We introduce a simple (one line of code) modification to the Generative Adversarial Network (GAN) training algorithm that materially improves results with no increase in computational cost: When updating the generator parameters, we simply zero out the gradient contributions from the elements of the batch that the critic scores as `least realistic'. Through experiments on many different GAN variants, we show that this `top-k update' procedure is a generally applicable improvement. In order to understand the nature of the improvement, we conduct extensive analysis on a simple mixture-of-Gaussians dataset and discover several interesting phenomena. Among these is that, when gradient updates are computed using the worst-scoring batch elements, samples can actually be pushed further away from the their nearest mode.

Optimization is a central ingredient of machine learning. First-order optimization algorithms, for instance, are particularly popular for deep learning tasks due to their scalabilities to highdimensional problems, because they employ gradient but not higher-order information of objective functions for iteratively approximating minimizers. Among first-order methods, arguably the most used is gradient descent method (GD), or rather one of its variants, stochastic gradient descent method (SGD). Designed for objective functions that sum a large amount of terms, which for instance can originate from big data, SGD introduces a randomization mechanism of gradient subsampling to improve the scalability of GD (e.g., Zhang [2004], Moulines and Bach [2011], Roux et al. [2012]). Consequently, the iteration of SGD, unlike GD, is not deterministic even when it is started at a fixed initial condition.

We introduce a sampling perspective to tackle the challenging task of training robust Reinforcement Learning (RL) agents. Leveraging the powerful Stochastic Gradient Langevin Dynamics, we present a novel, scalable two-player RL algorithm, which is a sampling variant of the two-player policy gradient method. Our algorithm consistently outperforms existing baselines, in terms of generalization across different training and testing conditions, on several MuJoCo environments. Our experiments also show that, even for objective functions that entirely ignore potential environmental shifts, our sampling approach remains highly robust in comparison to standard RL algorithms.

We introduce a novel and efficient algorithm called the stochastic approximate gradient descent (SAGD), as an alternative to the stochastic gradient descent for cases where unbiased stochastic gradients cannot be trivially obtained. Traditional methods for such problems rely on general-purpose sampling techniques such as Markov chain Monte Carlo, which typically requires manual intervention for tuning parameters and does not work efficiently in practice. Instead, SAGD makes use of the Langevin algorithm to construct stochastic gradients that are biased in finite steps but accurate asymptotically, enabling us to theoretically establish the convergence guarantee for SAGD. Inspired by our theoretical analysis, we also provide useful guidelines for its practical implementation. Finally, we show that SAGD performs well experimentally in popular statistical and machine learning problems such as the expectation-maximization algorithm and the variational autoencoders.

We propose a risk-averse statistical learning framework wherein the performance of a learning algorithm is evaluated by the conditional value-at-risk (CVaR) of losses rather than the expected loss. We devise algorithms based on stochastic gradient descent for this framework. While existing studies of CVaR optimization require direct access to the underlying distribution, our algorithms make a weaker assumption that only i.i.d.\ samples are given. For convex and Lipschitz loss functions, we show that our algorithm has $O(1/\sqrt{n})$-convergence to the optimal CVaR, where $n$ is the number of samples. For nonconvex and smooth loss functions, we show a generalization bound on CVaR. By conducting numerical experiments on various machine learning tasks, we demonstrate that our algorithms effectively minimize CVaR compared with other baseline algorithms.

Stochastic gradient descent with momentum (SGDm) is one of the most popular optimization algorithms in deep learning. While there is a rich theory of SGDm for convex problems, the theory is considerably less developed in the context of deep learning where the problem is non-convex and the gradient noise might exhibit a heavy-tailed behavior, as empirically observed in recent studies. In this study, we consider a \emph{continuous-time} variant of SGDm, known as the underdamped Langevin dynamics (ULD), and investigate its asymptotic properties under heavy-tailed perturbations. Supported by recent studies from statistical physics, we argue both theoretically and empirically that the heavy-tails of such perturbations can result in a bias even when the step-size is small, in the sense that \emph{the optima of stationary distribution} of the dynamics might not match \emph{the optima of the cost function to be optimized}. As a remedy, we develop a novel framework, which we coin as \emph{fractional} ULD (FULD), and prove that FULD targets the so-called Gibbs distribution, whose optima exactly match the optima of the original cost. We observe that the Euler discretization of FULD has noteworthy algorithmic similarities with \emph{natural gradient} methods and \emph{gradient clipping}, bringing a new perspective on understanding their role in deep learning. We support our theory with experiments conducted on a synthetic model and neural networks.

In this paper, we study the minimax optimization problem in the smooth and strongly convex-strongly concave setting when we have access to noisy estimates of gradients. In particular, we first analyze the stochastic Gradient Descent Ascent (GDA) method with constant stepsize, and show that it converges to a neighborhood of the solution of the minimax problem. We further provide tight bounds on the convergence rate and the size of this neighborhood. Next, we propose a multistage variant of stochastic GDA (M-GDA) that runs in multiple stages with a particular learning rate decay schedule and converges to the exact solution of the minimax problem. We show M-GDA achieves the lower bounds in terms of noise dependence without any assumptions on the knowledge of noise characteristics. We also show that M-GDA obtains a linear decay rate with respect to the error's dependence on the initial error, although the dependence on condition number is suboptimal. In order to improve this dependence, we apply the multistage machinery to the stochastic Optimistic Gradient Descent Ascent (OGDA) algorithm and propose the M-OGDA algorithm which also achieves the optimal linear decay rate with respect to the initial error. To the best of our knowledge, this method is the first to simultaneously achieve the best dependence on noise characteristic as well as the initial error and condition number.

In this paper, we study the problem of sampling from distributions of the form p(x) \propto e^{-\beta f(x)} for some function f whose values and gradients we can query. This mode of access to f is natural in the scenarios in which such problems arise, for instance sampling from posteriors in parametric Bayesian models. Classical results show that a natural random walk, Langevin diffusion, mixes rapidly when f is convex. Unfortunately, even in simple examples, the applications listed above will entail working with functions f that are nonconvex -- for which sampling from p may in general require an exponential number of queries. In this paper, we study one aspect of nonconvexity relevant for modern machine learning applications: existence of invariances (symmetries) in the function f, as a result of which the distribution p will have manifolds of points with equal probability. We give a recipe for proving mixing time bounds of Langevin dynamics in order to sample from manifolds of local optima of the function f in settings where the distribution is well-concentrated around them. We specialize our arguments to classic matrix factorization-like Bayesian inference problems where we get noisy measurements A(XX^T), X \in R^{d \times k} of a low-rank matrix, i.e. f(X) = \|A(XX^T) - b\|^2_2, X \in R^{d \times k}, and \beta the inverse of the variance of the noise. Such functions f are invariant under orthogonal transformations, and include problems like matrix factorization, sensing, completion. Beyond sampling, Langevin dynamics is a popular toy model for studying stochastic gradient descent. Along these lines, we believe that our work is an important first step towards understanding how SGD behaves when there is a high degree of symmetry in the space of parameters the produce the same output.

Decentralized methods to solve finite-sum minimization problems are important in many signal processing and machine learning tasks where the data is distributed over a network of nodes and raw data sharing is not permitted due to privacy and/or resource constraints. In this article, we review decentralized stochastic first-order methods and provide a unified algorithmic framework that combines variance-reduction with gradient tracking to achieve both robust performance and fast convergence. We provide explicit theoretical guarantees of the corresponding methods when the objective functions are smooth and strongly-convex, and show their applicability to non-convex problems via numerical experiments. Throughout the article, we provide intuitive illustrations of the main technical ideas by casting appropriate tradeoffs and comparisons among the methods of interest and by highlighting applications to decentralized training of machine learning models. I. INTRODUCTION In multi-agent networks and large-scale machine learning, when data is available at different devices with limited communication, it is often desirable to seek scalable learning methods that do not require bringing, storing, and processing data at one single location. In this article, we describe decentralized, stochastic first-order methods, which are particularly favorable to such ad-hoc and resource-constrained settings. Specifically, we describe a unified algorithmic framework for combining different variance reduction methods with gradient tracking in order to significantly improve upon the performance of the standard decentralized stochastic gradient descent (DSGD). However, this improvement comes at a price of losing the simplicity of DSGD and we study the added communication, computation, and storage requirements with the help of precise technical statements. For the ease of accessibility, we restrict the theoretical arguments to smooth and strongly-convex objectives, while the applicability to non-convex problems is shown with the help of numerical experiments.

Adaptivity is an important yet under-studied property in modern optimization theory. The gap between the state-of-the-art theory and the current practice is striking in that algorithms with desirable theoretical guarantees typically involve drastically different settings of hyperparameters, such as step-size schemes and batch sizes, in different regimes. Despite the appealing theoretical results, such divisive strategies provide little, if any, insight to practitioners to select algorithms that work broadly without tweaking the hyperparameters. In this work, blending the "geometrization" technique introduced by Lei & Jordan 2016 and the \texttt{SARAH} algorithm of Nguyen et al., 2017, we propose the Geometrized \texttt{SARAH} algorithm for non-convex finite-sum and stochastic optimization. Our algorithm is proved to achieve adaptivity to both the magnitude of the target accuracy and the Polyak-\L{}ojasiewicz (PL) constant if present. In addition, it achieves the best-available convergence rate for non-PL objectives simultaneously while outperforming existing algorithms for PL objectives.