Methods that align distributions by minimizing an adversarial distance between them have recently achieved impressive results. However, these approaches are difficult to optimize with gradient descent and they often do not converge well without careful hyperparameter tuning and proper initialization. We investigate whether turning the adversarial min-max problem into an optimization problem by replacing the maximization part with its dual improves the quality of the resulting alignment and explore its connections to Maximum Mean Discrepancy. Our empirical results suggest that using the dual formulation for the restricted family of linear discriminators results in a more stable convergence to a desirable solution when compared with the performance of a primal min-max GAN-like objective and an MMD objective under the same restrictions. We test our hypothesis on the problem of aligning two synthetic point clouds on a plane and on a real-image domain adaptation problem on digits. In both cases, the dual formulation yields an iterative procedure that gives more stable and monotonic improvement over time. Adversarial methods have recently become a popular choice for learning distributions of highdimensional data. The key idea is to learn a parametric representation of a distribution by aligning it with the empirical distribution of interest according to a distance given by a discriminative model. At the same time, the discriminative model is trained to differentiate between true and artificially obtained samples. Generative Adversarial Networks (GANs) that use neural networks to both discriminate samples and parameterize a learned distribution, have achieved particularly impressive results in many applications such as generative modeling of images BID9 BID4 BID26 , image super-resolution BID15 , and image-to-image translation BID13 . Adversarial matching of empirical distributions has also shown promise for aligning train and test data distributions n scenarios involving domain shift BID6 BID30 .However , GANs and related models have proved to be extremely difficult to optimize. It has been widely reported that training GANs is a tricky process that often diverges and requires very careful parameter initialization and tuning. Arjovsky and Bottou have recently identified several theoretical problems with loss functions used by GANs, and have analyzed how they contribute to instability and saturation during training.In this paper, we focus on one of the major barriers to stable optimization of adversarial methods, namely their min-max nature. Adversarial methods seek to match the generated and real distributions by minimizing some notion of statistical distance between the two, which is often defined as a maximal difference between values of certain test (witness) functions that could differentiate these distributions. More specifically in the case of GANs, the distance is usually considered to be equal to the likelihood of the best neural network classifier that discriminates between the distributions, assigning "real or generated?" labels to the input points. This way, in order to align these distributions one has to minimize this maximum likelihood w.r.t. the parameters of the learned or aligned distribution.Unfortunately, solving min-max problems using gradient descent is inherently very difficult. Below we use a simple example to demonstrate that different flavors of gradient descent are very unstable when it comes to solving problems of this kind.To address this issue, we explore the possibility of replacing the maximization part of the adversarial alignment problem with a dual minimization problem for linear and kernelized linear discriminators. The resulting dual problem turns out to be much easier to solve via gradient descent. Moreover, we make connections between our formulation and existing objectives such as the Maximum Mean Discrepancy (MMD) BID11 . We show that it is strongly related to the iteratively reweighted empirical estimator of MMD.We first evaluate how well our dual method can handle a point alignment problem on a lowdimensional synthetic dataset. Then, we compare its performance with the analogous primal method on a real-image domain adaptation problem using the Street View House Numbers (SVHN) and MNIST domain adaptation dataset pair. Here the goal is to align the feature distributions produced by the network on the two datasets so that a classifier trained to label digits on SVHN does not loose accuracy on MNIST due to the domain shift. In both cases, we show that our proposed dual formulation of the adversarial distance often shows improvement over time, whereas using the primal formulation results in drifting objective values and often does not converge to a solution.Our contributions can be summarized as follows:• we explore the dual formulation of the adversarial alignment objective for linear and kernelized linear discriminators and how they relate to the Maximum Mean Discrepancy; • we demonstrate experimentally on both synthetic and real datasets that the resulting objective leads to more stable convergence and better alignment quality; • we apply this idea to a domain adaptation scenario and show that the stability of reaching high target classification accuracy is also positively impacted by the dual formulation. While exact dual exists only in the logistic discriminator case, when one can use the duality and solve the inner problem in closed form, we want to stress that our paper presents a more general framework for alignment that can be extended to other classes of functions. More specifically, one can rewrite the quadratic form in kernel logistic regression (3) as a Frobenius inner product of a kernel matrix Q with a symmetric rank 1 alignment matrix S (outer product of alpha with itself). DISPLAYFORM0 The kernel matrix specifies distances between points and S chooses pairs that minimize the total distance. This way the problem reduces to "maximizing the maximum agreement between the alignment and similarity matrices" that in turn might be seen as replacing "adversity" in the original problem with "cooperation" in the dual maximization problem. In our paper, S is a rank 1 matrix, but we could choose different alignment matrix parameterizations and a corresponding regularizer that would correspond to having a neural network discriminator in the adversarial problem or a Wasserstein distance in Earth Mover's Distance form. The resulting problem is not dual to minimization of any existing adversarial distances, but exploits same underlying principle of "iteratively-reweighted alignment matrix fitting" discussed in this paper.We assert that in order to understand the basic properties of the resulting formulation, an in-depth discussion of the well-studied logistic case is no less important than the discussion involving complicated deep models, which deserves a paper of its own. This paper proposes a more stable "cooperative" problem reformulation rather than a new adversarial objective as many recent papers do. We presented an adversarial objective that does not lead to a min-max problem. We proposed using the dual of the discriminator objective to improve the stability of distribution alignment, showed its connection to MMD, and presented quantitative results for alignment of toy datasets and unsupervised domain adaptation results on real-image classification datasets. Our results suggest that the proposed dual optimization objective is indeed better suited for learning with gradient descent than the saddle point objective that naturally arises from the original primal formulation of adversarial alignment. Further attempts to use duality to reformulate other notions of statistical distances in adversarial settings as computationally feasible minimization problems may be promising. Bottom row: Evolution of target test accuracy over epochs. Our Dual objective (third column) clearly performs well under the majority of the learning rates. WGAN often performs better than MMD and ADDA, but experiences significant oscillations. Different validation heuristics, such as considering only runs that resulted in a significant drop in distance, did not significantly change these trends (Section 9.3). The proportion of runs that outperformed the source baseline after 40 epochs were: 52.3% for Dual, 21.5% for WGAN, 17.1% for MMD and 6.9% for ADDA.
 We propose a dual version of the logistic adversarial distance for feature alignment and show that it yields more stable gradient step iterations than the min-max objective. The paper deals with fixing GANs at the computational level This paper studies a dual formulation of an adversarial loss based on an upper-bound of the logistic loss, and turns the standard min max problem of adversarial training into a single minimization problem. Proposes to re-formulate the GAN saddle point objective (for a logistic regression discriminator) as a minimization problem by dualizing the maximum likelihood objective for regularized logistic regression