We close much of this theory/practice gap, and as a consequence are able to derive new problem-adaptive learning rate schedules. Our key technical contribution is a refined analysis of learning rate schedules for a wide class of optimization algorithms (including SGD). In contrast to most prior works that study the convergence of the average iterate, we study the last iterate, which is what most people use in practice. When considering only worst-case analysis, our theory predicts that the best choice is the linear decay schedule: a popular choice in practice that sets the stepsize proportionally to 1 - t/T, where t is the current iteration and T is the total number of steps. To go beyond this worst-case analysis, we use the observed gradient norms to derive schedules refined for any particular task. These refined schedules exhibit learning rate warm-up and rapid learning rate annealing near the end of training. Ours is the first systematic approach to automatically yield both of these properties. We perform the most comprehensive evaluation of learning rate schedules to date, evaluating across 10 diverse deep learning problems, a series of LLMs, and a suite of logistic regression problems. We validate that overall, the linear-decay schedule matches or outperforms all commonly used default schedules including cosine annealing, and that our schedule refinement method gives further improvements.

Grokking, the sudden generalization that occurs after prolonged overfitting, is a surprising phenomenon challenging our understanding of deep learning. Although significant progress has been made in understanding grokking, the reasons behind the delayed generalization and its dependence on regularization remain unclear. In this work, we argue that without regularization, grokking tasks push models to the edge of numerical stability, introducing floating point errors in the Softmax function, which we refer to as Softmax Collapse (SC). We demonstrate that SC prevents grokking and that mitigating SC enables grokking without regularization. Investigating the root cause of SC, we find that beyond the point of overfitting, the gradients strongly align with what we call the na\"ive loss minimization (NLM) direction. This component of the gradient does not alter the model's predictions but decreases the loss by scaling the logits, typically by scaling the weights along their current direction. We show that this scaling of the logits explains the delay in generalization characteristic of grokking and eventually leads to SC, halting further learning. To validate our hypotheses, we introduce two key contributions that address the challenges in grokking tasks: StableMax, a new activation function that prevents SC and enables grokking without regularization, and perpGrad, a training algorithm that promotes quick generalization in grokking tasks by preventing NLM altogether. These contributions provide new insights into grokking, elucidating its delayed generalization, reliance on regularization, and the effectiveness of existing grokking-inducing methods. Code for this paper is available at https://github.com/LucasPrietoAl/grokking-at-the-edge-of-numerical-stability.

Generative Adversarial Networks (GANs) are powerful generative models, but suffer from training instability. The recently proposed Wasserstein GAN (WGAN) makes progress toward stable training of GANs, but sometimes can still generate only low-quality samples or fail to converge. We find that these problems are often due to the use of weight clipping in WGAN to enforce a Lipschitz constraint on the critic, which can lead to undesired behavior. We propose an alternative to clipping weights: penalize the norm of gradient of the critic with respect to its input. Our proposed method performs better than standard WGAN and enables stable training of a wide variety of GAN architectures with almost no hyperparameter tuning, including 101-layer ResNets and language models over discrete data. We also achieve high quality generations on CIFAR-10 and LSUN bedrooms.

Curriculum Learning is the presentation of samples to the machine learning model in a meaningful order instead of a random order. The main challenge of Curriculum Learning is determining how to rank these samples. The ranking of the samples is expressed by the scoring function. In this study, scoring functions were compared using data set features, using the model to be trained, and using another model and their ensemble versions. Experiments were performed for 4 images and 4 text datasets. No significant differences were found between scoring functions for text datasets, but significant improvements were obtained in scoring functions created using transfer learning compared to classical model training and other scoring functions for image datasets. It shows that different new scoring functions are waiting to be found for text classification tasks.

While implicit feedback is foundational to modern recommender systems, factors such as human error, uncertainty, and ambiguity in user behavior inevitably introduce significant noise into this feedback, adversely affecting the accuracy and robustness of recommendations. To address this issue, existing methods typically aim to reduce the training weight of noisy feedback or discard it entirely, based on the observation that noisy interactions often exhibit higher losses in the overall loss distribution. However, we identify two key issues: (1) there is a significant overlap between normal and noisy interactions in the overall loss distribution, and (2) this overlap becomes even more pronounced when transitioning from pointwise loss functions (e.g., BCE loss) to pairwise loss functions (e.g., BPR loss). This overlap leads traditional methods to misclassify noisy interactions as normal, and vice versa. To tackle these challenges, we further investigate the loss overlap and find that for a given user, there is a clear distinction between normal and noisy interactions in the user's personal loss distribution. Based on this insight, we propose a resampling strategy to Denoise using the user's Personal Loss distribution, named PLD, which reduces the probability of noisy interactions being optimized. Specifically, during each optimization iteration, we create a candidate item pool for each user and resample the items from this pool based on the user's personal loss distribution, prioritizing normal interactions. Additionally, we conduct a theoretical analysis to validate PLD's effectiveness and suggest ways to further enhance its performance. Extensive experiments conducted on three datasets with varying noise ratios demonstrate PLD's efficacy and robustness.

We consider a class of assortment optimization problems in an offline data-driven setting. A firm does not know the underlying customer choice model but has access to an offline dataset consisting of the historically offered assortment set, customer choice, and revenue. The objective is to use the offline dataset to find an optimal assortment. Due to the combinatorial nature of assortment optimization, the problem of insufficient data coverage is likely to occur in the offline dataset. Therefore, designing a provably efficient offline learning algorithm becomes a significant challenge. To this end, we propose an algorithm referred to as Pessimistic ASsortment opTimizAtion (PASTA for short) designed based on the principle of pessimism, that can correctly identify the optimal assortment by only requiring the offline data to cover the optimal assortment under general settings. In particular, we establish a regret bound for the offline assortment optimization problem under the celebrated multinomial logit model. We also propose an efficient computational procedure to solve our pessimistic assortment optimization problem. Numerical studies demonstrate the superiority of the proposed method over the existing baseline method.

Diffusion models are known to be vulnerable to outliers in training data. In this paper we study an alternative diffusion loss function, which can preserve the high quality of generated data like the original squared L_{2} loss while at the same time being robust to outliers. We propose to use pseudo-Huber loss function with a time-dependent parameter to allow for the trade-off between robustness on the most vulnerable early reverse-diffusion steps and fine details restoration on the final steps. We show that pseudo-Huber loss with the time-dependent parameter exhibits better performance on corrupted datasets in both image and audio domains. In addition, the loss function we propose can potentially help diffusion models to resist dataset corruption while not requiring data filtering or purification compared to conventional training algorithms.

Aligning large language models (LLMs) to human preferences is challenging in domains where preference data is unavailable. We address the problem of learning reward models for such target domains by leveraging feedback collected from simpler source domains, where human preferences are easier to obtain. Our key insight is that, while domains may differ significantly, human preferences convey domain-agnostic concepts that can be effectively captured by a reward model. We propose \method, a framework that trains domain-invariant reward models by optimizing a dual loss: a domain loss that minimizes the divergence between source and target distribution, and a source loss that optimizes preferences on the source domain. We show \method is a general approach that we evaluate and analyze across 4 distinct settings: (1) Cross-lingual transfer (accuracy: 0.621 rightarrow 0.661), (2) Clean-to-noisy (accuracy: 0.671 rightarrow 0.703), (3) Few-shot-to-full transfer (accuracy: 0.845 rightarrow 0.920), and (4) Simple-to-complex tasks transfer (correlation: 0.508 rightarrow 0.556). Our code, models and data are available at https://github.com/portal-cornell/dial.

Understanding how overparameterized neural networks generalize despite perfect interpolation of noisy training data is a fundamental question. Mallinar et. al. 2022 noted that neural networks seem to often exhibit ``tempered overfitting'', wherein the population risk does not converge to the Bayes optimal error, but neither does it approach infinity, yielding non-trivial generalization. However, this has not been studied rigorously. We provide the first rigorous analysis of the overfitting behavior of regression with minimum norm (ell_2 of weights), focusing on univariate two-layer ReLU networks. We show overfitting is tempered (with high probability) when measured with respect to the L_1 loss, but also show that the situation is more complex than suggested by Mallinar et. al., and overfitting is catastrophic with respect to the L_2 loss, or when taking an expectation over the training set.

Sparse autoencoders (SAEs) are a promising approach to extracting features from neural networks, enabling model interpretability as well as causal interventions on model internals. SAEs generate sparse feature representations using a sparsifying activation function that implicitly defines a set of token-feature matches. We frame the token-feature matching as a resource allocation problem constrained by a total sparsity upper bound. For example, TopK SAEs solve this allocation problem with the additional constraint that each token matches with at most k features. In TopK SAEs, the k active features per token constraint is the same across tokens, despite some tokens being more difficult to reconstruct than others. To address this limitation, we propose two novel SAE variants, Feature Choice SAEs and Mutual Choice SAEs, which each allow for a variable number of active features per token. Feature Choice SAEs solve the sparsity allocation problem under the additional constraint that each feature matches with at most m tokens. Mutual Choice SAEs solve the unrestricted allocation problem where the total sparsity budget can be allocated freely between tokens and features. Additionally, we introduce a new auxiliary loss function, aux_zipf_loss, which generalises the aux_k_loss to mitigate dead and underutilised features. Our methods result in SAEs with fewer dead features and improved reconstruction loss at equivalent sparsity levels as a result of the inherent adaptive computation. More accurate and scalable feature extraction methods provide a path towards better understanding and more precise control of foundation models.

A primary cost driver for training large models is wall-clock training time. We show that popular time estimates based on FLOPs are poor estimates, and construct a more accurate proxy based on memory copies. We show that with some simple accounting, we can estimate the training speed of a transformer model from its hyperparameters. Combined with a scaling law curve like Chinchilla, this lets us estimate the final loss of the model. We fit our estimate to real data with a linear regression, and apply the result to rewrite Chinchilla in terms of a model's estimated training time as opposed to the amount of training data. This gives an expression for the loss in terms of the model's hyperparameters alone. We show that this expression is accurate across a wide range of model hyperparameter values, enabling us to analytically make architectural decisions and train models more efficiently.

In this work, we present a variety of novel information-theoretic generalization bounds for learning algorithms, from the supersample setting of Steinke & Zakynthinou (2020)-the setting of the "conditional mutual information" framework. Our development exploits projecting the loss pair (obtained from a training instance and a testing instance) down to a single number and correlating loss values with a Rademacher sequence (and its shifted variants). The presented bounds include square-root bounds, fast-rate bounds, including those based on variance and sharpness, and bounds for interpolating algorithms etc. We show theoretically or empirically that these bounds are tighter than all information-theoretic bounds known to date on the same supersample setting.

We consider the problem of subset selection where one is given multiple rankings of items and the goal is to select the highest ``quality'' subset. Score functions from the multiwinner voting literature have been used to aggregate rankings into quality scores for subsets. We study this setting of subset selection problems when, in addition, rankings may contain systemic or unconscious biases toward a group of items. For a general model of input rankings and biases, we show that requiring the selected subset to satisfy group fairness constraints can improve the quality of the selection with respect to unbiased rankings. Importantly, we show that for fairness constraints to be effective, different multiwinner score functions may require a drastically different number of rankings: While for some functions, fairness constraints need an exponential number of rankings to recover a close-to-optimal solution, for others, this dependency is only polynomial. This result relies on a novel notion of ``smoothness'' of submodular functions in this setting that quantifies how well a function can ``correctly'' assess the quality of items in the presence of bias. The results in this paper can be used to guide the choice of multiwinner score functions for the subset selection setting considered here; we additionally provide a tool to empirically enable this.

Recent work have demonstrated that robustness (to "corruption") can be at odds with generalization. Adversarial training, for instance, aims to reduce the problematic susceptibility of modern neural networks to small data perturbations. Surprisingly, overfitting is a major concern in adversarial training despite being mostly absent in standard training. We provide here theoretical evidence for this peculiar "robust overfitting" phenomenon. Subsequently, we advance a novel distributionally robust loss function bridging robustness and generalization. We demonstrate both theoretically as well as empirically the loss to enjoy a certified level of robustness against two common types of corruption--data evasion and poisoning attacks--while ensuring guaranteed generalization. We show through careful numerical experiments that our resulting holistic robust (HR) training procedure yields SOTA performance. Finally, we indicate that HR training can be interpreted as a direct extension of adversarial training and comes with a negligible additional computational burden. A ready-to-use python library implementing our algorithm is available at https://github.com/RyanLucas3/HR_Neural_Networks.

This paper addresses a regression problem in which output label values are the results of sensing the magnitude of a phenomenon. A low value of such labels can mean either that the actual magnitude of the phenomenon was low or that the sensor made an incomplete observation. This leads to a bias toward lower values in labels and the resultant learning because labels may have lower values due to incomplete observations, even if the actual magnitude of the phenomenon was high. Moreover, because an incomplete observation does not provide any tags indicating incompleteness, we cannot eliminate or impute them. To address this issue, we propose a learning algorithm that explicitly models incomplete observations corrupted with an asymmetric noise that always has a negative value. We show that our algorithm is unbiased as if it were learned from uncorrupted data that does not involve incomplete observations. We demonstrate the advantages of our algorithm through numerical experiments.

What do auto-encoders learn about the underlying data generating distribution? Recent work suggests that some auto-encoder variants do a good job of capturing the local manifold structure of data. This paper clarifies some of these previous observations by showing that minimizing a particular form of regularized reconstruction error yields a reconstruction function that locally characterizes the shape of the data generating density. We show that the auto-encoder captures the score (derivative of the log-density with respect to the input). It contradicts previous interpretations of reconstruction error as an energy function. Unlike previous results, the theorems provided here are completely generic and do not depend on the parametrization of the auto-encoder: they show what the auto-encoder would tend to if given enough capacity and examples. These results are for a contractive training criterion we show to be similar to the denoising auto-encoder training criterion with small corruption noise, but with contraction applied on the whole reconstruction function rather than just encoder. Similarly to score matching, one can consider the proposed training criterion as a convenient alternative to maximum likelihood because it does not involve a partition function. Finally, we show how an approximate Metropolis-Hastings MCMC can be setup to recover samples from the estimated distribution, and this is confirmed in sampling experiments.