Daily Papers

Machine learning-based interatomic potentials and force fields depend critically on accurate atomic structures, yet such data are scarce due to the limited availability of experimentally resolved crystals. Although atomic-resolution electron microscopy offers a potential source of structural data, converting these images into simulation-ready formats remains labor-intensive and error-prone, creating a bottleneck for model training and validation. We introduce AutoMat, an end-to-end, agent-assisted pipeline that automatically transforms scanning transmission electron microscopy (STEM) images into atomic crystal structures and predicts their physical properties. AutoMat combines pattern-adaptive denoising, physics-guided template retrieval, symmetry-aware atomic reconstruction, fast relaxation and property prediction via MatterSim, and coordinated orchestration across all stages. We propose the first dedicated STEM2Mat-Bench for this task and evaluate performance using lattice RMSD, formation energy MAE, and structure-matching success rate. By orchestrating external tool calls, AutoMat enables a text-only LLM to outperform vision-language models in this domain, achieving closed-loop reasoning throughout the pipeline. In large-scale experiments over 450 structure samples, AutoMat substantially outperforms existing multimodal large language models and tools. These results validate both AutoMat and STEM2Mat-Bench, marking a key step toward bridging microscopy and atomistic simulation in materials science.The code and dataset are publicly available at https://github.com/yyt-2378/AutoMat and https://huggingface.co/datasets/yaotianvector/STEM2Mat.

Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.

We describe MGARD, a software providing MultiGrid Adaptive Reduction for floating-point scientific data on structured and unstructured grids. With exceptional data compression capability and precise error control, MGARD addresses a wide range of requirements, including storage reduction, high-performance I/O, and in-situ data analysis. It features a unified application programming interface (API) that seamlessly operates across diverse computing architectures. MGARD has been optimized with highly-tuned GPU kernels and efficient memory and device management mechanisms, ensuring scalable and rapid operations.

We present 360Zhinao models with 7B parameter size and context lengths spanning 4K, 32K and 360K, all available at https://github.com/Qihoo360/360zhinao. For rapid development in pretraining, we establish a stable and sensitive ablation environment to evaluate and compare experiment runs with minimal model size. Under such guidance, we perfect our data cleaning and composition strategies to pretrain 360Zhinao-7B-Base on 3.4T tokens. We also mainly emphasize data during alignment, where we strive to balance quantity and quality with filtering and reformatting. With tailored data, 360Zhinao-7B's context window is easily extended to 32K and 360K. RMs and RLHF are trained following SFT and credibly applied to specific tasks. All together these contributions lead to 360Zhinao-7B's competitive performance among models of similar size.

Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-m rate, where m is the number of simulated samples. This can lead to significant computational challenges since a large m is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.

Approximating Stochastic Gradient Descent (SGD) as a Stochastic Differential Equation (SDE) has allowed researchers to enjoy the benefits of studying a continuous optimization trajectory while carefully preserving the stochasticity of SGD. Analogous study of adaptive gradient methods, such as RMSprop and Adam, has been challenging because there were no rigorously proven SDE approximations for these methods. This paper derives the SDE approximations for RMSprop and Adam, giving theoretical guarantees of their correctness as well as experimental validation of their applicability to common large-scaling vision and language settings. A key practical result is the derivation of a square root scaling rule to adjust the optimization hyperparameters of RMSprop and Adam when changing batch size, and its empirical validation in deep learning settings.

We propose using Normalizing Flows as a trainable kernel within the molecular dynamics update of Hamiltonian Monte Carlo (HMC). By learning (invertible) transformations that simplify our dynamics, we can outperform traditional methods at generating independent configurations. We show that, using a carefully constructed network architecture, our approach can be easily scaled to large lattice volumes with minimal retraining effort. The source code for our implementation is publicly available online at https://github.com/nftqcd/fthmc.

Layer normalization (LayerNorm) has been successfully applied to various deep neural networks to help stabilize training and boost model convergence because of its capability in handling re-centering and re-scaling of both inputs and weight matrix. However, the computational overhead introduced by LayerNorm makes these improvements expensive and significantly slows the underlying network, e.g. RNN in particular. In this paper, we hypothesize that re-centering invariance in LayerNorm is dispensable and propose root mean square layer normalization, or RMSNorm. RMSNorm regularizes the summed inputs to a neuron in one layer according to root mean square (RMS), giving the model re-scaling invariance property and implicit learning rate adaptation ability. RMSNorm is computationally simpler and thus more efficient than LayerNorm. We also present partial RMSNorm, or pRMSNorm where the RMS is estimated from p% of the summed inputs without breaking the above properties. Extensive experiments on several tasks using diverse network architectures show that RMSNorm achieves comparable performance against LayerNorm but reduces the running time by 7%~64% on different models. Source code is available at https://github.com/bzhangGo/rmsnorm.

Lattice-based planning techniques simplify the motion planning problem for autonomous vehicles by limiting available motions to a pre-computed set of primitives. These primitives are then combined online to generate more complex maneuvers. A set of motion primitives t-span a lattice if, given a real number t at least 1, any configuration in the lattice can be reached via a sequence of motion primitives whose cost is no more than a factor of t from optimal. Computing a minimal t-spanning set balances a trade-off between computed motion quality and motion planning performance. In this work, we formulate this problem for an arbitrary lattice as a mixed integer linear program. We also propose an A*-based algorithm to solve the motion planning problem using these primitives. Finally, we present an algorithm that removes the excessive oscillations from planned motions -- a common problem in lattice-based planning. Our method is validated for autonomous driving in both parking lot and highway scenarios.

Recent advances in 3D X-ray Computed Tomographic (CT) sensors have stimulated research efforts to unveil the extremely complex micro-scale processes that control the activity of soil microorganisms. Voxel-based description (up to hundreds millions voxels) of the pore space can be extracted, from grey level 3D CT scanner images, by means of simple image processing tools. Classical methods for numerical simulation of biological dynamics using mesh of voxels, such as Lattice Boltzmann Model (LBM), are too much time consuming. Thus, the use of more compact and reliable geometrical representations of pore space can drastically decrease the computational cost of the simulations. Several recent works propose basic analytic volume primitives (e.g. spheres, generalized cylinders, ellipsoids) to define a piece-wise approximation of pore space for numerical simulation of draining, diffusion and microbial decomposition. Such approaches work well but the drawback is that it generates approximation errors. In the present work, we study another alternative where pore space is described by means of geometrically relevant connected subsets of voxels (regions) computed from the curvilinear skeleton. Indeed, many works use the curvilinear skeleton (3D medial axis) for analyzing and partitioning 3D shapes within various domains (medicine, material sciences, petroleum engineering, etc.) but only a few ones in soil sciences. Within the context of soil sciences, most studies dealing with 3D medial axis focus on the determination of pore throats. Here, we segment pore space using curvilinear skeleton in order to achieve numerical simulation of microbial decomposition (including diffusion processes). We validate simulation outputs by comparison with other methods using different pore space geometrical representations (balls, voxels).

Radio frequency (RF) signal mapping, which is the process of analyzing and predicting the RF signal strength and distribution across specific areas, is crucial for cellular network planning and deployment. Traditional approaches to RF signal mapping rely on statistical models constructed based on measurement data, which offer low complexity but often lack accuracy, or ray tracing tools, which provide enhanced precision for the target area but suffer from increased computational complexity. Recently, machine learning (ML) has emerged as a data-driven method for modeling RF signal propagation, which leverages models trained on synthetic datasets to perform RF signal mapping in "unseen" areas. In this paper, we present Geo2SigMap, an ML-based framework for efficient and high-fidelity RF signal mapping using geographic databases. First, we develop an automated framework that seamlessly integrates three open-source tools: OpenStreetMap (geographic databases), Blender (computer graphics), and Sionna (ray tracing), enabling the efficient generation of large-scale 3D building maps and ray tracing models. Second, we propose a cascaded U-Net model, which is pre-trained on synthetic datasets and employed to generate detailed RF signal maps, leveraging environmental information and sparse measurement data. Finally, we evaluate the performance of Geo2SigMap via a real-world measurement campaign, where three types of user equipment (UE) collect over 45,000 data points related to cellular information from six LTE cells operating in the citizens broadband radio service (CBRS) band. Our results show that Geo2SigMap achieves an average root-mean-square-error (RMSE) of 6.04 dB for predicting the reference signal received power (RSRP) at the UE, representing an average RMSE improvement of 3.59 dB compared to existing methods.

Motivation: Ab initio protein docking represents a major challenge for optimizing a noisy and costly "black box"-like function in a high-dimensional space. Despite progress in this field, there is no docking method available for rigorous uncertainty quantification (UQ) of its solution quality (e.g. interface RMSD or iRMSD). Results: We introduce a novel algorithm, Bayesian Active Learning (BAL), for optimization and UQ of such black-box functions and flexible protein docking. BAL directly models the posterior distribution of the global optimum (or native structures for protein docking) with active sampling and posterior estimation iteratively feeding each other. Furthermore, we use complex normal modes to represent a homogeneous Euclidean conformation space suitable for high-dimension optimization and construct funnel-like energy models for encounter complexes. Over a protein docking benchmark set and a CAPRI set including homology docking, we establish that BAL significantly improve against both starting points by rigid docking and refinements by particle swarm optimization, providing for one third targets a top-3 near-native prediction. BAL also generates tight confidence intervals with half range around 25% of iRMSD and confidence level at 85%. Its estimated probability of a prediction being native or not achieves binary classification AUROC at 0.93 and AUPRC over 0.60 (compared to 0.14 by chance); and also found to help ranking predictions. To the best of our knowledge, this study represents the first uncertainty quantification solution for protein docking, with theoretical rigor and comprehensive assessment. Source codes are available at https://github.com/Shen-Lab/BAL.

Attention mechanisms have revolutionized sequence learning but suffer from quadratic computational complexity. This paper introduces Lattice, a novel recurrent neural network (RNN) mechanism that leverages the inherent low-rank structure of K-V matrices to efficiently compress the cache into a fixed number of memory slots, achieving sub-quadratic complexity. We formulate this compression as an online optimization problem and derive a dynamic memory update rule based on a single gradient descent step. The resulting recurrence features a state- and input-dependent gating mechanism, offering an interpretable memory update process. The core innovation is the orthogonal update: each memory slot is updated exclusively with information orthogonal to its current state hence incorporation of only novel, non-redundant data, which minimizes the interference with previously stored information. The experimental results show that Lattice achieves the best perplexity compared to all baselines across diverse context lengths, with performance improvement becoming more pronounced as the context length increases.

MATSim (Multi-Agent Transport Simulation Toolkit) is an open source large-scale agent-based transportation planning project applied to various areas like road transport, public transport, freight transport, regional evacuation, etc. BEAM (Behavior, Energy, Autonomy, and Mobility) framework extends MATSim to enable powerful and scalable analysis of urban transportation systems. The agents from the BEAM simulation exhibit 'mode choice' behavior based on multinomial logit model. In our study, we consider eight mode choices viz. bike, car, walk, ride hail, driving to transit, walking to transit, ride hail to transit, and ride hail pooling. The 'alternative specific constants' for each mode choice are critical hyperparameters in a configuration file related to a particular scenario under experimentation. We use the 'Urbansim-10k' BEAM scenario (with 10,000 population size) for all our experiments. Since these hyperparameters affect the simulation in complex ways, manual calibration methods are time consuming. We present a parallel Bayesian optimization method with early stopping rule to achieve fast convergence for the given multi-in-multi-out problem to its optimal configurations. Our model is based on an open source HpBandSter package. This approach combines hierarchy of several 1D Kernel Density Estimators (KDE) with a cheap evaluator (Hyperband, a single multidimensional KDE). Our model has also incorporated extrapolation based early stopping rule. With our model, we could achieve a 25% L1 norm for a large-scale BEAM simulation in fully autonomous manner. To the best of our knowledge, our work is the first of its kind applied to large-scale multi-agent transportation simulations. This work can be useful for surrogate modeling of scenarios with very large populations.

Crystalline materials are a fundamental component in next-generation technologies, yet modeling their distribution presents unique computational challenges. Of the plausible arrangements of atoms in a periodic lattice only a vanishingly small percentage are thermodynamically stable, which is a key indicator of the materials that can be experimentally realized. Two fundamental tasks in this area are to (a) predict the stable crystal structure of a known composition of elements and (b) propose novel compositions along with their stable structures. We present FlowMM, a pair of generative models that achieve state-of-the-art performance on both tasks while being more efficient and more flexible than competing methods. We generalize Riemannian Flow Matching to suit the symmetries inherent to crystals: translation, rotation, permutation, and periodic boundary conditions. Our framework enables the freedom to choose the flow base distributions, drastically simplifying the problem of learning crystal structures compared with diffusion models. In addition to standard benchmarks, we validate FlowMM's generated structures with quantum chemistry calculations, demonstrating that it is about 3x more efficient, in terms of integration steps, at finding stable materials compared to previous open methods.

We present the design, implementation and engineering experience in building and deploying MegaScale, a production system for training large language models (LLMs) at the scale of more than 10,000 GPUs. Training LLMs at this scale brings unprecedented challenges to training efficiency and stability. We take a full-stack approach that co-designs the algorithmic and system components across model block and optimizer design, computation and communication overlapping, operator optimization, data pipeline, and network performance tuning. Maintaining high efficiency throughout the training process (i.e., stability) is an important consideration in production given the long extent of LLM training jobs. Many hard stability issues only emerge at large scale, and in-depth observability is the key to address them. We develop a set of diagnosis tools to monitor system components and events deep in the stack, identify root causes, and derive effective techniques to achieve fault tolerance and mitigate stragglers. MegaScale achieves 55.2% Model FLOPs Utilization (MFU) when training a 175B LLM model on 12,288 GPUs, improving the MFU by 1.34x compared to Megatron-LM. We share our operational experience in identifying and fixing failures and stragglers. We hope by articulating the problems and sharing our experience from a systems perspective, this work can inspire future LLM systems research.

This paper presents MetricGrids, a novel grid-based neural representation that combines elementary metric grids in various metric spaces to approximate complex nonlinear signals. While grid-based representations are widely adopted for their efficiency and scalability, the existing feature grids with linear indexing for continuous-space points can only provide degenerate linear latent space representations, and such representations cannot be adequately compensated to represent complex nonlinear signals by the following compact decoder. To address this problem while keeping the simplicity of a regular grid structure, our approach builds upon the standard grid-based paradigm by constructing multiple elementary metric grids as high-order terms to approximate complex nonlinearities, following the Taylor expansion principle. Furthermore, we enhance model compactness with hash encoding based on different sparsities of the grids to prevent detrimental hash collisions, and a high-order extrapolation decoder to reduce explicit grid storage requirements. experimental results on both 2D and 3D reconstructions demonstrate the superior fitting and rendering accuracy of the proposed method across diverse signal types, validating its robustness and generalizability. Code is available at https://github.com/wangshu31/MetricGrids}{https://github.com/wangshu31/MetricGrids.

Crystals are the foundation of numerous scientific and industrial applications. While various learning-based approaches have been proposed for crystal generation, existing methods seldom consider the space group constraint which is crucial in describing the geometry of crystals and closely relevant to many desirable properties. However, considering space group constraint is challenging owing to its diverse and nontrivial forms. In this paper, we reduce the space group constraint into an equivalent formulation that is more tractable to be handcrafted into the generation process. In particular, we translate the space group constraint into two parts: the basis constraint of the invariant logarithmic space of the lattice matrix and the Wyckoff position constraint of the fractional coordinates. Upon the derived constraints, we then propose DiffCSP++, a novel diffusion model that has enhanced a previous work DiffCSP by further taking space group constraint into account. Experiments on several popular datasets verify the benefit of the involvement of the space group constraint, and show that our DiffCSP++ achieves promising performance on crystal structure prediction, ab initio crystal generation and controllable generation with customized space groups.

This paper revisits datasets and evaluation criteria for Symbolic Regression, a task of expressing given data using mathematical equations, specifically focused on its potential for scientific discovery. Focused on a set of formulas used in the existing datasets based on Feynman Lectures on Physics, we recreate 120 datasets to discuss the performance of symbolic regression for scientific discovery (SRSD). For each of the 120 SRSD datasets, we carefully review the properties of the formula and its variables to design reasonably realistic sampling range of values so that our new SRSD datasets can be used for evaluating the potential of SRSD such as whether or not an SR method can (re)discover physical laws from such datasets. As an evaluation metric, we also propose to use normalized edit distances between a predicted equation and the ground-truth equation trees. While existing metrics are either binary or errors between the target values and an SR model's predicted values for a given input, normalized edit distances evaluate a sort of similarity between the ground-truth and predicted equation trees. We have conducted experiments on our new SRSD datasets using five state-of-the-art SR methods in SRBench and a simple baseline based on a recent Transformer architecture. The results show that we provide a more realistic performance evaluation and open up a new machine learning-based approach for scientific discovery. Our datasets and code repository are publicly available.

Linear time-invariant state space models (SSM) are a classical model from engineering and statistics, that have recently been shown to be very promising in machine learning through the Structured State Space sequence model (S4). A core component of S4 involves initializing the SSM state matrix to a particular matrix called a HiPPO matrix, which was empirically important for S4's ability to handle long sequences. However, the specific matrix that S4 uses was actually derived in previous work for a particular time-varying dynamical system, and the use of this matrix as a time-invariant SSM had no known mathematical interpretation. Consequently, the theoretical mechanism by which S4 models long-range dependencies actually remains unexplained. We derive a more general and intuitive formulation of the HiPPO framework, which provides a simple mathematical interpretation of S4 as a decomposition onto exponentially-warped Legendre polynomials, explaining its ability to capture long dependencies. Our generalization introduces a theoretically rich class of SSMs that also lets us derive more intuitive S4 variants for other bases such as the Fourier basis, and explains other aspects of training S4, such as how to initialize the important timescale parameter. These insights improve S4's performance to 86% on the Long Range Arena benchmark, with 96% on the most difficult Path-X task.

Maximum mean discrepancy (MMD) flows suffer from high computational costs in large scale computations. In this paper, we show that MMD flows with Riesz kernels K(x,y) = - |x-y|^r, r in (0,2) have exceptional properties which allow their efficient computation. We prove that the MMD of Riesz kernels, which is also known as energy distance, coincides with the MMD of their sliced version. As a consequence, the computation of gradients of MMDs can be performed in the one-dimensional setting. Here, for r=1, a simple sorting algorithm can be applied to reduce the complexity from O(MN+N^2) to O((M+N)log(M+N)) for two measures with M and N support points. As another interesting follow-up result, the MMD of compactly supported measures can be estimated from above and below by the Wasserstein-1 distance. For the implementations we approximate the gradient of the sliced MMD by using only a finite number P of slices. We show that the resulting error has complexity O(d/P), where d is the data dimension. These results enable us to train generative models by approximating MMD gradient flows by neural networks even for image applications. We demonstrate the efficiency of our model by image generation on MNIST, FashionMNIST and CIFAR10.

Molecular dynamics (MD) simulations play a crucial role in scientific research. Yet their computational cost often limits the timescales and system sizes that can be explored. Most data-driven efforts have been focused on reducing the computational cost of accurate interatomic forces required for solving the equations of motion. Despite their success, however, these machine learning interatomic potentials (MLIPs) are still bound to small time-steps. In this work, we introduce TrajCast, a transferable and data-efficient framework based on autoregressive equivariant message passing networks that directly updates atomic positions and velocities lifting the constraints imposed by traditional numerical integration. We benchmark our framework across various systems, including a small molecule, crystalline material, and bulk liquid, demonstrating excellent agreement with reference MD simulations for structural, dynamical, and energetic properties. Depending on the system, TrajCast allows for forecast intervals up to 30times larger than traditional MD time-steps, generating over 15 ns of trajectory data per day for a solid with more than 4,000 atoms. By enabling efficient large-scale simulations over extended timescales, TrajCast can accelerate materials discovery and explore physical phenomena beyond the reach of traditional simulations and experiments. An open-source implementation of TrajCast is accessible under https://github.com/IBM/trajcast.

In several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-moment estimators requires memory equal to the number of parameters. For the case of neural network weight matrices, we propose maintaining only the per-row and per-column sums of these moving averages, and estimating the per-parameter second moments based on these sums. We demonstrate empirically that this method produces similar results to the baseline. Secondly, we show that adaptive methods can produce larger-than-desired updates when the decay rate of the second moment accumulator is too slow. We propose update clipping and a gradually increasing decay rate scheme as remedies. Combining these methods and dropping momentum, we achieve comparable results to the published Adam regime in training the Transformer model on the WMT 2014 English-German machine translation task, while using very little auxiliary storage in the optimizer. Finally, we propose scaling the parameter updates based on the scale of the parameters themselves.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

In nature, the behaviors of many complex systems can be described by parsimonious math equations. Automatically distilling these equations from limited data is cast as a symbolic regression process which hitherto remains a grand challenge. Keen efforts in recent years have been placed on tackling this issue and demonstrated success in symbolic regression. However, there still exist bottlenecks that current methods struggle to break when the discrete search space tends toward infinity and especially when the underlying math formula is intricate. To this end, we propose a novel Reinforcement Symbolic Regression Machine (RSRM) that masters the capability of uncovering complex math equations from only scarce data. The RSRM model is composed of three key modules: (1) a Monte Carlo tree search (MCTS) agent that explores optimal math expression trees consisting of pre-defined math operators and variables, (2) a Double Q-learning block that helps reduce the feasible search space of MCTS via properly understanding the distribution of reward, and (3) a modulated sub-tree discovery block that heuristically learns and defines new math operators to improve representation ability of math expression trees. Biding of these modules yields the state-of-the-art performance of RSRM in symbolic regression as demonstrated by multiple sets of benchmark examples. The RSRM model shows clear superiority over several representative baseline models.

Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph representation learning. Specifically, constructing graphs that effectively capture the complete geometric information of crystals and handle chiral crystals remains an unsolved and challenging problem. In this paper, we introduce a novel approach that utilizes the periodic patterns of unit cells to establish the lattice-based representation for each atom, enabling efficient and expressive graph representations of crystals. Furthermore, we propose ComFormer, a SE(3) transformer designed specifically for crystalline materials. ComFormer includes two variants; namely, iComFormer that employs invariant geometric descriptors of Euclidean distances and angles, and eComFormer that utilizes equivariant vector representations. Experimental results demonstrate the state-of-the-art predictive accuracy of ComFormer variants on various tasks across three widely-used crystal benchmarks. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Hyperbolic geometry is gaining traction in machine learning for its effectiveness at capturing hierarchical structures in real-world data. Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial advantages and consistently deliver state-of-the-art results across diverse applications. However, hyperbolic classifiers often grapple with computational challenges. Methods reliant on Riemannian optimization frequently exhibit sluggishness, stemming from the increased computational demands of operations on Riemannian manifolds. In response to these challenges, we present hyperDT, a novel extension of decision tree algorithms into hyperbolic space. Crucially, hyperDT eliminates the need for computationally intensive Riemannian optimization, numerically unstable exponential and logarithmic maps, or pairwise comparisons between points by leveraging inner products to adapt Euclidean decision tree algorithms to hyperbolic space. Our approach is conceptually straightforward and maintains constant-time decision complexity while mitigating the scalability issues inherent in high-dimensional Euclidean spaces. Building upon hyperDT we introduce hyperRF, a hyperbolic random forest model. Extensive benchmarking across diverse datasets underscores the superior performance of these models, providing a swift, precise, accurate, and user-friendly toolkit for hyperbolic data analysis.

We present XLand-MiniGrid, a suite of tools and grid-world environments for meta-reinforcement learning research inspired by the diversity and depth of XLand and the simplicity and minimalism of MiniGrid. XLand-Minigrid is written in JAX, designed to be highly scalable, and can potentially run on GPU or TPU accelerators, democratizing large-scale experimentation with limited resources. To demonstrate the generality of our library, we have implemented some well-known single-task environments as well as new meta-learning environments capable of generating 10^8 distinct tasks. We have empirically shown that the proposed environments can scale up to 2^{13} parallel instances on the GPU, reaching tens of millions of steps per second.

Achieving a balance between computational speed, prediction accuracy, and universal applicability in molecular simulations has been a persistent challenge. This paper presents substantial advancements in the TorchMD-Net software, a pivotal step forward in the shift from conventional force fields to neural network-based potentials. The evolution of TorchMD-Net into a more comprehensive and versatile framework is highlighted, incorporating cutting-edge architectures such as TensorNet. This transformation is achieved through a modular design approach, encouraging customized applications within the scientific community. The most notable enhancement is a significant improvement in computational efficiency, achieving a very remarkable acceleration in the computation of energy and forces for TensorNet models, with performance gains ranging from 2-fold to 10-fold over previous iterations. Other enhancements include highly optimized neighbor search algorithms that support periodic boundary conditions and the smooth integration with existing molecular dynamics frameworks. Additionally, the updated version introduces the capability to integrate physical priors, further enriching its application spectrum and utility in research. The software is available at https://github.com/torchmd/torchmd-net.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

Ligand molecule conformation generation is a critical challenge in drug discovery. Deep learning models have been developed to tackle this problem, particularly through the use of generative models in recent years. However, these models often generate conformations that lack meaningful structure and randomness due to the absence of essential side information. Examples of such side information include the chemical and geometric features of the target protein, ligand-target compound interactions, and ligand chemical properties. Without these constraints, the generated conformations may not be suitable for further selection and design of new drugs. To address this limitation, we propose a novel method for generating ligand conformations that leverage side information and incorporate flexible constraints into standard diffusion models. Drawing inspiration from the concept of message passing, we introduce ligand-target massage passing block, a mechanism that facilitates the exchange of information between target nodes and ligand nodes, thereby incorporating target node features. To capture non-covalent interactions, we introduce ligand-target compound inter and intra edges. To further improve the biological relevance of the generated conformations, we train energy models using scalar chemical features. These models guide the progress of the standard Denoising Diffusion Probabilistic Models, resulting in more biologically meaningful conformations. We evaluate the performance of SIDEGEN using the PDBBind-2020 dataset, comparing it against other methods. The results demonstrate improvements in both Aligned RMSD and Ligand RMSD evaluations. Specifically, our model outperforms GeoDiff (trained on PDBBind-2020) by 20% in terms of the median aligned RMSD metric.

Diffusion models for super-resolution (SR) produce high-quality visual results but require expensive computational costs. Despite the development of several methods to accelerate diffusion-based SR models, some (e.g., SinSR) fail to produce realistic perceptual details, while others (e.g., OSEDiff) may hallucinate non-existent structures. To overcome these issues, we present RSD, a new distillation method for ResShift, one of the top diffusion-based SR models. Our method is based on training the student network to produce such images that a new fake ResShift model trained on them will coincide with the teacher model. RSD achieves single-step restoration and outperforms the teacher by a large margin. We show that our distillation method can surpass the other distillation-based method for ResShift - SinSR - making it on par with state-of-the-art diffusion-based SR distillation methods. Compared to SR methods based on pre-trained text-to-image models, RSD produces competitive perceptual quality, provides images with better alignment to degraded input images, and requires fewer parameters and GPU memory. We provide experimental results on various real-world and synthetic datasets, including RealSR, RealSet65, DRealSR, ImageNet, and DIV2K.

Statistics of grain sizes and orientations in metals correlate to the material's mechanical properties. Reproducing representative volume elements for further analysis of deformation and failure in metals, like 316L stainless steel, is particularly important due to their wide use in manufacturing goods today. Two approaches, initially created for video games, were considered for the procedural generation of representative grain microstructures. The first is the Wave Function Collapse (WFC) algorithm, and the second is constraint propagation and probabilistic inference through Markov Junior, a free and open-source software. This study aimed to investigate these two algorithms' effectiveness in using reference electron backscatter diffraction (EBSD) maps and recreating a statistically similar one that could be used in further research. It utilized two stainless steel EBSD maps as references to test both algorithms. First, the WFC algorithm was too constricting and, thus, incapable of producing images that resembled EBSDs. The second, MarkovJunior, was much more effective in creating a Voronoi tessellation that could be used to create an EBSD map in Python. When comparing the results between the reference and the generated EBSD, we discovered that the orientation and volume fractions were extremely similar. With the study, it was concluded that MarkovJunior is an effective machine learning tool that can reproduce representative grain microstructures.

We generalize the Hamiltonian Monte Carlo algorithm with a stack of neural network layers and evaluate its ability to sample from different topologies in a two dimensional lattice gauge theory. We demonstrate that our model is able to successfully mix between modes of different topologies, significantly reducing the computational cost required to generated independent gauge field configurations. Our implementation is available at https://github.com/saforem2/l2hmc-qcd .

Obesity, the leading cause of many non-communicable diseases, occurs mainly for eating more than our body requirements and lack of proper activity. So, being healthy requires heathy diet plans, especially for patients with comorbidities. But it is difficult to figure out the exact quantity of each nutrient because nutrients requirement varies based on physical and disease conditions. In our study we proposed a novel machine learning based system to predict the amount of nutrients one individual requires for being healthy. We applied different machine learning algorithms: linear regression, support vector machine (SVM), decision tree, random forest, XGBoost, LightGBM on fluid and 3 other major micronutrients: carbohydrate, protein, fat consumption prediction. We achieved high accuracy with low root mean square error (RMSE) by using linear regression in fluid prediction, random forest in carbohydrate prediction and LightGBM in protein and fat prediction. We believe our diet recommender system, OBESEYE, is the only of its kind which recommends diet with the consideration of comorbidities and physical conditions and promote encouragement to get rid of obesity.

Integral field units (IFU) have extended our knowledge of galactic properties to kpc (or, sometimes, even smaller) patches of galaxies. These scales are where the physics driving galaxy evolution (feedback, chemical enrichment, etc.) take place. Quantifying the spatially-resolved properties of galaxies, both observationally and theoretically, is therefore critical to our understanding of galaxy evolution. To this end, we investigate spatially-resolved scaling relations within central galaxies (M_star>10^{9.0}) at z=0 in IllustrisTNG. We examine both the resolved star-forming main sequence (rSFMS) and the resolved mass-metallicity relation (rMZR) using 1~{rm kpc}times1~{rm kpc} maps of galaxies. We find that the rSFMS in IllustrisTNG is well-described by a power-law, but has some dependence on the host galaxy's mass. Conversely, the rMZR for IllustrisTNG can be described by a single power-law at low stellar mass surface density that flattens at high surface densities and is independent of host galaxy mass. We find quantitative agreement in both the rSFMS and rMZR with recent IFU observational campaigns. Furthermore, we argue that the rSFMS is an indirect result of the Schmidt-Kennicutt (SK) law and local gas fraction relation, which are both independent of host galaxy properties. Finally, we expand upon a localized leaky-box model to study the evolution of idealized spaxels and find that it provides a good description of these resolved relations. The degree of agreement, however, between idealized spaxels and simulated spaxels depends on the `net' outflow rate for the spaxel, and the observed scaling relations indicate a preference for a low net outflow rate.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Deep neural networks such as AlphaFold and RoseTTAFold predict remarkably accurate structures of proteins compared to other algorithmic approaches. It is known that biologically small perturbations in the protein sequence do not lead to drastic changes in the protein structure. In this paper, we demonstrate that RoseTTAFold does not exhibit such a robustness despite its high accuracy, and biologically small perturbations for some input sequences result in radically different predicted protein structures. This raises the challenge of detecting when these predicted protein structures cannot be trusted. We define the robustness measure for the predicted structure of a protein sequence to be the inverse of the root-mean-square distance (RMSD) in the predicted structure and the structure of its adversarially perturbed sequence. We use adversarial attack methods to create adversarial protein sequences, and show that the RMSD in the predicted protein structure ranges from 0.119A to 34.162A when the adversarial perturbations are bounded by 20 units in the BLOSUM62 distance. This demonstrates very high variance in the robustness measure of the predicted structures. We show that the magnitude of the correlation (0.917) between our robustness measure and the RMSD between the predicted structure and the ground truth is high, that is, the predictions with low robustness measure cannot be trusted. This is the first paper demonstrating the susceptibility of RoseTTAFold to adversarial attacks.

We examine in detail the accuracy, efficiency and implementation issues that arise when a simplified code structure is employed to evaluate the partition function of the two-dimensional square Ising model on periodic lattices though repeated tensor contractions.

The conventional evaluation protocols on machine learning models rely heavily on a labeled, i.i.d-assumed testing dataset, which is not often present in real world applications. The Automated Model Evaluation (AutoEval) shows an alternative to this traditional workflow, by forming a proximal prediction pipeline of the testing performance without the presence of ground-truth labels. Despite its recent successes, the AutoEval frameworks still suffer from an overconfidence issue, substantial storage and computational cost. In that regard, we propose a novel measure -- Meta-Distribution Energy (MDE) -- that allows the AutoEval framework to be both more efficient and effective. The core of the MDE is to establish a meta-distribution statistic, on the information (energy) associated with individual samples, then offer a smoother representation enabled by energy-based learning. We further provide our theoretical insights by connecting the MDE with the classification loss. We provide extensive experiments across modalities, datasets and different architectural backbones to validate MDE's validity, together with its superiority compared with prior approaches. We also prove MDE's versatility by showing its seamless integration with large-scale models, and easy adaption to learning scenarios with noisy- or imbalanced- labels. Code and data are available: https://github.com/pengr/Energy_AutoEval

This research presents a comprehensive approach to predicting the duration of traffic incidents and classifying them as short-term or long-term across the Sydney Metropolitan Area. Leveraging a dataset that encompasses detailed records of traffic incidents, road network characteristics, and socio-economic indicators, we train and evaluate a variety of advanced machine learning models including Gradient Boosted Decision Trees (GBDT), Random Forest, LightGBM, and XGBoost. The models are assessed using Root Mean Square Error (RMSE) for regression tasks and F1 score for classification tasks. Our experimental results demonstrate that XGBoost and LightGBM outperform conventional models with XGBoost achieving the lowest RMSE of 33.7 for predicting incident duration and highest classification F1 score of 0.62 for a 30-minute duration threshold. For classification, the 30-minute threshold balances performance with 70.84% short-term duration classification accuracy and 62.72% long-term duration classification accuracy. Feature importance analysis, employing both tree split counts and SHAP values, identifies the number of affected lanes, traffic volume, and types of primary and secondary vehicles as the most influential features. The proposed methodology not only achieves high predictive accuracy but also provides stakeholders with vital insights into factors contributing to incident durations. These insights enable more informed decision-making for traffic management and response strategies. The code is available by the link: https://github.com/Future-Mobility-Lab/SydneyIncidents

Air pollution constitutes a global problem of paramount importance that affects not only human health, but also the environment. The existence of spatial and temporal data regarding the concentrations of pollutants is crucial for performing air pollution studies and monitor emissions. However, although observation data presents great temporal coverage, the number of stations is very limited and they are usually built in more populated areas. The main objective of this work is to create models capable of inferring pollutant concentrations in locations where no observation data exists. A machine learning model, more specifically the random forest model, was developed for predicting concentrations in the Iberian Peninsula in 2019 for five selected pollutants: NO_2, O_3 SO_2, PM10, and PM2.5. Model features include satellite measurements, meteorological variables, land use classification, temporal variables (month, day of year), and spatial variables (latitude, longitude, altitude). The models were evaluated using various methods, including station 10-fold cross-validation, in which in each fold observations from 10\% of the stations are used as testing data and the rest as training data. The R^2, RMSE and mean bias were determined for each model. The NO_2 and O_3 models presented good values of R^2, 0.5524 and 0.7462, respectively. However, the SO_2, PM10, and PM2.5 models performed very poorly in this regard, with R^2 values of -0.0231, 0.3722, and 0.3303, respectively. All models slightly overestimated the ground concentrations, except the O_3 model. All models presented acceptable cross-validation RMSE, except the O_3 and PM10 models where the mean value was a little higher (12.5934 mu g/m^3 and 10.4737 mu g/m^3, respectively).

Matbench Discovery simulates the deployment of machine learning (ML) energy models in a high-throughput search for stable inorganic crystals. We address the disconnect between (i) thermodynamic stability and formation energy and (ii) in-domain vs out-of-distribution performance. Alongside this paper, we publish a Python package to aid with future model submissions and a growing online leaderboard with further insights into trade-offs between various performance metrics. To answer the question which ML methodology performs best at materials discovery, our initial release explores a variety of models including random forests, graph neural networks (GNN), one-shot predictors, iterative Bayesian optimizers and universal interatomic potentials (UIP). Ranked best-to-worst by their test set F1 score on thermodynamic stability prediction, we find CHGNet > M3GNet > MACE > ALIGNN > MEGNet > CGCNN > CGCNN+P > Wrenformer > BOWSR > Voronoi tessellation fingerprints with random forest. The top 3 models are UIPs, the winning methodology for ML-guided materials discovery, achieving F1 scores of ~0.6 for crystal stability classification and discovery acceleration factors (DAF) of up to 5x on the first 10k most stable predictions compared to dummy selection from our test set. We also highlight a sharp disconnect between commonly used global regression metrics and more task-relevant classification metrics. Accurate regressors are susceptible to unexpectedly high false-positive rates if those accurate predictions lie close to the decision boundary at 0 eV/atom above the convex hull where most materials are. Our results highlight the need to focus on classification metrics that actually correlate with improved stability hit rate.

In this work, we propose a framework that constructs reduced order models for nonlinear structural mechanics in a nonintrusive fashion, and can handle large scale simulations. We identify three steps that are carried out separately in time, and possibly on different devices: (i) the production of high-fidelity solutions by a commercial software, (ii) the offline stage of the model reduction and (iii) the online stage where the reduced order model is exploited. The nonintrusivity assumes that only the displacement field solution is known, and relies on operations on simulation data during the offline phase by using an in-house code. The compatibility with a new commercial code only needs the implementation of a routine converting the mesh and result format into our in-house data format. The nonintrusive capabilities of the framework are demonstrated on numerical experiments using commercial versions of the finite element softwares Zset and Ansys Mechanical. The nonlinear constitutive equations are evaluated by using the same external plugins as for Zset or Ansys Mechanical. The large scale simulations are handled using domain decomposition and parallel computing with distributed memory. The features and performances of the framework are evaluated on two numerical applications involving elastoviscoplastic materials: the second one involves a model of high-pressure blade, where the framework is used to extrapolate cyclic loadings in 6.5 hours, whereas the reference high-fidelity computation would take 9.5 days.

We report a flexible multi-modal mechanics language model, MeLM, applied to solve various nonlinear forward and inverse problems, that can deal with a set of instructions, numbers and microstructure data. The framework is applied to various examples including bio-inspired hierarchical honeycomb design, carbon nanotube mechanics, and protein unfolding. In spite of the flexible nature of the model-which allows us to easily incorporate diverse materials, scales, and mechanical features-it performs well across disparate forward and inverse tasks. Based on an autoregressive attention-model, MeLM effectively represents a large multi-particle system consisting of hundreds of millions of neurons, where the interaction potentials are discovered through graph-forming self-attention mechanisms that are then used to identify relationships from emergent structures, while taking advantage of synergies discovered in the training data. We show that the model can solve complex degenerate mechanics design problems and determine novel material architectures across a range of hierarchical levels, providing an avenue for materials discovery and analysis. Looking beyond the demonstrations reported in this paper, we discuss other opportunities in applied mechanics and general considerations about the use of large language models in modeling, design, and analysis that can span a broad spectrum of material properties from mechanical, thermal, optical, to electronic.

Generative models have advanced significantly in sampling material systems with continuous variables, such as atomistic structures. However, their application to discrete variables, like atom types or spin states, remains underexplored. In this work, we introduce a Boltzmann generator built on discrete flow matching, specifically tailored for systems with discrete phase-space coordinates (e.g., the Ising model or crystalline compounds). This approach enables a single model to sample free energy surfaces over a wide temperature range with minimal training overhead. In addition, the model generation is scalable to larger lattice sizes than those in the training set. We demonstrate the effectiveness of our approach on the 2D Ising model, showing efficient and reliable free energy sampling. This framework provides a scalable and computationally efficient solution for discrete coordinate systems and can be extended to sample the alchemical degrees of freedom in crystalline compounds.

Rayleigh-B\'enard convection (RBC) is a recurrent phenomenon in several industrial and geoscience flows and a well-studied system from a fundamental fluid-mechanics viewpoint. However, controlling RBC, for example by modulating the spatial distribution of the bottom-plate heating in the canonical RBC configuration, remains a challenging topic for classical control-theory methods. In the present work, we apply deep reinforcement learning (DRL) for controlling RBC. We show that effective RBC control can be obtained by leveraging invariant multi-agent reinforcement learning (MARL), which takes advantage of the locality and translational invariance inherent to RBC flows inside wide channels. The MARL framework applied to RBC allows for an increase in the number of control segments without encountering the curse of dimensionality that would result from a naive increase in the DRL action-size dimension. This is made possible by the MARL ability for re-using the knowledge generated in different parts of the RBC domain. We show in a case study that MARL DRL is able to discover an advanced control strategy that destabilizes the spontaneous RBC double-cell pattern, changes the topology of RBC by coalescing adjacent convection cells, and actively controls the resulting coalesced cell to bring it to a new stable configuration. This modified flow configuration results in reduced convective heat transfer, which is beneficial in several industrial processes. Therefore, our work both shows the potential of MARL DRL for controlling large RBC systems, as well as demonstrates the possibility for DRL to discover strategies that move the RBC configuration between different topological configurations, yielding desirable heat-transfer characteristics. These results are useful for both gaining further understanding of the intrinsic properties of RBC, as well as for developing industrial applications.

Inspired by the latest developments in multilevel Monte Carlo (MLMC) methods and randomised sketching for linear algebra problems we propose a MLMC estimator for real-time processing of matrix structured random data. Our algorithm is particularly effective in handling high-dimensional inner products and matrix multiplication, in applications of image analysis and large-scale supervised learning.

Introduction: The paper addresses the challenging problem of predicting the short-term realized volatility of the Bitcoin price using order flow information. The inherent stochastic nature and anti-persistence of price pose difficulties in accurate prediction. Methods: To address this, we propose a method that transforms order flow data over a fixed time interval (snapshots) into images. The order flow includes trade sizes, trade directions, and limit order book, and is mapped into image colour channels. These images are then used to train both a simple 3-layer Convolutional Neural Network (CNN) and more advanced ResNet-18 and ConvMixer, with additionally supplementing them with hand-crafted features. The models are evaluated against classical GARCH, Multilayer Perceptron trained on raw data, and a naive guess method that considers current volatility as a prediction. Results: The experiments are conducted using price data from January 2021 and evaluate model performance in terms of root mean square error (RMSPE). The results show that our order flow representation with a CNN as a predictive model achieves the best performance, with an RMSPE of 0.85+/-1.1 for the model with aggregated features and 1.0+/-1.4 for the model without feature supplementation. ConvMixer with feature supplementation follows closely. In comparison, the RMSPE for the naive guess method was 1.4+/-3.0.

This paper provides the first tight convergence analyses for RMSProp and Adam in non-convex optimization under the most relaxed assumptions of coordinate-wise generalized smoothness and affine noise variance. We first analyze RMSProp, which is a special case of Adam with adaptive learning rates but without first-order momentum. Specifically, to solve the challenges due to dependence among adaptive update, unbounded gradient estimate and Lipschitz constant, we demonstrate that the first-order term in the descent lemma converges and its denominator is upper bounded by a function of gradient norm. Based on this result, we show that RMSProp with proper hyperparameters converges to an epsilon-stationary point with an iteration complexity of mathcal O(epsilon^{-4}). We then generalize our analysis to Adam, where the additional challenge is due to a mismatch between the gradient and first-order momentum. We develop a new upper bound on the first-order term in the descent lemma, which is also a function of the gradient norm. We show that Adam with proper hyperparameters converges to an epsilon-stationary point with an iteration complexity of mathcal O(epsilon^{-4}). Our complexity results for both RMSProp and Adam match with the complexity lower bound established in arjevani2023lower.

We present GeoGrid-Bench, a benchmark designed to evaluate the ability of foundation models to understand geo-spatial data in the grid structure. Geo-spatial datasets pose distinct challenges due to their dense numerical values, strong spatial and temporal dependencies, and unique multimodal representations including tabular data, heatmaps, and geographic visualizations. To assess how foundation models can support scientific research in this domain, GeoGrid-Bench features large-scale, real-world data covering 16 climate variables across 150 locations and extended time frames. The benchmark includes approximately 3,200 question-answer pairs, systematically generated from 8 domain expert-curated templates to reflect practical tasks encountered by human scientists. These range from basic queries at a single location and time to complex spatiotemporal comparisons across regions and periods. Our evaluation reveals that vision-language models perform best overall, and we provide a fine-grained analysis of the strengths and limitations of different foundation models in different geo-spatial tasks. This benchmark offers clearer insights into how foundation models can be effectively applied to geo-spatial data analysis and used to support scientific research.

Recent studies have demonstrated the effectiveness of LLM test-time scaling. However, existing approaches to incentivize LLMs' deep thinking abilities generally require large-scale data or significant training efforts. Meanwhile, it remains unclear how to improve the thinking abilities of less powerful base models. In this work, we introduce S^2R, an efficient framework that enhances LLM reasoning by teaching models to self-verify and self-correct during inference. Specifically, we first initialize LLMs with iterative self-verification and self-correction behaviors through supervised fine-tuning on carefully curated data. The self-verification and self-correction skills are then further strengthened by both outcome-level and process-level reinforcement learning, with minimized resource requirements, enabling the model to adaptively refine its reasoning process during inference. Our results demonstrate that, with only 3.1k self-verifying and self-correcting behavior initialization samples, Qwen2.5-math-7B achieves an accuracy improvement from 51.0\% to 81.6\%, outperforming models trained on an equivalent amount of long-CoT distilled data. Extensive experiments and analysis based on three base models across both in-domain and out-of-domain benchmarks validate the effectiveness of S^2R. Our code and data are available at https://github.com/NineAbyss/S2R.

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.

We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.

Diffusion models excel in high-quality generation but suffer from slow inference due to iterative sampling. While recent methods have successfully transformed diffusion models into one-step generators, they neglect model size reduction, limiting their applicability in compute-constrained scenarios. This paper aims to develop small, efficient one-step diffusion models based on the powerful rectified flow framework, by exploring joint compression of inference steps and model size. The rectified flow framework trains one-step generative models using two operations, reflow and distillation. Compared with the original framework, squeezing the model size brings two new challenges: (1) the initialization mismatch between large teachers and small students during reflow; (2) the underperformance of naive distillation on small student models. To overcome these issues, we propose Annealing Reflow and Flow-Guided Distillation, which together comprise our SlimFlow framework. With our novel framework, we train a one-step diffusion model with an FID of 5.02 and 15.7M parameters, outperforming the previous state-of-the-art one-step diffusion model (FID=6.47, 19.4M parameters) on CIFAR10. On ImageNet 64times64 and FFHQ 64times64, our method yields small one-step diffusion models that are comparable to larger models, showcasing the effectiveness of our method in creating compact, efficient one-step diffusion models.

We investigate spectral features of bottomonium at high temperature, in particular the thermal mass shift and width of ground state S-wave and P-wave state. We employ and compare a range of methods for determining these features from lattice NRQCD correlators, including direct correlator analyses (multi-exponential fits and moments of spectral functions), linear methods (Backus-Gilbert, Tikhonov and HLT methods), and Bayesian methods for spectral function reconstruction (MEM and BR). We comment on the reliability and limitations of the various methods.

Molecular docking is an important tool for structure-based drug design, accelerating the efficiency of drug development. Complex and dynamic binding processes between proteins and small molecules require searching and sampling over a wide spatial range. Traditional docking by searching for possible binding sites and conformations is computationally complex and results poorly under blind docking. Quantum-inspired algorithms combining quantum properties and annealing show great advantages in solving combinatorial optimization problems. Inspired by this, we achieve an improved in blind docking by using quantum-inspired combined with gradients learned by deep learning in the encoded molecular space. Numerical simulation shows that our method outperforms traditional docking algorithms and deep learning-based algorithms over 10\%. Compared to the current state-of-the-art deep learning-based docking algorithm DiffDock, the success rate of Top-1 (RMSD<2) achieves an improvement from 33\% to 35\% in our same setup. In particular, a 6\% improvement is realized in the high-precision region(RMSD<1) on molecules data unseen in DiffDock, which demonstrates the well-generalized of our method.

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable "perturb-then-diagonalize" (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

Grid-based structures are commonly used to encode explicit features for graphics primitives such as images, signed distance functions (SDF), and neural radiance fields (NeRF) due to their simple implementation. However, in n-dimensional space, calculating the value of a sampled point requires interpolating the values of its 2^n neighboring vertices. The exponential scaling with dimension leads to significant computational overheads. To address this issue, we propose a simplex-based approach for encoding graphics primitives. The number of vertices in a simplex-based structure increases linearly with dimension, making it a more efficient and generalizable alternative to grid-based representations. Using the non-axis-aligned simplicial structure property, we derive and prove a coordinate transformation, simplicial subdivision, and barycentric interpolation scheme for efficient sampling, which resembles transformation procedures in the simplex noise algorithm. Finally, we use hash tables to store multiresolution features of all interest points in the simplicial grid, which are passed into a tiny fully connected neural network to parameterize graphics primitives. We implemented a detailed simplex-based structure encoding algorithm in C++ and CUDA using the methods outlined in our approach. In the 2D image fitting task, the proposed method is capable of fitting a giga-pixel image with 9.4% less time compared to the baseline method proposed by instant-ngp, while maintaining the same quality and compression rate. In the volumetric rendering setup, we observe a maximum 41.2% speedup when the samples are dense enough.

We present rStar-Math to demonstrate that small language models (SLMs) can rival or even surpass the math reasoning capability of OpenAI o1, without distillation from superior models. rStar-Math achieves this by exercising "deep thinking" through Monte Carlo Tree Search (MCTS), where a math policy SLM performs test-time search guided by an SLM-based process reward model. rStar-Math introduces three innovations to tackle the challenges in training the two SLMs: (1) a novel code-augmented CoT data sythesis method, which performs extensive MCTS rollouts to generate step-by-step verified reasoning trajectories used to train the policy SLM; (2) a novel process reward model training method that avoids na\"ive step-level score annotation, yielding a more effective process preference model (PPM); (3) a self-evolution recipe in which the policy SLM and PPM are built from scratch and iteratively evolved to improve reasoning capabilities. Through 4 rounds of self-evolution with millions of synthesized solutions for 747k math problems, rStar-Math boosts SLMs' math reasoning to state-of-the-art levels. On the MATH benchmark, it improves Qwen2.5-Math-7B from 58.8% to 90.0% and Phi3-mini-3.8B from 41.4% to 86.4%, surpassing o1-preview by +4.5% and +0.9%. On the USA Math Olympiad (AIME), rStar-Math solves an average of 53.3% (8/15) of problems, ranking among the top 20% the brightest high school math students. Code and data will be available at https://github.com/microsoft/rStar.

It has recently been conjectured that neural network solution sets reachable via stochastic gradient descent (SGD) are convex, considering permutation invariances (Entezari et al., 2022). This means that a linear path can connect two independent solutions with low loss, given the weights of one of the models are appropriately permuted. However, current methods to test this theory often require very wide networks to succeed. In this work, we conjecture that more generally, the SGD solution set is a "star domain" that contains a "star model" that is linearly connected to all the other solutions via paths with low loss values, modulo permutations. We propose the Starlight algorithm that finds a star model of a given learning task. We validate our claim by showing that this star model is linearly connected with other independently found solutions. As an additional benefit of our study, we demonstrate better uncertainty estimates on the Bayesian Model Averaging over the obtained star domain. Further, we demonstrate star models as potential substitutes for model ensembles. Our code is available at https://github.com/aktsonthalia/starlight.

We propose an adaptive step size with an energy approach for a suitable class of preconditioned gradient descent methods. We focus on settings where the preconditioning is applied to address the constraints in optimization problems, such as the Hessian-Riemannian and natural gradient descent methods. More specifically, we incorporate these preconditioned gradient descent algorithms in the recently introduced Adaptive Energy Gradient Descent (AEGD) framework. In particular, we discuss theoretical results on the unconditional energy-stability and convergence rates across three classes of objective functions. Furthermore, our numerical results demonstrate excellent performance of the proposed method on several test bed optimization problems.

Modularization is a cornerstone of computer science, abstracting complex functions into atomic building blocks. In this paper, we introduce a new level of modularization by abstracting generative models into atomic generative modules. Analogous to fractals in mathematics, our method constructs a new type of generative model by recursively invoking atomic generative modules, resulting in self-similar fractal architectures that we call fractal generative models. As a running example, we instantiate our fractal framework using autoregressive models as the atomic generative modules and examine it on the challenging task of pixel-by-pixel image generation, demonstrating strong performance in both likelihood estimation and generation quality. We hope this work could open a new paradigm in generative modeling and provide a fertile ground for future research. Code is available at https://github.com/LTH14/fractalgen.

Following the success of the in-context learning paradigm in large-scale language and computer vision models, the recently emerging field of in-context reinforcement learning is experiencing a rapid growth. However, its development has been held back by the lack of challenging benchmarks, as all the experiments have been carried out in simple environments and on small-scale datasets. We present XLand-100B, a large-scale dataset for in-context reinforcement learning based on the XLand-MiniGrid environment, as a first step to alleviate this problem. It contains complete learning histories for nearly 30,000 different tasks, covering 100B transitions and 2.5B episodes. It took 50,000 GPU hours to collect the dataset, which is beyond the reach of most academic labs. Along with the dataset, we provide the utilities to reproduce or expand it even further. With this substantial effort, we aim to democratize research in the rapidly growing field of in-context reinforcement learning and provide a solid foundation for further scaling. The code is open-source and available under Apache 2.0 licence at https://github.com/dunno-lab/xland-minigrid-datasets.

When considering a model architecture, there are several ways to reduce its memory footprint. Historically, popular approaches included selecting smaller architectures and creating sparse networks through pruning. More recently, randomized parameter-sharing (RPS) methods have gained traction for model compression at start of training. In this paper, we comprehensively assess the trade-off between memory and accuracy across RPS, pruning techniques, and building smaller models. Our findings demonstrate that RPS, which is both data and model-agnostic, consistently outperforms/matches smaller models and all moderately informed pruning strategies, such as MAG, SNIP, SYNFLOW, and GRASP, across the entire compression range. This advantage becomes particularly pronounced in higher compression scenarios. Notably, even when compared to highly informed pruning techniques like Lottery Ticket Rewinding (LTR), RPS exhibits superior performance in high compression settings. This points out inherent capacity advantage that RPS enjoys over sparse models. Theoretically, we establish RPS as a superior technique in terms of memory-efficient representation when compared to pruning for linear models. This paper argues in favor of paradigm shift towards RPS based models. During our rigorous evaluation of RPS, we identified issues in the state-of-the-art RPS technique ROAST, specifically regarding stability (ROAST's sensitivity to initialization hyperparameters, often leading to divergence) and Pareto-continuity (ROAST's inability to recover the accuracy of the original model at zero compression). We provably address both of these issues. We refer to the modified RPS, which incorporates our improvements, as STABLE-RPS.

It is widely acknowledged that large models have the potential to deliver superior performance across a broad range of domains. Despite the remarkable progress made in the field of machine learning systems research, which has enabled the development and exploration of large models, such abilities remain confined to a small group of advanced users and industry leaders, resulting in an implicit technical barrier for the wider community to access and leverage these technologies. In this paper, we introduce PyTorch Fully Sharded Data Parallel (FSDP) as an industry-grade solution for large model training. FSDP has been closely co-designed with several key PyTorch core components including Tensor implementation, dispatcher system, and CUDA memory caching allocator, to provide non-intrusive user experiences and high training efficiency. Additionally, FSDP natively incorporates a range of techniques and settings to optimize resource utilization across a variety of hardware configurations. The experimental results demonstrate that FSDP is capable of achieving comparable performance to Distributed Data Parallel while providing support for significantly larger models with near-linear scalability in terms of TFLOPS.

The generation of plausible crystal structures is often the first step in predicting the structure and properties of a material from its chemical composition. Quickly generating and predicting inorganic crystal structures is important for the discovery of new materials, which can target applications such as energy or electronic devices. However, most current methods for crystal structure prediction are computationally expensive, slowing the pace of innovation. Seeding structure prediction algorithms with quality generated candidates can overcome a major bottleneck. Here, we introduce CrystaLLM, a methodology for the versatile generation of crystal structures, based on the autoregressive large language modeling (LLM) of the Crystallographic Information File (CIF) format. Trained on millions of CIF files, CrystaLLM focuses on modeling crystal structures through text. CrystaLLM can produce plausible crystal structures for a wide range of inorganic compounds unseen in training, as demonstrated by ab initio simulations. The integration with predictors of formation energy permits the use of a Monte Carlo Tree Search algorithm to improve the generation of meaningful structures. Our approach challenges conventional representations of crystals, and demonstrates the potential of LLMs for learning effective 'world models' of crystal chemistry, which will lead to accelerated discovery and innovation in materials science.

Machine learning is a prevalent approach to tame the complexity of design space exploration for domain-specific architectures. Using ML for design space exploration poses challenges. First, it's not straightforward to identify the suitable algorithm from an increasing pool of ML methods. Second, assessing the trade-offs between performance and sample efficiency across these methods is inconclusive. Finally, lack of a holistic framework for fair, reproducible, and objective comparison across these methods hinders progress of adopting ML-aided architecture design space exploration and impedes creating repeatable artifacts. To mitigate these challenges, we introduce ArchGym, an open-source gym and easy-to-extend framework that connects diverse search algorithms to architecture simulators. To demonstrate utility, we evaluate ArchGym across multiple vanilla and domain-specific search algorithms in designing custom memory controller, deep neural network accelerators, and custom SoC for AR/VR workloads, encompassing over 21K experiments. Results suggest that with unlimited samples, ML algorithms are equally favorable to meet user-defined target specification if hyperparameters are tuned; no solution is necessarily better than another (e.g., reinforcement learning vs. Bayesian methods). We coin the term hyperparameter lottery to describe the chance for a search algorithm to find an optimal design provided meticulously selected hyperparameters. The ease of data collection and aggregation in ArchGym facilitates research in ML-aided architecture design space exploration. As a case study, we show this advantage by developing a proxy cost model with an RMSE of 0.61% that offers a 2,000-fold reduction in simulation time. Code and data for ArchGym is available at https://bit.ly/ArchGym.

Robust Markov Decision Processes (MDPs) and risk-sensitive MDPs are both powerful tools for making decisions in the presence of uncertainties. Previous efforts have aimed to establish their connections, revealing equivalences in specific formulations. This paper introduces a new formulation for risk-sensitive MDPs, which assesses risk in a slightly different manner compared to the classical Markov risk measure (Ruszczy\'nski 2010), and establishes its equivalence with a class of regularized robust MDP (RMDP) problems, including the standard RMDP as a special case. Leveraging this equivalence, we further derive the policy gradient theorem for both problems, proving gradient domination and global convergence of the exact policy gradient method under the tabular setting with direct parameterization. This forms a sharp contrast to the Markov risk measure, known to be potentially non-gradient-dominant (Huang et al. 2021). We also propose a sample-based offline learning algorithm, namely the robust fitted-Z iteration (RFZI), for a specific regularized RMDP problem with a KL-divergence regularization term (or equivalently the risk-sensitive MDP with an entropy risk measure). We showcase its streamlined design and less stringent assumptions due to the equivalence and analyze its sample complexity.

In this paper, we present a complete refractive Structure-from-Motion (RSfM) framework for underwater 3D reconstruction using refractive camera setups (for both, flat- and dome-port underwater housings). Despite notable achievements in refractive multi-view geometry over the past decade, a robust, complete and publicly available solution for such tasks is not available at present, and often practical applications have to resort to approximating refraction effects by the intrinsic (distortion) parameters of a pinhole camera model. To fill this gap, we have integrated refraction considerations throughout the entire SfM process within the state-of-the-art, open-source SfM framework COLMAP. Numerical simulations and reconstruction results on synthetically generated but photo-realistic images with ground truth validate that enabling refraction does not compromise accuracy or robustness as compared to in-air reconstructions. Finally, we demonstrate the capability of our approach for large-scale refractive scenarios using a dataset consisting of nearly 6000 images. The implementation is released as open-source at: https://cau-git.rz.uni-kiel.de/inf-ag-koeser/colmap_underwater.

We prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Unlike previous analyses, our approach accelerates any stochastic gradient method which makes sufficient progress in expectation. The proof, which proceeds using the estimating sequences framework, applies to both convex and strongly convex functions and is easily specialized to accelerated SGD under the strong growth condition. In this special case, our analysis reduces the dependence on the strong growth constant from rho to rho as compared to prior work. This improvement is comparable to a square-root of the condition number in the worst case and address criticism that guarantees for stochastic acceleration could be worse than those for SGD.

Rydberg atom arrays have emerged as a powerful platform to simulate a number of exotic quantum ground states and phase transitions. To verify these capabilities numerically, we develop a versatile quantum Monte Carlo sampling technique which operates in the reduced Hilbert space generated by enforcing the constraint of a Rydberg blockade. We use the framework of stochastic series expansion and show that in the restricted space, the configuration space of operator strings can be understood as a hard rod gas in d+1 dimensions. We use this mapping to develop cluster algorithms which can be visualized as various non-local movements of rods. We study the efficiency of each of our updates individually and collectively. To elucidate the utility of the algorithm, we show that it can efficiently generate the phase diagram of a Rydberg atom array, to temperatures much smaller than all energy scales involved, on a Kagom\'e link lattice. This is of broad interest as the presence of a Z_2 spin liquid has been hypothesized recently.

Diffusion-based language models offer a compelling alternative to autoregressive (AR) models by enabling parallel and controllable generation. Among this family of models, Masked Diffusion Models (MDMs) achieve the strongest performance but still underperform AR models in perplexity and lack key inference-time efficiency features--most notably, KV caching. In this work, we introduce Eso-LMs, a new family of models that fuses AR and MDM paradigms, enabling smooth interpolation between their perplexities while overcoming their respective limitations. Eso-LMs set a new state of the art on standard language modeling benchmarks. Crucially, we are the **first to introduce KV caching for MDMs** while preserving parallel generation, significantly improving inference efficiency. Combined with an optimized sampling schedule, our method achieves up to **65x** faster inference than standard MDMs and **4x** faster inference than prior semi-autoregressive approaches. We provide the code and model checkpoints on the project page: [http://s-sahoo.github.io/Eso-LMs](http://s-sahoo.github.io/Eso-LMs)

Singly-TASE operators for the numerical solution of stiff differential equations were proposed by Calvo et al. in J.Sci. Comput. 2023 to reduce the computational cost of Runge-Kutta-TASE (RKTASE) methods when the involved linear systems are solved by some LU factorization. In this paper we propose a modification of these methods to improve the efficiency by considering different TASE operators for each stage of the Runge-Kutta. We prove that the resulting RKTASE methods are equivalent to W-methods (Steihaug and Wolfbrandt, Mathematics of Computation,1979) and this allows us to obtain the order conditions of the proposed Modified Singly-RKTASE methods (MSRKTASE) through the theory developed for the W-methods. We construct new MSRKTASE methods of order two and three and demonstrate their effectiveness through numerical experiments on both linear and nonlinear stiff systems. The results show that the MSRKTASE schemes significantly enhance efficiency and accuracy compared to previous Singly-RKTASE schemes.

Dynamic Mode Decomposition (DMD) is an equation-free method that aims at reconstructing the best linear fit from temporal datasets. In this paper, we show that DMD does not provide accurate approximation for datasets describing oscillatory dynamics, like spiral waves and relaxation oscillations, or spatio-temporal Turing instability. Inspired from the classical "divide and conquer" approach, we propose a piecewise version of DMD (pDMD) to overcome this problem. The main idea is to split the original dataset in N submatrices and then apply the exact (randomized) DMD method in each subset of the obtained partition. We describe the pDMD algorithm in detail and we introduce some error indicators to evaluate its performance when N is increased. Numerical experiments show that very accurate reconstructions are obtained by pDMD for datasets arising from time snapshots of some reaction-diffusion PDE systems, like the FitzHugh-Nagumo model, the lambda-omega system and the DIB morpho-chemical system for battery modeling.

Generative models trained on internet-scale data are capable of generating novel and realistic texts, images, and videos. A natural next question is whether these models can advance science, for example by generating novel stable materials. Traditionally, models with explicit structures (e.g., graphs) have been used in modeling structural relationships in scientific data (e.g., atoms and bonds in crystals), but generating structures can be difficult to scale to large and complex systems. Another challenge in generating materials is the mismatch between standard generative modeling metrics and downstream applications. For instance, common metrics such as the reconstruction error do not correlate well with the downstream goal of discovering stable materials. In this work, we tackle the scalability challenge by developing a unified crystal representation that can represent any crystal structure (UniMat), followed by training a diffusion probabilistic model on these UniMat representations. Our empirical results suggest that despite the lack of explicit structure modeling, UniMat can generate high fidelity crystal structures from larger and more complex chemical systems, outperforming previous graph-based approaches under various generative modeling metrics. To better connect the generation quality of materials to downstream applications, such as discovering novel stable materials, we propose additional metrics for evaluating generative models of materials, including per-composition formation energy and stability with respect to convex hulls through decomposition energy from Density Function Theory (DFT). Lastly, we show that conditional generation with UniMat can scale to previously established crystal datasets with up to millions of crystals structures, outperforming random structure search (the current leading method for structure discovery) in discovering new stable materials.

Classification of time series data is an important task for many application domains. One of the best existing methods for this task, in terms of accuracy and computation time, is MiniROCKET. In this work, we extend this approach to provide better global temporal encodings using hyperdimensional computing (HDC) mechanisms. HDC (also known as Vector Symbolic Architectures, VSA) is a general method to explicitly represent and process information in high-dimensional vectors. It has previously been used successfully in combination with deep neural networks and other signal processing algorithms. We argue that the internal high-dimensional representation of MiniROCKET is well suited to be complemented by the algebra of HDC. This leads to a more general formulation, HDC-MiniROCKET, where the original algorithm is only a special case. We will discuss and demonstrate that HDC-MiniROCKET can systematically overcome catastrophic failures of MiniROCKET on simple synthetic datasets. These results are confirmed by experiments on the 128 datasets from the UCR time series classification benchmark. The extension with HDC can achieve considerably better results on datasets with high temporal dependence without increasing the computational effort for inference.

Crystalline materials often exhibit a high level of symmetry. However, most generative models do not account for symmetry, but rather model each atom without any constraints on its position or element. We propose a generative model, Wyckoff Diffusion (WyckoffDiff), which generates symmetry-based descriptions of crystals. This is enabled by considering a crystal structure representation that encodes all symmetry, and we design a novel neural network architecture which enables using this representation inside a discrete generative model framework. In addition to respecting symmetry by construction, the discrete nature of our model enables fast generation. We additionally present a new metric, Fr\'echet Wrenformer Distance, which captures the symmetry aspects of the materials generated, and we benchmark WyckoffDiff against recently proposed generative models for crystal generation. Code is available online at https://github.com/httk/wyckoffdiff

We introduce a principled way of computing the Wasserstein distance between two distributions in a federated manner. Namely, we show how to estimate the Wasserstein distance between two samples stored and kept on different devices/clients whilst a central entity/server orchestrates the computations (again, without having access to the samples). To achieve this feat, we take advantage of the geometric properties of the Wasserstein distance -- in particular, the triangle inequality -- and that of the associated {\em geodesics}: our algorithm, FedWad (for Federated Wasserstein Distance), iteratively approximates the Wasserstein distance by manipulating and exchanging distributions from the space of geodesics in lieu of the input samples. In addition to establishing the convergence properties of FedWad, we provide empirical results on federated coresets and federate optimal transport dataset distance, that we respectively exploit for building a novel federated model and for boosting performance of popular federated learning algorithms.

Benchmarks that closely reflect downstream application scenarios are essential for the streamlined adoption of new research in tabular machine learning (ML). In this work, we examine existing tabular benchmarks and find two common characteristics of industry-grade tabular data that are underrepresented in the datasets available to the academic community. First, tabular data often changes over time in real-world deployment scenarios. This impacts model performance and requires time-based train and test splits for correct model evaluation. Yet, existing academic tabular datasets often lack timestamp metadata to enable such evaluation. Second, a considerable portion of datasets in production settings stem from extensive data acquisition and feature engineering pipelines. For each specific dataset, this can have a different impact on the absolute and relative number of predictive, uninformative, and correlated features, which in turn can affect model selection. To fill the aforementioned gaps in academic benchmarks, we introduce TabReD -- a collection of eight industry-grade tabular datasets covering a wide range of domains from finance to food delivery services. We assess a large number of tabular ML models in the feature-rich, temporally-evolving data setting facilitated by TabReD. We demonstrate that evaluation on time-based data splits leads to different methods ranking, compared to evaluation on random splits more common in academic benchmarks. Furthermore, on the TabReD datasets, MLP-like architectures and GBDT show the best results, while more sophisticated DL models are yet to prove their effectiveness.

Reinforcement learning has been applied in operation research and has shown promise in solving large combinatorial optimization problems. However, existing works focus on developing neural network architectures for certain problems. These works lack the flexibility to incorporate recent advances in reinforcement learning, as well as the flexibility of customizing model architectures for operation research problems. In this work, we analyze the end-to-end autoregressive models for vehicle routing problems and show that these models can benefit from the recent advances in reinforcement learning with a careful re-implementation of the model architecture. In particular, we re-implemented the Attention Model and trained it with Proximal Policy Optimization (PPO) in CleanRL, showing at least 8 times speed up in training time. We hereby introduce RLOR, a flexible framework for Deep Reinforcement Learning for Operation Research. We believe that a flexible framework is key to developing deep reinforcement learning models for operation research problems. The code of our work is publicly available at https://github.com/cpwan/RLOR.

Fine-tuning large pre-trained LLMs generally demands extensive GPU memory. Traditional first-order optimizers like SGD encounter substantial difficulties due to increased memory requirements from storing activations and gradients during both the forward and backward phases as the model size expands. Alternatively, zeroth-order (ZO) techniques can compute gradients using just forward operations, eliminating the need to store activations. Furthermore, by leveraging CPU capabilities, it's feasible to enhance both the memory and processing power available to a single GPU. We propose a novel framework, ZO2 (Zeroth-Order Offloading), for efficient zeroth-order fine-tuning of LLMs with only limited GPU memory. Our framework dynamically shifts model parameters between the CPU and GPU as required, optimizing computation flow and maximizing GPU usage by minimizing downtime. This integration of parameter adjustments with ZO's double forward operations reduces unnecessary data movement, enhancing the fine-tuning efficacy. Additionally, our framework supports an innovative low-bit precision approach in AMP mode to streamline data exchanges between the CPU and GPU. Employing this approach allows us to fine-tune extraordinarily large models, such as the OPT-175B with more than 175 billion parameters, on a mere 18GB GPU--achievements beyond the reach of traditional methods. Moreover, our framework achieves these results with almost no additional time overhead and absolutely no accuracy loss compared to standard zeroth-order methods. ZO2's code has been open-sourced in https://github.com/liangyuwang/zo2.

The Maximal Update Parametrization (muP) aims to make the optimal hyperparameters (HPs) of a model independent of its size, allowing them to be swept using a cheap proxy model rather than the full-size target model. We present a new scheme, u-muP, which improves upon muP by combining it with Unit Scaling, a method for designing models that makes them easy to train in low-precision. The two techniques have a natural affinity: muP ensures that the scale of activations is independent of model size, and Unit Scaling ensures that activations, weights and gradients begin training with a scale of one. This synthesis opens the door to a simpler scheme, whose default values are near-optimal. This in turn facilitates a more efficient sweeping strategy, with u-muP models reaching a lower loss than comparable muP models and working out-of-the-box in FP8.

Current Vehicle-to-Everything (V2X) systems have significantly enhanced 3D object detection using LiDAR and camera data. However, these methods suffer from performance degradation in adverse weather conditions. The weather-robust 4D radar provides Doppler and additional geometric information, raising the possibility of addressing this challenge. To this end, we present V2X-R, the first simulated V2X dataset incorporating LiDAR, camera, and 4D radar. V2X-R contains 12,079 scenarios with 37,727 frames of LiDAR and 4D radar point clouds, 150,908 images, and 170,859 annotated 3D vehicle bounding boxes. Subsequently, we propose a novel cooperative LiDAR-4D radar fusion pipeline for 3D object detection and implement it with various fusion strategies. To achieve weather-robust detection, we additionally propose a Multi-modal Denoising Diffusion (MDD) module in our fusion pipeline. MDD utilizes weather-robust 4D radar feature as a condition to prompt the diffusion model to denoise noisy LiDAR features. Experiments show that our LiDAR-4D radar fusion pipeline demonstrates superior performance in the V2X-R dataset. Over and above this, our MDD module further improved the performance of basic fusion model by up to 5.73%/6.70% in foggy/snowy conditions with barely disrupting normal performance. The dataset and code will be publicly available at: https://github.com/ylwhxht/V2X-R.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

This paper presents a differentially private approach to Kaplan-Meier estimation that achieves accurate survival probability estimates while safeguarding individual privacy. The Kaplan-Meier estimator is widely used in survival analysis to estimate survival functions over time, yet applying it to sensitive datasets, such as clinical records, risks revealing private information. To address this, we introduce a novel algorithm that applies time-indexed Laplace noise, dynamic clipping, and smoothing to produce a privacy-preserving survival curve while maintaining the cumulative structure of the Kaplan-Meier estimator. By scaling noise over time, the algorithm accounts for decreasing sensitivity as fewer individuals remain at risk, while dynamic clipping and smoothing prevent extreme values and reduce fluctuations, preserving the natural shape of the survival curve. Our results, evaluated on the NCCTG lung cancer dataset, show that the proposed method effectively lowers root mean squared error (RMSE) and enhances accuracy across privacy budgets (epsilon). At epsilon = 10, the algorithm achieves an RMSE as low as 0.04, closely approximating non-private estimates. Additionally, membership inference attacks reveal that higher epsilon values (e.g., epsilon geq 6) significantly reduce influential points, particularly at higher thresholds, lowering susceptibility to inference attacks. These findings confirm that our approach balances privacy and utility, advancing privacy-preserving survival analysis.

This paper presents a new open-source high-fidelity dataset for Machine Learning (ML) containing 355 geometric variants of the Windsor body, to help the development and testing of ML surrogate models for external automotive aerodynamics. Each Computational Fluid Dynamics (CFD) simulation was run with a GPU-native high-fidelity Wall-Modeled Large-Eddy Simulations (WMLES) using a Cartesian immersed-boundary method using more than 280M cells to ensure the greatest possible accuracy. The dataset contains geometry variants that exhibits a wide range of flow characteristics that are representative of those observed on road-cars. The dataset itself contains the 3D time-averaged volume & boundary data as well as the geometry and force & moment coefficients. This paper discusses the validation of the underlying CFD methods as well as contents and structure of the dataset. To the authors knowledge, this represents the first, large-scale high-fidelity CFD dataset for the Windsor body with a permissive open-source license (CC-BY-SA).

Recent years have witnessed the rapid progress and broad application of diffusion probabilistic models (DPMs). Sampling from DPMs can be viewed as solving an ordinary differential equation (ODE). Despite the promising performance, the generation of DPMs usually consumes much time due to the large number of function evaluations (NFE). Though recent works have accelerated the sampling to around 20 steps with high-order solvers, the sample quality with less than 10 NFE can still be improved. In this paper, we propose a unified sampling framework (USF) to study the optional strategies for solver. Under this framework, we further reveal that taking different solving strategies at different timesteps may help further decrease the truncation error, and a carefully designed solver schedule has the potential to improve the sample quality by a large margin. Therefore, we propose a new sampling framework based on the exponential integral formulation that allows free choices of solver strategy at each step and design specific decisions for the framework. Moreover, we propose S^3, a predictor-based search method that automatically optimizes the solver schedule to get a better time-quality trade-off of sampling. We demonstrate that S^3 can find outstanding solver schedules which outperform the state-of-the-art sampling methods on CIFAR-10, CelebA, ImageNet, and LSUN-Bedroom datasets. Specifically, we achieve 2.69 FID with 10 NFE and 6.86 FID with 5 NFE on CIFAR-10 dataset, outperforming the SOTA method significantly. We further apply S^3 to Stable-Diffusion model and get an acceleration ratio of 2times, showing the feasibility of sampling in very few steps without retraining the neural network.

Dynamic Sparse Training (DST) methods achieve state-of-the-art results in sparse neural network training, matching the generalization of dense models while enabling sparse training and inference. Although the resulting models are highly sparse and theoretically less computationally expensive, achieving speedups with unstructured sparsity on real-world hardware is challenging. In this work, we propose a sparse-to-sparse DST method, Structured RigL (SRigL), to learn a variant of fine-grained structured N:M sparsity by imposing a constant fan-in constraint. Using our empirical analysis of existing DST methods at high sparsity, we additionally employ a neuron ablation method which enables SRigL to achieve state-of-the-art sparse-to-sparse structured DST performance on a variety of Neural Network (NN) architectures. We demonstrate reduced real-world timings on CPU for online inference -- 3.6x/2x faster at 90% sparsity than equivalent dense/unstructured sparse layers, respectively. Our source code is available at https://github.com/calgaryml/condensed-sparsity

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

State space models (SSM) have recently been shown to be very effective as a deep learning layer as a promising alternative to sequence models such as RNNs, CNNs, or Transformers. The first version to show this potential was the S4 model, which is particularly effective on tasks involving long-range dependencies by using a prescribed state matrix called the HiPPO matrix. While this has an interpretable mathematical mechanism for modeling long dependencies, it introduces a custom representation and algorithm that can be difficult to implement. On the other hand, a recent variant of S4 called DSS showed that restricting the state matrix to be fully diagonal can still preserve the performance of the original model when using a specific initialization based on approximating S4's matrix. This work seeks to systematically understand how to parameterize and initialize such diagonal state space models. While it follows from classical results that almost all SSMs have an equivalent diagonal form, we show that the initialization is critical for performance. We explain why DSS works mathematically, by showing that the diagonal restriction of S4's matrix surprisingly recovers the same kernel in the limit of infinite state dimension. We also systematically describe various design choices in parameterizing and computing diagonal SSMs, and perform a controlled empirical study ablating the effects of these choices. Our final model S4D is a simple diagonal version of S4 whose kernel computation requires just 2 lines of code and performs comparably to S4 in almost all settings, with state-of-the-art results for image, audio, and medical time-series domains, and averaging 85\% on the Long Range Arena benchmark.

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

We present GSPMD, an automatic, compiler-based parallelization system for common machine learning computations. It allows users to write programs in the same way as for a single device, then give hints through a few annotations on how to distribute tensors, based on which GSPMD will parallelize the computation. Its representation of partitioning is simple yet general, allowing it to express different or mixed paradigms of parallelism on a wide variety of models. GSPMD infers the partitioning for every operator based on limited user annotations, making it convenient to scale existing single-device programs. It solves several technical challenges for production usage, allowing GSPMD to achieve 50% to 62% compute utilization on up to 2048 Cloud TPUv3 cores for models with up to one trillion parameters.

Large language models (LLMs) such as GPT-3, OPT, and LLaMA have demonstrated remarkable accuracy in a wide range of tasks. However, training these models can incur significant expenses, often requiring tens of thousands of GPUs for months of continuous operation. Typically, this training is carried out in specialized GPU clusters equipped with homogeneous high-speed Remote Direct Memory Access (RDMA) network interface cards (NICs). The acquisition and maintenance of such dedicated clusters is challenging. Current LLM training frameworks, like Megatron-LM and Megatron-DeepSpeed, focus primarily on optimizing training within homogeneous cluster settings. In this paper, we introduce Holmes, a training framework for LLMs that employs thoughtfully crafted data and model parallelism strategies over the heterogeneous NIC environment. Our primary technical contribution lies in a novel scheduling method that intelligently allocates distinct computational tasklets in LLM training to specific groups of GPU devices based on the characteristics of their connected NICs. Furthermore, our proposed framework, utilizing pipeline parallel techniques, demonstrates scalability to multiple GPU clusters, even in scenarios without high-speed interconnects between nodes in distinct clusters. We conducted comprehensive experiments that involved various scenarios in the heterogeneous NIC environment. In most cases, our framework achieves performance levels close to those achievable with homogeneous RDMA-capable networks (InfiniBand or RoCE), significantly exceeding training efficiency within the pure Ethernet environment. Additionally, we verified that our framework outperforms other mainstream LLM frameworks under heterogeneous NIC environment in terms of training efficiency and can be seamlessly integrated with them.

In location estimation, we are given n samples from a known distribution f shifted by an unknown translation lambda, and want to estimate lambda as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cram\'er-Rao bound of error mathcal N(0, 1{nmathcal I}), where mathcal I is the Fisher information of f. However, the n required for convergence depends on f, and may be arbitrarily large. We build on the theory using smoothed estimators to bound the error for finite n in terms of mathcal I_r, the Fisher information of the r-smoothed distribution. As n to infty, r to 0 at an explicit rate and this converges to the Cram\'er-Rao bound. We (1) improve the prior work for 1-dimensional f to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

In this paper, we introduce a new class of score-based generative models (SGMs) designed to handle high-cardinality data distributions by leveraging concepts from mean-field theory. We present mean-field chaos diffusion models (MF-CDMs), which address the curse of dimensionality inherent in high-cardinality data by utilizing the propagation of chaos property of interacting particles. By treating high-cardinality data as a large stochastic system of interacting particles, we develop a novel score-matching method for infinite-dimensional chaotic particle systems and propose an approximation scheme that employs a subdivision strategy for efficient training. Our theoretical and empirical results demonstrate the scalability and effectiveness of MF-CDMs for managing large high-cardinality data structures, such as 3D point clouds.

Scaling laws are typically fit using a family of models with a narrow range of frozen hyper-parameter choices. In this work we study scaling laws using a wide range of architecture and hyper-parameter choices, and highlight their impact on resulting prescriptions. As a primary artifact of our research, we release the Gemstones: the most comprehensive open-source scaling law dataset to date, consisting of over 4000 checkpoints from transformers with up to 2 billion parameters; these models have been trained with different learning rates, cooldown schedules, and architectural shapes. Our checkpoints enable more complex studies of scaling, such as a law that predicts language modeling performance as a function of model width and depth. By examining the various facets of our model suite, we find that the prescriptions of scaling laws can be highly sensitive to the experimental design process and the specific model checkpoints used during fitting. Code: https://github.com/mcleish7/gemstone-scaling-laws

The fastrerandomize R package provides hardware-accelerated tools for performing rerandomization and randomization testing in experimental research. Using a JAX backend, the package enables exact rerandomization inference even for large experiments with hundreds of billions of possible randomizations. Key functionalities include generating pools of acceptable rerandomizations based on covariate balance, conducting exact randomization tests, and performing pre-analysis evaluations to determine optimal rerandomization acceptance thresholds. Through batched processing and GPU acceleration, fastrerandomize achieves substantial performance gains compared to existing implementations, making previously intractable designs computationally feasible. The package therefore extends the randomization-based inference toolkit in R, allowing researchers to efficiently implement more stringent rerandomization designs and conduct valid inference even with large sample sizes or in high-dimensional settings.

Amid mounting concern about the reliability and credibility of machine learning research, we present a principled framework for making robust and generalizable claims: the multiverse analysis. Our framework builds upon the multiverse analysis (Steegen et al., 2016) introduced in response to psychology's own reproducibility crisis. To efficiently explore high-dimensional and often continuous ML search spaces, we model the multiverse with a Gaussian Process surrogate and apply Bayesian experimental design. Our framework is designed to facilitate drawing robust scientific conclusions about model performance, and thus our approach focuses on exploration rather than conventional optimization. In the first of two case studies, we investigate disputed claims about the relative merit of adaptive optimizers. Second, we synthesize conflicting research on the effect of learning rate on the large batch training generalization gap. For the machine learning community, the multiverse analysis is a simple and effective technique for identifying robust claims, for increasing transparency, and a step toward improved reproducibility.

We present a trainable framework for efficiently generating gauge configurations, and discuss ongoing work in this direction. In particular, we consider the problem of sampling configurations from a 4D SU(3) lattice gauge theory, and consider a generalized leapfrog integrator in the molecular dynamics update that can be trained to improve sampling efficiency. Code is available online at https://github.com/saforem2/l2hmc-qcd.

Model selection/optimization in conformal inference is challenging, since it may break the exchangeability between labeled and unlabeled data. We study this problem in the context of conformal selection, which uses conformal p-values to select ``interesting'' instances with large unobserved labels from a pool of unlabeled data, while controlling the FDR in finite sample. For validity, existing solutions require the model choice to be independent of the data used to construct the p-values and calibrate the selection set. However, when presented with many model choices and limited labeled data, it is desirable to (i) select the best model in a data-driven manner, and (ii) mitigate power loss due to sample splitting. This paper presents OptCS, a general framework that allows valid statistical testing (selection) after flexible data-driven model optimization. We introduce general conditions under which OptCS constructs valid conformal p-values despite substantial data reuse and handles complex p-value dependencies to maintain finite-sample FDR control via a novel multiple testing procedure. We instantiate this general recipe to propose three FDR-controlling procedures, each optimizing the models differently: (i) selecting the most powerful one among multiple pre-trained candidate models, (ii) using all data for model fitting without sample splitting, and (iii) combining full-sample model fitting and selection. We demonstrate the efficacy of our methods via simulation studies and real applications in drug discovery and alignment of large language models in radiology report generation.

Analytical framework for predicting General Matrix Multiplication (GEMM) performance on modern GPUs, focusing on runtime, power consumption, and energy efficiency. Our study employs two approaches: a custom-implemented tiled matrix multiplication kernel for fundamental analysis, and NVIDIA's CUTLASS library for comprehensive performance data collection across advanced configurations. Using the NVIDIA RTX 4070 as our experimental platform, we developed a Random Forest-based prediction model with multi-output regression capability. Through analysis of both naive tiled matrix multiplication with varying tile sizes (1 to 32) and 16,128 CUTLASS GEMM operations across diverse configurations, we identified critical performance patterns related to matrix dimensions, thread block configurations, and memory access patterns. Our framework achieved exceptional accuracy with an R^2 score of 0.98 for runtime prediction (mean error 15.57%) and 0.78 for power prediction (median error 5.42%). The system successfully predicts performance across matrix sizes, demonstrating robust scaling behavior. Our results show that optimal tile size selection can improve performance by up to 3.2x while reducing power consumption by 22% compared to baseline configurations. Analysis of shared memory utilization and SM occupancy reveals that tile sizes of 16x16 achieve the best balance between parallelism and resource usage. The implementation of our framework, including prediction models and analysis tools, is available as an open-source project at GPPerf [https://github.com/pavlyhalim/GPPerf].

We study a family of distributed stochastic optimization algorithms where gradients are sampled by a token traversing a network of agents in random-walk fashion. Typically, these random-walks are chosen to be Markov chains that asymptotically sample from a desired target distribution, and play a critical role in the convergence of the optimization iterates. In this paper, we take a novel approach by replacing the standard linear Markovian token by one which follows a nonlinear Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar {\alpha}, is less likely to transition to states that were highly visited in the past, thus the name. In the context of MCMC sampling on a graph, a recent breakthrough in Doshi et al. (2023) shows that the SRRW achieves O(1/{\alpha}) decrease in the asymptotic variance for sampling. We propose the use of a 'generalized' version of the SRRW to drive token algorithms for distributed stochastic optimization in the form of stochastic approximation, termed SA-SRRW. We prove that the optimization iterate errors of the resulting SA-SRRW converge to zero almost surely and prove a central limit theorem, deriving the explicit form of the resulting asymptotic covariance matrix corresponding to iterate errors. This asymptotic covariance is always smaller than that of an algorithm driven by the base Markov chain and decreases at rate O(1/{\alpha}^2) - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings.

The analysis of parametric and non-parametric uncertainties of very large dynamical systems requires the construction of a stochastic model of said system. Linear approaches relying on random matrix theory and principal componant analysis can be used when systems undergo low-frequency vibrations. In the case of fast dynamics and wave propagation, we investigate a random generator of boundary conditions for fast submodels by using machine learning. We show that the use of non-linear techniques in machine learning and data-driven methods is highly relevant. Physics-informed neural networks is a possible choice for a data-driven method to replace linear modal analysis. An architecture that support a random component is necessary for the construction of the stochastic model of the physical system for non-parametric uncertainties, since the goal is to learn the underlying probabilistic distribution of uncertainty in the data. Generative Adversarial Networks (GANs) are suited for such applications, where the Wasserstein-GAN with gradient penalty variant offers improved convergence results for our problem. The objective of our approach is to train a GAN on data from a finite element method code (Fenics) so as to extract stochastic boundary conditions for faster finite element predictions on a submodel. The submodel and the training data have both the same geometrical support. It is a zone of interest for uncertainty quantification and relevant to engineering purposes. In the exploitation phase, the framework can be viewed as a randomized and parametrized simulation generator on the submodel, which can be used as a Monte Carlo estimator.

Hyperdimensional computing (HD), also known as vector symbolic architectures (VSA), is a framework for computing with distributed representations by exploiting properties of random high-dimensional vector spaces. The commitment of the scientific community to aggregate and disseminate research in this particularly multidisciplinary area has been fundamental for its advancement. Joining these efforts, we present Torchhd, a high-performance open source Python library for HD/VSA. Torchhd seeks to make HD/VSA more accessible and serves as an efficient foundation for further research and application development. The easy-to-use library builds on top of PyTorch and features state-of-the-art HD/VSA functionality, clear documentation, and implementation examples from well-known publications. Comparing publicly available code with their corresponding Torchhd implementation shows that experiments can run up to 100x faster. Torchhd is available at: https://github.com/hyperdimensional-computing/torchhd.

Crystal Structure Prediction (CSP) is crucial in various scientific disciplines. While CSP can be addressed by employing currently-prevailing generative models (e.g. diffusion models), this task encounters unique challenges owing to the symmetric geometry of crystal structures -- the invariance of translation, rotation, and periodicity. To incorporate the above symmetries, this paper proposes DiffCSP, a novel diffusion model to learn the structure distribution from stable crystals. To be specific, DiffCSP jointly generates the lattice and atom coordinates for each crystal by employing a periodic-E(3)-equivariant denoising model, to better model the crystal geometry. Notably, different from related equivariant generative approaches, DiffCSP leverages fractional coordinates other than Cartesian coordinates to represent crystals, remarkably promoting the diffusion and the generation process of atom positions. Extensive experiments verify that our DiffCSP significantly outperforms existing CSP methods, with a much lower computation cost in contrast to DFT-based methods. Moreover, the superiority of DiffCSP is also observed when it is extended for ab initio crystal generation.

Masked diffusion models (MDMs) have emerged as a popular research topic for generative modeling of discrete data, thanks to their superior performance over other discrete diffusion models, and are rivaling the auto-regressive models (ARMs) for language modeling tasks. The recent effort in simplifying the masked diffusion framework further leads to alignment with continuous-space diffusion models and more principled training and sampling recipes. In this paper, however, we reveal that both training and sampling of MDMs are theoretically free from the time variable, arguably the key signature of diffusion models, and are instead equivalent to masked models. The connection on the sampling aspect is drawn by our proposed first-hitting sampler (FHS). Specifically, we show that the FHS is theoretically equivalent to MDMs' original generation process while significantly alleviating the time-consuming categorical sampling and achieving a 20times speedup. In addition, our investigation raises doubts about whether MDMs can truly beat ARMs. We identify, for the first time, an underlying numerical issue, even with the commonly used 32-bit floating-point precision, which results in inaccurate categorical sampling. We show that the numerical issue lowers the effective temperature both theoretically and empirically, and the resulting decrease in token diversity makes previous evaluations, which assess the generation quality solely through the incomplete generative perplexity metric, somewhat unfair.

A growing literature in computational neuroscience leverages gradient descent and learning algorithms that approximate it to study synaptic plasticity in the brain. However, the vast majority of this work ignores a critical underlying assumption: the choice of distance for synaptic changes - i.e. the geometry of synaptic plasticity. Gradient descent assumes that the distance is Euclidean, but many other distances are possible, and there is no reason that biology necessarily uses Euclidean geometry. Here, using the theoretical tools provided by mirror descent, we show that the distribution of synaptic weights will depend on the geometry of synaptic plasticity. We use these results to show that experimentally-observed log-normal weight distributions found in several brain areas are not consistent with standard gradient descent (i.e. a Euclidean geometry), but rather with non-Euclidean distances. Finally, we show that it should be possible to experimentally test for different synaptic geometries by comparing synaptic weight distributions before and after learning. Overall, our work shows that the current paradigm in theoretical work on synaptic plasticity that assumes Euclidean synaptic geometry may be misguided and that it should be possible to experimentally determine the true geometry of synaptic plasticity in the brain.

Methods for carefully selecting or generating a small set of training data to learn from, i.e., data pruning, coreset selection, and data distillation, have been shown to be effective in reducing the ever-increasing cost of training neural networks. Behind this success are rigorously designed strategies for identifying informative training examples out of large datasets. However, these strategies come with additional computational costs associated with subset selection or data distillation before training begins, and furthermore, many are shown to even under-perform random sampling in high data compression regimes. As such, many data pruning, coreset selection, or distillation methods may not reduce 'time-to-accuracy', which has become a critical efficiency measure of training deep neural networks over large datasets. In this work, we revisit a powerful yet overlooked random sampling strategy to address these challenges and introduce an approach called Repeated Sampling of Random Subsets (RSRS or RS2), where we randomly sample the subset of training data for each epoch of model training. We test RS2 against thirty state-of-the-art data pruning and data distillation methods across four datasets including ImageNet. Our results demonstrate that RS2 significantly reduces time-to-accuracy compared to existing techniques. For example, when training on ImageNet in the high-compression regime (using less than 10% of the dataset each epoch), RS2 yields accuracy improvements up to 29% compared to competing pruning methods while offering a runtime reduction of 7x. Beyond the above meta-study, we provide a convergence analysis for RS2 and discuss its generalization capability. The primary goal of our work is to establish RS2 as a competitive baseline for future data selection or distillation techniques aimed at efficient training.

The recent progress of AI can be largely attributed to large language models (LLMs). However, their escalating memory requirements introduce challenges for machine learning (ML) researchers and engineers. Addressing this requires developers to partition a large model to distribute it across multiple GPUs or TPUs. This necessitates considerable coding and intricate configuration efforts with existing model parallel tools, such as Megatron-LM, DeepSpeed, and Alpa. These tools require users' expertise in machine learning systems (MLSys), creating a bottleneck in LLM development, particularly for developers without MLSys background. In this work, we present Redco, a lightweight and user-friendly tool crafted to automate distributed training and inference for LLMs, as well as to simplify ML pipeline development. The design of Redco emphasizes two key aspects. Firstly, to automate model parallism, our study identifies two straightforward rules to generate tensor parallel strategies for any given LLM. Integrating these rules into Redco facilitates effortless distributed LLM training and inference, eliminating the need of additional coding or complex configurations. We demonstrate the effectiveness by applying Redco on a set of LLM architectures, such as GPT-J, LLaMA, T5, and OPT, up to the size of 66B. Secondly, we propose a mechanism that allows for the customization of diverse ML pipelines through the definition of merely three functions, eliminating redundant and formulaic code like multi-host related processing. This mechanism proves adaptable across a spectrum of ML algorithms, from foundational language modeling to complex algorithms like meta-learning and reinforcement learning. Consequently, Redco implementations exhibit much fewer code lines compared to their official counterparts.

We provide a theoretical framework for Reinforcement Learning with Human Feedback (RLHF). Our analysis shows that when the true reward function is linear, the widely used maximum likelihood estimator (MLE) converges under both the Bradley-Terry-Luce (BTL) model and the Plackett-Luce (PL) model. However, we show that when training a policy based on the learned reward model, MLE fails while a pessimistic MLE provides policies with improved performance under certain coverage assumptions. Additionally, we demonstrate that under the PL model, the true MLE and an alternative MLE that splits the K-wise comparison into pairwise comparisons both converge. Moreover, the true MLE is asymptotically more efficient. Our results validate the empirical success of existing RLHF algorithms in InstructGPT and provide new insights for algorithm design. Furthermore, our results unify the problem of RLHF and max-entropy Inverse Reinforcement Learning (IRL), and provide the first sample complexity bound for max-entropy IRL.

We present a novel type of neural fields that uses general radial bases for signal representation. State-of-the-art neural fields typically rely on grid-based representations for storing local neural features and N-dimensional linear kernels for interpolating features at continuous query points. The spatial positions of their neural features are fixed on grid nodes and cannot well adapt to target signals. Our method instead builds upon general radial bases with flexible kernel position and shape, which have higher spatial adaptivity and can more closely fit target signals. To further improve the channel-wise capacity of radial basis functions, we propose to compose them with multi-frequency sinusoid functions. This technique extends a radial basis to multiple Fourier radial bases of different frequency bands without requiring extra parameters, facilitating the representation of details. Moreover, by marrying adaptive radial bases with grid-based ones, our hybrid combination inherits both adaptivity and interpolation smoothness. We carefully designed weighting schemes to let radial bases adapt to different types of signals effectively. Our experiments on 2D image and 3D signed distance field representation demonstrate the higher accuracy and compactness of our method than prior arts. When applied to neural radiance field reconstruction, our method achieves state-of-the-art rendering quality, with small model size and comparable training speed.

Scaling laws offer valuable insights into the design of time series foundation models (TSFMs). However, previous research has largely focused on the scaling laws of TSFMs for in-distribution (ID) data, leaving their out-of-distribution (OOD) scaling behavior and the influence of model architectures less explored. In this work, we examine two common TSFM architectures, encoder-only and decoder-only Transformers, and investigate their scaling behavior on both ID and OOD data. These models are trained and evaluated across varying parameter counts, compute budgets, and dataset sizes. Our experiments reveal that the log-likelihood loss of TSFMs exhibits similar scaling behavior in both OOD and ID settings. We further compare the scaling properties across different architectures, incorporating two state-of-the-art TSFMs as case studies, showing that model architecture plays a significant role in scaling. The encoder-only Transformers demonstrate better scalability than the decoder-only Transformers, while the architectural enhancements in the two advanced TSFMs primarily improve ID performance but reduce OOD scalability. While scaling up TSFMs is expected to drive performance breakthroughs, the lack of a comprehensive understanding of TSFM scaling laws has hindered the development of a robust framework to guide model scaling. We fill this gap in this work by synthesizing our findings and providing practical guidelines for designing and scaling larger TSFMs with enhanced model capabilities.

The past few years have witnessed the great success of Diffusion models~(DMs) in generating high-fidelity samples in generative modeling tasks. A major limitation of the DM is its notoriously slow sampling procedure which normally requires hundreds to thousands of time discretization steps of the learned diffusion process to reach the desired accuracy. Our goal is to develop a fast sampling method for DMs with a much less number of steps while retaining high sample quality. To this end, we systematically analyze the sampling procedure in DMs and identify key factors that affect the sample quality, among which the method of discretization is most crucial. By carefully examining the learned diffusion process, we propose Diffusion Exponential Integrator Sampler~(DEIS). It is based on the Exponential Integrator designed for discretizing ordinary differential equations (ODEs) and leverages a semilinear structure of the learned diffusion process to reduce the discretization error. The proposed method can be applied to any DMs and can generate high-fidelity samples in as few as 10 steps. In our experiments, it takes about 3 minutes on one A6000 GPU to generate 50k images from CIFAR10. Moreover, by directly using pre-trained DMs, we achieve the state-of-art sampling performance when the number of score function evaluation~(NFE) is limited, e.g., 4.17 FID with 10 NFEs, 3.37 FID, and 9.74 IS with only 15 NFEs on CIFAR10. Code is available at https://github.com/qsh-zh/deis

This paper presents our approach to accelerate computer architecture simulation by leveraging machine learning techniques. Traditional computer architecture simulations are time-consuming, making it challenging to explore different design choices efficiently. Our proposed model utilizes a combination of application features and micro-architectural features to predict the performance of an application. These features are derived from simulations of a small portion of the application. We demonstrate the effectiveness of our approach by building and evaluating a machine learning model that offers significant speedup in architectural exploration. This model demonstrates the ability to predict IPC values for the testing data with a root mean square error of less than 0.1.

Prediction of stock prices plays a significant role in aiding the decision-making of investors. Considering its importance, a growing literature has emerged trying to forecast stock prices with improved accuracy. In this study, we introduce an innovative approach for forecasting stock prices with greater accuracy. We incorporate external economic environment-related information along with stock prices. In our novel approach, we improve the performance of stock price prediction by taking into account variations due to future expected macroeconomic policy changes as investors adjust their current behavior ahead of time based on expected future macroeconomic policy changes. Furthermore, we incorporate macroeconomic variables along with historical stock prices to make predictions. Results from this strongly support the inclusion of future economic policy changes along with current macroeconomic information. We confirm the supremacy of our method over the conventional approach using several tree-based machine-learning algorithms. Results are strongly conclusive across various machine learning models. Our preferred model outperforms the conventional approach with an RMSE value of 1.61 compared to an RMSE value of 1.75 from the conventional approach.

We present a novel approach to accelerate stochastic gradient descent (SGD) by utilizing curvature information obtained from Hessian-vector products or finite differences of parameters and gradients, similar to the BFGS algorithm. Our approach involves two preconditioners: a matrix-free preconditioner and a low-rank approximation preconditioner. We update both preconditioners online using a criterion that is robust to stochastic gradient noise and does not require line search or damping. To preserve the corresponding symmetry or invariance, our preconditioners are constrained to certain connected Lie groups. The Lie group's equivariance property simplifies the preconditioner fitting process, while its invariance property eliminates the need for damping, which is commonly required in second-order optimizers. As a result, the learning rate for parameter updating and the step size for preconditioner fitting are naturally normalized, and their default values work well in most scenarios. Our proposed approach offers a promising direction for improving the convergence of SGD with low computational overhead. We demonstrate that Preconditioned SGD (PSGD) outperforms SoTA on Vision, NLP, and RL tasks across multiple modern deep-learning architectures. We have provided code for reproducing toy and large scale experiments in this paper.

In machine learning applications, it is well known that carefully designed learning rate (step size) schedules can significantly improve the convergence of commonly used first-order optimization algorithms. Therefore how to set step size adaptively becomes an important research question. A popular and effective method is the Polyak step size, which sets step size adaptively for gradient descent or stochastic gradient descent without the need to estimate the smoothness parameter of the objective function. However, there has not been a principled way to generalize the Polyak step size for algorithms with momentum accelerations. This paper presents a general framework to set the learning rate adaptively for first-order optimization methods with momentum, motivated by the derivation of Polyak step size. It is shown that the resulting methods are much less sensitive to the choice of momentum parameter and may avoid the oscillation of the heavy-ball method on ill-conditioned problems. These adaptive step sizes are further extended to the stochastic settings, which are attractive choices for stochastic gradient descent with momentum. Our methods are demonstrated to be more effective for stochastic gradient methods than prior adaptive step size algorithms in large-scale machine learning tasks.

Low precision (LP) datatypes such as MXFP4 can accelerate matrix multiplications (GEMMs) and reduce training costs. However, directly using MXFP4 instead of BF16 during training significantly degrades model quality. In this work, we present the first near-lossless training recipe that uses MXFP4 GEMMs, which are 2times faster than FP8 on supported hardware. Our key insight is to compute unbiased gradient estimates with stochastic rounding (SR), resulting in more accurate model updates. However, directly applying SR to MXFP4 can result in high variance from block-level outliers, harming convergence. To overcome this, we use the random Hadamard tranform to theoretically bound the variance of SR. We train GPT models up to 6.7B parameters and find that our method induces minimal degradation over mixed-precision BF16 training. Our recipe computes >1/2 the training FLOPs in MXFP4, enabling an estimated speedup of >1.3times over FP8 and >1.7times over BF16 during backpropagation.

This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an O(d/epsilon) mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures. This bound improves the best previously known O(d/epsilon) result and is optimal (in terms of order) in both dimension d and accuracy tolerance epsilon for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.

Federated learning (FL) is a distributed machine learning framework where the global model of a central server is trained via multiple collaborative steps by participating clients without sharing their data. While being a flexible framework, where the distribution of local data, participation rate, and computing power of each client can greatly vary, such flexibility gives rise to many new challenges, especially in the hyperparameter tuning on the client side. We propose Delta-SGD, a simple step size rule for SGD that enables each client to use its own step size by adapting to the local smoothness of the function each client is optimizing. We provide theoretical and empirical results where the benefit of the client adaptivity is shown in various FL scenarios.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Time series forecasting is widely used in extensive applications, such as traffic planning and weather forecasting. However, real-world time series usually present intricate temporal variations, making forecasting extremely challenging. Going beyond the mainstream paradigms of plain decomposition and multiperiodicity analysis, we analyze temporal variations in a novel view of multiscale-mixing, which is based on an intuitive but important observation that time series present distinct patterns in different sampling scales. The microscopic and the macroscopic information are reflected in fine and coarse scales respectively, and thereby complex variations can be inherently disentangled. Based on this observation, we propose TimeMixer as a fully MLP-based architecture with Past-Decomposable-Mixing (PDM) and Future-Multipredictor-Mixing (FMM) blocks to take full advantage of disentangled multiscale series in both past extraction and future prediction phases. Concretely, PDM applies the decomposition to multiscale series and further mixes the decomposed seasonal and trend components in fine-to-coarse and coarse-to-fine directions separately, which successively aggregates the microscopic seasonal and macroscopic trend information. FMM further ensembles multiple predictors to utilize complementary forecasting capabilities in multiscale observations. Consequently, TimeMixer is able to achieve consistent state-of-the-art performances in both long-term and short-term forecasting tasks with favorable run-time efficiency.

Hoffmann et al. (2022) propose three methods for estimating a compute-optimal scaling law. We attempt to replicate their third estimation procedure, which involves fitting a parametric loss function to a reconstruction of data from their plots. We find that the reported estimates are inconsistent with their first two estimation methods, fail at fitting the extracted data, and report implausibly narrow confidence intervals--intervals this narrow would require over 600,000 experiments, while they likely only ran fewer than 500. In contrast, our rederivation of the scaling law using the third approach yields results that are compatible with the findings from the first two estimation procedures described by Hoffmann et al.

The kernel thinning (KT) algorithm of Dwivedi and Mackey (2021) compresses a probability distribution more effectively than independent sampling by targeting a reproducing kernel Hilbert space (RKHS) and leveraging a less smooth square-root kernel. Here we provide four improvements. First, we show that KT applied directly to the target RKHS yields tighter, dimension-free guarantees for any kernel, any distribution, and any fixed function in the RKHS. Second, we show that, for analytic kernels like Gaussian, inverse multiquadric, and sinc, target KT admits maximum mean discrepancy (MMD) guarantees comparable to or better than those of square-root KT without making explicit use of a square-root kernel. Third, we prove that KT with a fractional power kernel yields better-than-Monte-Carlo MMD guarantees for non-smooth kernels, like Laplace and Mat\'ern, that do not have square-roots. Fourth, we establish that KT applied to a sum of the target and power kernels (a procedure we call KT+) simultaneously inherits the improved MMD guarantees of power KT and the tighter individual function guarantees of target KT. In our experiments with target KT and KT+, we witness significant improvements in integration error even in 100 dimensions and when compressing challenging differential equation posteriors.

General full-wave electromagnetic solvers, such as those utilizing the finite-difference time-domain (FDTD) method, are computationally demanding for simulating practical GPR problems. We explore the performance of a near-real-time, forward modeling approach for GPR that is based on a machine learning (ML) architecture. To ease the process, we have developed a framework that is capable of generating these ML-based forward solvers automatically. The framework uses an innovative training method that combines a predictive dimensionality reduction technique and a large data set of modeled GPR responses from our FDTD simulation software, gprMax. The forward solver is parameterized for a specific GPR application, but the framework can be extended in a straightforward manner to different electromagnetic problems.

In this work, we question the necessity of adaptive gradient methods for training deep neural networks. SGD-SaI is a simple yet effective enhancement to stochastic gradient descent with momentum (SGDM). SGD-SaI performs learning rate Scaling at Initialization (SaI) to distinct parameter groups, guided by their respective gradient signal-to-noise ratios (g-SNR). By adjusting learning rates without relying on adaptive second-order momentum, SGD-SaI helps prevent training imbalances from the very first iteration and cuts the optimizer's memory usage by half compared to AdamW. Despite its simplicity and efficiency, SGD-SaI consistently matches or outperforms AdamW in training a variety of Transformer-based tasks, effectively overcoming a long-standing challenge of using SGD for training Transformers. SGD-SaI excels in ImageNet-1K classification with Vision Transformers(ViT) and GPT-2 pretraining for large language models (LLMs, transformer decoder-only), demonstrating robustness to hyperparameter variations and practicality for diverse applications. We further tested its robustness on tasks like LoRA fine-tuning for LLMs and diffusion models, where it consistently outperforms state-of-the-art optimizers. From a memory efficiency perspective, SGD-SaI achieves substantial memory savings for optimizer states, reducing memory usage by 5.93 GB for GPT-2 (1.5B parameters) and 25.15 GB for Llama2-7B compared to AdamW in full-precision training settings.

MRI is an inherently slow process, which leads to long scan time for high-resolution imaging. The speed of acquisition can be increased by ignoring parts of the data (undersampling). Consequently, this leads to the degradation of image quality, such as loss of resolution or introduction of image artefacts. This work aims to reconstruct highly undersampled Cartesian or radial MR acquisitions, with better resolution and with less to no artefact compared to conventional techniques like compressed sensing. In recent times, deep learning has emerged as a very important area of research and has shown immense potential in solving inverse problems, e.g. MR image reconstruction. In this paper, a deep learning based MR image reconstruction framework is proposed, which includes a modified regularised version of ResNet as the network backbone to remove artefacts from the undersampled image, followed by data consistency steps that fusions the network output with the data already available from undersampled k-space in order to further improve reconstruction quality. The performance of this framework for various undersampling patterns has also been tested, and it has been observed that the framework is robust to deal with various sampling patterns, even when mixed together while training, and results in very high quality reconstruction, in terms of high SSIM (highest being 0.990pm0.006 for acceleration factor of 3.5), while being compared with the fully sampled reconstruction. It has been shown that the proposed framework can successfully reconstruct even for an acceleration factor of 20 for Cartesian (0.968pm0.005) and 17 for radially (0.962pm0.012) sampled data. Furthermore, it has been shown that the framework preserves brain pathology during reconstruction while being trained on healthy subjects.

Heavy-ball momentum with decaying learning rates is widely used with SGD for optimizing deep learning models. In contrast to its empirical popularity, the understanding of its theoretical property is still quite limited, especially under the standard anisotropic gradient noise condition for quadratic regression problems. Although it is widely conjectured that heavy-ball momentum method can provide accelerated convergence and should work well in large batch settings, there is no rigorous theoretical analysis. In this paper, we fill this theoretical gap by establishing a non-asymptotic convergence bound for stochastic heavy-ball methods with step decay scheduler on quadratic objectives, under the anisotropic gradient noise condition. As a direct implication, we show that heavy-ball momentum can provide mathcal{O}(kappa) accelerated convergence of the bias term of SGD while still achieving near-optimal convergence rate with respect to the stochastic variance term. The combined effect implies an overall convergence rate within log factors from the statistical minimax rate. This means SGD with heavy-ball momentum is useful in the large-batch settings such as distributed machine learning or federated learning, where a smaller number of iterations can significantly reduce the number of communication rounds, leading to acceleration in practice.

We present a new regular grid search algorithm for quick fixed-radius nearest-neighbor lookup developed in Python. This module indexes a set of k-dimensional points in a regular grid, with optional periodic conditions, providing a fast approach for nearest neighbors queries. In this first installment we provide three types of queries: bubble, shell and the nth-nearest; as well as three different metrics of interest in astronomy: the euclidean and two distance functions in spherical coordinates of varying precision, haversine and Vincenty; and the possibility of providing a custom distance function. This package results particularly useful for large datasets where a brute-force search turns impractical.

We consider the problem of estimating the optimal transport map between two probability distributions, P and Q in mathbb R^d, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points in mathbb R^d. We study a computationally efficient estimator initially proposed by Pooladian and Niles-Weed (2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate n^{-1/2}, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.

Large language models (LLMs) have demonstrated remarkable success as foundational models, benefiting various downstream applications through fine-tuning. Recent studies on loss scaling have demonstrated the superior performance of larger LLMs compared to their smaller counterparts. Nevertheless, training LLMs with billions of parameters poses significant challenges and requires considerable computational resources. For example, training a one trillion parameter GPT-style model on 20 trillion tokens requires a staggering 120 million exaflops of computation. This research explores efficient distributed training strategies to extract this computation from Frontier, the world's first exascale supercomputer dedicated to open science. We enable and investigate various model and data parallel training techniques, such as tensor parallelism, pipeline parallelism, and sharded data parallelism, to facilitate training a trillion-parameter model on Frontier. We empirically assess these techniques and their associated parameters to determine their impact on memory footprint, communication latency, and GPU's computational efficiency. We analyze the complex interplay among these techniques and find a strategy to combine them to achieve high throughput through hyperparameter tuning. We have identified efficient strategies for training large LLMs of varying sizes through empirical analysis and hyperparameter tuning. For 22 Billion, 175 Billion, and 1 Trillion parameters, we achieved GPU throughputs of 38.38%, 36.14%, and 31.96%, respectively. For the training of the 175 Billion parameter model and the 1 Trillion parameter model, we achieved 100% weak scaling efficiency on 1024 and 3072 MI250X GPUs, respectively. We also achieved strong scaling efficiencies of 89% and 87% for these two models.

Recent studies suggest that with sufficiently wide models, most SGD solutions can, up to permutation, converge into the same basin. This phenomenon, known as the model re-basin regime, has significant implications for model averaging by ensuring the linear mode connectivity. However, current re-basin strategies are ineffective in many scenarios due to a lack of comprehensive understanding of underlying mechanisms. Addressing this gap, this paper provides novel insights into understanding and improving the standard practice. Firstly, we decompose re-normalization into rescaling and reshift, uncovering that rescaling plays a crucial role in re-normalization while re-basin performance is sensitive to shifts in model activation. The finding calls for a more nuanced handling of the activation shift. Secondly, we identify that the merged model suffers from the issue of activation collapse and magnitude collapse. Varying the learning rate, weight decay, and initialization method can mitigate the issues and improve model performance. Lastly, we propose a new perspective to unify the re-basin and pruning, under which a lightweight yet effective post-pruning technique is derived, which can significantly improve the model performance after pruning. Our implementation is available at https://github.com/XingyuQu/rethink-re-basin.

In the era of proliferation of large language and image generation models, the phenomenon of "model collapse" refers to the situation whereby as a model is trained recursively on data generated from previous generations of itself over time, its performance degrades until the model eventually becomes completely useless, i.e the model collapses. In this work, we study this phenomenon in the setting of high-dimensional regression and obtain analytic formulae which quantitatively outline this phenomenon in a broad range of regimes. In the special case of polynomial decaying spectral and source conditions, we obtain modified scaling laws which exhibit new crossover phenomena from fast to slow rates. We also propose a simple strategy based on adaptive regularization to mitigate model collapse. Our theoretical results are validated with experiments.

A new class of affine scaling matrices for the interior point Newton-type methods is considered to solve the nonlinear systems with simple bounds. We review the essential properties of a scaling matrix and consider several well-known scaling matrices proposed in the literature. We define a new scaling matrix that is the convex combination of these matrices. The proposed scaling matrix inherits those interesting properties of the individual matrices and satisfies additional desired requirements. The numerical experiments demonstrate the superiority of the new scaling matrix in solving several important test problems.

This work explores the in-context learning capabilities of State Space Models (SSMs) and presents, to the best of our knowledge, the first theoretical explanation of a possible underlying mechanism. We introduce a novel weight construction for SSMs, enabling them to predict the next state of any dynamical system after observing previous states without parameter fine-tuning. This is accomplished by extending the HiPPO framework to demonstrate that continuous SSMs can approximate the derivative of any input signal. Specifically, we find an explicit weight construction for continuous SSMs and provide an asymptotic error bound on the derivative approximation. The discretization of this continuous SSM subsequently yields a discrete SSM that predicts the next state. Finally, we demonstrate the effectiveness of our parameterization empirically. This work should be an initial step toward understanding how sequence models based on SSMs learn in context.

Updating a truncated Singular Value Decomposition (SVD) is crucial in representation learning, especially when dealing with large-scale data matrices that continuously evolve in practical scenarios. Aligning SVD-based models with fast-paced updates becomes increasingly important. Existing methods for updating truncated SVDs employ Rayleigh-Ritz projection procedures, where projection matrices are augmented based on original singular vectors. However, these methods suffer from inefficiency due to the densification of the update matrix and the application of the projection to all singular vectors. To address these limitations, we introduce a novel method for dynamically approximating the truncated SVD of a sparse and temporally evolving matrix. Our approach leverages sparsity in the orthogonalization process of augmented matrices and utilizes an extended decomposition to independently store projections in the column space of singular vectors. Numerical experiments demonstrate a remarkable efficiency improvement of an order of magnitude compared to previous methods. Remarkably, this improvement is achieved while maintaining a comparable precision to existing approaches.

Recently, neural networks have been extensively employed to solve partial differential equations (PDEs) in physical system modeling. While major studies focus on learning system evolution on predefined static mesh discretizations, some methods utilize reinforcement learning or supervised learning techniques to create adaptive and dynamic meshes, due to the dynamic nature of these systems. However, these approaches face two primary challenges: (1) the need for expensive optimal mesh data, and (2) the change of the solution space's degree of freedom and topology during mesh refinement. To address these challenges, this paper proposes a neural PDE solver with a neural mesh adapter. To begin with, we introduce a novel data-free neural mesh adaptor, called Data-free Mesh Mover (DMM), with two main innovations. Firstly, it is an operator that maps the solution to adaptive meshes and is trained using the Monge-Amp\`ere equation without optimal mesh data. Secondly, it dynamically changes the mesh by moving existing nodes rather than adding or deleting nodes and edges. Theoretical analysis shows that meshes generated by DMM have the lowest interpolation error bound. Based on DMM, to efficiently and accurately model dynamic systems, we develop a moving mesh based neural PDE solver (MM-PDE) that embeds the moving mesh with a two-branch architecture and a learnable interpolation framework to preserve information within the data. Empirical experiments demonstrate that our method generates suitable meshes and considerably enhances accuracy when modeling widely considered PDE systems. The code can be found at: https://github.com/Peiyannn/MM-PDE.git.

Iterative methods are ubiquitous in large-scale scientific computing applications, and a number of approaches based on meta-learning have been recently proposed to accelerate them. However, a systematic study of these approaches and how they differ from meta-learning is lacking. In this paper, we propose a framework to analyze such learning-based acceleration approaches, where one can immediately identify a departure from classical meta-learning. We show that this departure may lead to arbitrary deterioration of model performance. Based on our analysis, we introduce a novel training method for learning-based acceleration of iterative methods. Furthermore, we theoretically prove that the proposed method improves upon the existing methods, and demonstrate its significant advantage and versatility through various numerical applications.

Over the past three decades, numerous controllers have been developed to regulate complex chemical processes, but they have certain limitations. Traditional PI/PID controllers often require customized tuning for various set-point scenarios. On the other hand, MPC frameworks involve resource-intensive steps, and the utilization of black-box machine learning (ML) models can lead to issues such as local minima and infeasibility. Thus, there is a need for an alternative controller paradigm that combines the simplicity of a PI controller with the grade-to-grade (G2G) transferability of an MPC approach. To this end, we developed a novel LSTM controller (LSTMc) as a model-free data-driven controller framework. The LSTMc considers an augmented input tensor that incorporates information on state evolution and error dynamics for the current and previous W time steps, to predict the manipulated input at the next step (u_{t+1}). To demonstrate LSTMc, batch crystallization of dextrose was taken as a representative case study. The desired output for set-point tracking was the mean crystal size (L), with the manipulated input being the jacket temperature (T_j). Extensive training data, encompassing 7000+ different operating conditions, was compiled to ensure comprehensive training of LSTMc across a wide state space region. For comparison, we also designed a PI controller and an LSTM-MPC for different set-point tracking cases. The results consistently showed that LSTMc achieved the lowest set-point deviation (<2\%), three times lower than the MPC. Remarkably, LSTMc maintained this superior performance across all set points, even when sensor measurements contained noise levels of 10\% to 15\%. In summary, by effectively leveraging process data and utilizing sequential ML models, LSTMc offers a superior controller design approach.

We consider the dictionary-based ROM-net (Reduced Order Model) framework [T. Daniel, F. Casenave, N. Akkari, D. Ryckelynck, Model order reduction assisted by deep neural networks (ROM-net), Advanced modeling and Simulation in Engineering Sciences 7 (16), 2020] and summarize the underlying methodologies and their recent improvements. The main contribution of this work is the application of the complete workflow to a real-life industrial model of an elastoviscoplastic high-pressure turbine blade subjected to thermal, centrifugal and pressure loadings, for the quantification of the uncertainty on dual quantities (such as the accumulated plastic strain and the stress tensor), generated by the uncertainty on the temperature loading field. The dictionary-based ROM-net computes predictions of dual quantities of interest for 1008 Monte Carlo draws of the temperature loading field in 2 hours and 48 minutes, which corresponds to a speedup greater than 600 with respect to a reference parallel solver using domain decomposition, with a relative error in the order of 2%. Another contribution of this work consists in the derivation of a meta-model to reconstruct the dual quantities of interest over the complete mesh from their values on the reduced integration points.

The R-Mode system, an advanced terrestrial integrated navigation system, is designed to address the vulnerabilities of global navigation satellite systems (GNSS) and explore the potential of a complementary navigation system. This study aims to enhance the accuracy of performance simulation for the medium frequency (MF) R-Mode system by modeling the variance of time-of-arrival (TOA) measurements based on actual data. Drawing inspiration from the method used to calculate the standard deviation of time-of-reception (TOR) measurements in Loran, we adapted and applied this approach to the MF R-Mode system. Data were collected from transmitters in Palmi and Chungju, South Korea, and the parameters for modeling the variance of TOA were estimated.

This paper rigorously shows how over-parameterization changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where M^* in R^{n times n} is a positive semi-definite unknown matrix of rank r ll n, and one uses a symmetric parameterization XX^top to learn M^*. Here X in R^{n times k} with k > r is the factor matrix. We give a novel Omega (1/T^2) lower bound of randomly initialized GD for the over-parameterized case (k >r) where T is the number of iterations. This is in stark contrast to the exact-parameterization scenario (k=r) where the convergence rate is exp (-Omega (T)). Next, we study asymmetric setting where M^* in R^{n_1 times n_2} is the unknown matrix of rank r ll min{n_1,n_2}, and one uses an asymmetric parameterization FG^top to learn M^* where F in R^{n_1 times k} and G in R^{n_2 times k}. Building on prior work, we give a global exact convergence result of randomly initialized GD for the exact-parameterization case (k=r) with an exp (-Omega(T)) rate. Furthermore, we give the first global exact convergence result for the over-parameterization case (k>r) with an exp(-Omega(alpha^2 T)) rate where alpha is the initialization scale. This linear convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from Omega (1/T^2) to linear convergence. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of alpha, recovering the rate in the exact-parameterization case.

We propose ScaledGD(\lambda), a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, ScaledGD(\lambda) starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, ScaledGD(\lambda) is remarkably robust to ill-conditioning compared to vanilla gradient descent (GD) even with overprameterization. Specifically, we show that, under the Gaussian design, ScaledGD(\lambda) converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla GD which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

We consider the solvable neural scaling model with three parameters: data complexity, target complexity, and model-parameter-count. We use this neural scaling model to derive new predictions about the compute-limited, infinite-data scaling law regime. To train the neural scaling model, we run one-pass stochastic gradient descent on a mean-squared loss. We derive a representation of the loss curves which holds over all iteration counts and improves in accuracy as the model parameter count grows. We then analyze the compute-optimal model-parameter-count, and identify 4 phases (+3 subphases) in the data-complexity/target-complexity phase-plane. The phase boundaries are determined by the relative importance of model capacity, optimizer noise, and embedding of the features. We furthermore derive, with mathematical proof and extensive numerical evidence, the scaling-law exponents in all of these phases, in particular computing the optimal model-parameter-count as a function of floating point operation budget.

We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation through a Dyck tiling on a ribbon. We introduce a new combinatorial object similar to a tree-like tableau, which we call a forest. A forest is shown to give a permutation, and be bijective to a network corresponding to the inverse of the permutation. We show that the poset of networks is a finite graded lattice and admits an EL-labeling. By use of this EL-labeling, we show the lattice is supersolvable and compute the M\"obius function of an interval of the poset.

Stars on the AGB can exhibit acoustic pulsation modes of different radial orders, along with non-radial modes. These pulsations are essential to the mass-loss process and influence the evolutionary pathways of AGB stars. P-L relations serve as a valuable diagnostic for understanding stellar evolution along the AGB. 3D RHD simulations provide a powerful tool for investigating pulsation phenomena driven by convective processes and their non-linear coupling with stellar oscillations. We investigate multi-mode pulsations in AGB stars using advanced 3D 'star-in-a-box' simulations with the CO5BOLD code. Signatures of these multi-mode pulsations were weak in our previous 3D models. Our focus is on identifying and characterising the various pulsation modes, examining their persistence and transitions, and comparing the results with 1D model predictions and observational data where applicable. We produced a new model grid comprising AGB stars with current masses of 0.7, 0.8, and 1,M_{odot}. Fourier analysis was applied to dynamic, time-dependent quantities to extract dominant pulsation modes and their corresponding periods. Additionally, wavelet transforms were employed to identify mode-switching behaviour over time. The models successfully reproduce the P-L sequences found in AGB stars. Mode-switching phenomena are found in both the models and wavelet analyses of observational data, allowing us to infer similarities in the underlying pulsation dynamics. These 3D simulations highlight the natural emergence of multi-mode pulsations, including both radial and non-radial modes, driven by the self-consistent interplay of convection and oscillations. Our findings underscore the value of 3D RHD models in capturing the non-linear behaviour of AGB pulsations, providing insights into mode switching, envelope structures, and potential links to episodic mass-loss events.

Robust reinforcement learning is essential for deploying reinforcement learning algorithms in real-world scenarios where environmental uncertainty predominates. Traditional robust reinforcement learning often depends on rectangularity assumptions, where adverse probability measures of outcome states are assumed to be independent across different states and actions. This assumption, rarely fulfilled in practice, leads to overly conservative policies. To address this problem, we introduce a new time-constrained robust MDP (TC-RMDP) formulation that considers multifactorial, correlated, and time-dependent disturbances, thus more accurately reflecting real-world dynamics. This formulation goes beyond the conventional rectangularity paradigm, offering new perspectives and expanding the analytical framework for robust RL. We propose three distinct algorithms, each using varying levels of environmental information, and evaluate them extensively on continuous control benchmarks. Our results demonstrate that these algorithms yield an efficient tradeoff between performance and robustness, outperforming traditional deep robust RL methods in time-constrained environments while preserving robustness in classical benchmarks. This study revisits the prevailing assumptions in robust RL and opens new avenues for developing more practical and realistic RL applications.

Adaptive optimization methods such as AdaGrad, RMSprop and Adam have been proposed to achieve a rapid training process with an element-wise scaling term on learning rates. Though prevailing, they are observed to generalize poorly compared with SGD or even fail to converge due to unstable and extreme learning rates. Recent work has put forward some algorithms such as AMSGrad to tackle this issue but they failed to achieve considerable improvement over existing methods. In our paper, we demonstrate that extreme learning rates can lead to poor performance. We provide new variants of Adam and AMSGrad, called AdaBound and AMSBound respectively, which employ dynamic bounds on learning rates to achieve a gradual and smooth transition from adaptive methods to SGD and give a theoretical proof of convergence. We further conduct experiments on various popular tasks and models, which is often insufficient in previous work. Experimental results show that new variants can eliminate the generalization gap between adaptive methods and SGD and maintain higher learning speed early in training at the same time. Moreover, they can bring significant improvement over their prototypes, especially on complex deep networks. The implementation of the algorithm can be found at https://github.com/Luolc/AdaBound .