Epoch gradient descent method (a.k.a. Epoch-GD) proposed by Hazan and Kale (2011) was deemed a breakthrough for stochastic strongly convex minimization, which achieves the optimal convergence rate of $O(1/T)$ with $T$ iterative updates for the {\it objective gap}. However, its extension to solving stochastic min-max problems with strong convexity and strong concavity still remains open, and it is still unclear whether a fast rate of $O(1/T)$ for the {\it duality gap} is achievable for stochastic min-max optimization under strong convexity and strong concavity. Although some recent studies have proposed stochastic algorithms with fast convergence rates for min-max problems, they require additional assumptions about the problem, e.g., smoothness, bi-linear structure, etc. In this paper, we bridge this gap by providing a sharp analysis of epoch-wise stochastic gradient descent ascent method (referred to as Epoch-GDA) for solving strongly convex strongly concave (SCSC) min-max problems, without imposing any additional assumption about smoothness or the function's structure. To the best of our knowledge, our result is the first one that shows Epoch-GDA can achieve the optimal rate of $O(1/T)$ for the duality gap of general SCSC min-max problems. We emphasize that such generalization of Epoch-GD for strongly convex minimization problems to Epoch-GDA for SCSC min-max problems is non-trivial and requires novel technical analysis. Moreover, we notice that the key lemma can also be used for proving the convergence of Epoch-GDA for weakly-convex strongly-concave min-max problems, leading to a nearly optimal complexity without resorting to smoothness or other structural conditions.
Stochastic Gradient Descent (SGD) has become popular for solving large scale supervised machine learning optimization problems such as SVM, due to their strong theoretical guarantees. While the closely related Dual Coordinate Ascent (DCA) method has been implemented in various software packages, it has so far lacked good convergence analysis. This paper presents a new analysis of Stochastic Dual Coordinate Ascent (SDCA) showing that this class of methods enjoy strong theoretical guarantees that are comparable or better than SGD. This analysis justifies the effectiveness of SDCA for practical a…
An increasing number of machine learning problems, such as robust or adversarial variants of existing algorithms, require minimizing a loss function that is itself defined as a maximum. Carrying a loop of stochastic gradient ascent (SGA) steps on the (inner) maximization problem, followed by an SGD step on the (outer) minimization, is known as Epoch Stochastic Gradient \textit{Descent Ascent} (ESGDA). While successful in practice, the theoretical analysis of ESGDA remains challenging, with no clear guidance on choices for the inner loop size nor on the interplay between inner/outer step sizes.…
Stochastic dual coordinate ascent (SDCA) is an effective technique for solving regularized loss minimization problems in machine learning. This paper considers an extension of SDCA under the mini-batch setting that is often used in practice. Our main contribution is to introduce an accelerated mini-batch version of SDCA and prove a fast convergence rate for this method. We discuss an implementation of our method over a parallel computing system, and compare the results to both the vanilla stochastic dual coordinate ascent and to the accelerated deterministic gradient descent method of \cite{ne…
We introduce a proximal version of dual coordinate ascent method. We demonstrate how the derived algorithmic framework can be used for numerous regularized loss minimization problems, including $\ell_1$ regularization and structured output SVM. The convergence rates we obtain match, and sometimes improve, state-of-the-art results.
Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems
Shanghai University（上海大学）；Newtouch Center for Mathematics of Shanghai University（上海大学纽崔克数学中心）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
University of Toronto（多伦多大学）；Princeton University（普林斯顿大学）；Northeastern University（东北大学）
Smooth minimax games often proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice, the majority of existing theoretical analyses focus on simultaneous algorithms for convenience of analysis. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart~(Sim-GDA) in many settings. We prove that Alt-GDA achieves a near-optimal local convergence rate for strongly convex-strongly concave (SCSC) problems while Sim-GDA converges at a …
The Hong Kong University of Science and Technology（香港科技大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）
We consider nonconvex-concave minimax optimization problems of the form $\min_{\bf x}\max_{\bf y\in{\mathcal Y}} f({\bf x},{\bf y})$, where $f$ is strongly-concave in $\bf y$ but possibly nonconvex in $\bf x$ and ${\mathcal Y}$ is a convex and compact set. We focus on the stochastic setting, where we can only access an unbiased stochastic gradient estimate of $f$ at each iteration. This formulation includes many machine learning applications as special cases such as robust optimization and adversary training. We are interested in finding an ${\mathcal O}(\varepsilon)$-stationary point of the f…
There are much recent interests in solving noncovnex min-max optimization problems due to its broad applications in many areas including machine learning, networked resource allocations, and distributed optimization. Perhaps, the most popular first-order method in solving min-max optimization is the so-called simultaneous (or single-loop) gradient descent-ascent algorithm due to its simplicity in implementation. However, theoretical guarantees on the convergence of this algorithm is very sparse since it can diverge even in a simple bilinear problem. In this paper, our focus is to characterize …
This paper introduces AdaSDCA: an adaptive variant of stochastic dual coordinate ascent (SDCA) for solving the regularized empirical risk minimization problems. Our modification consists in allowing the method adaptively change the probability distribution over the dual variables throughout the iterative process. AdaSDCA achieves provably better complexity bound than SDCA with the best fixed probability distribution, known as importance sampling. However, it is of a theoretical character as it is expensive to implement. We also propose AdaSDCA+: a practical variant which in our experiments out…
Columbia University（哥伦比亚大学）；Singapore University of Technology and Design（新加坡科技设计大学）
We study a wide class of non-convex non-concave min-max games that generalizes over standard bilinear zero-sum games. In this class, players control the inputs of a smooth function whose output is being applied to a bilinear zero-sum game. This class of games is motivated by the indirect nature of the competition in Generative Adversarial Networks, where players control the parameters of a neural network while the actual competition happens between the distributions that the generator and discriminator capture. We establish theoretically, that depending on the specific instance of the problem …
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；University of Pittsburgh（匹兹堡大学）
In the paper, we study a class of useful minimax problems on Riemanian manifolds and propose a class of effective Riemanian gradient-based methods to solve these minimax problems. Specifically, we propose an effective Riemannian gradient descent ascent (RGDA) algorithm for the deterministic minimax optimization. Moreover, we prove that our RGDA has a sample complexity of $O(κ^2ε^{-2})$ for finding an $ε$-stationary solution of the Geodesically-Nonconvex Strongly-Concave (GNSC) minimax problems, where $κ$ denotes the condition number. At the same time, we present an effective Riemannian stochas…
Columbia University（哥伦比亚大学）；Singapore University of Technology and Design（新加坡科技设计大学）
Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convex non-concave zero-sum games, that we call hidden zero-sum games. In this class, players control the inputs of smooth but possibly non-linear functions whose outputs are being applied as inputs to a convex-concave game. Unlike general zero-sum games, these games have a well-defined notion of solution; outcomes that implement the von-Neumann equilibrium of the …
Local SGD is a promising approach to overcome the communication overhead in distributed learning by reducing the synchronization frequency among worker nodes. Despite the recent theoretical advances of local SGD in empirical risk minimization, the efficiency of its counterpart in minimax optimization remains unexplored. Motivated by large scale minimax learning problems, such as adversarial robust learning and training generative adversarial networks (GANs), we propose local Stochastic Gradient Descent Ascent (local SGDA), where the primal and dual variables can be trained locally and averaged…
Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aims to solve a nonconvex minimax optimization problem. However, the existing studies show that it suffers from a high computation complexity in nonconvex minimax optimization. In this paper, we develop a single-loop and fast AltGDA-type algorithm that leverages proximal gradient updates and momentum acceleration to solve regularized nonconvex minimax optimization problems. By leveraging the momentum acceleration technique, we pr…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；University of Pittsburgh（匹兹堡大学）
In the paper, we propose a class of faster adaptive Gradient Descent Ascent (GDA) methods for solving the nonconvex-strongly-concave minimax problems by using the unified adaptive matrices, which include almost all existing coordinate-wise and global adaptive learning rates. In particular, we provide an effective convergence analysis framework for our adaptive GDA methods. Specifically, we propose a fast Adaptive Gradient Descent Ascent (AdaGDA) method based on the basic momentum technique, which reaches a lower gradient complexity of $\tilde{O}(κ^4ε^{-4})$ for finding an $ε$-stationary point …
MIT（麻省理工学院）；CUHK（香港中文大学）；MIT-IBM AI Lab（麻省理工学院-IBM人工智能实验室）；IBM Research（IBM研究院）
Gradient Descent Ascent (GDA) methods are the mainstream algorithms for minimax optimization in generative adversarial networks (GANs). Convergence properties of GDA have drawn significant interest in the recent literature. Specifically, for $\min_{\mathbf{x}} \max_{\mathbf{y}} f(\mathbf{x};\mathbf{y})$ where $f$ is strongly-concave in $\mathbf{y}$ and possibly nonconvex in $\mathbf{x}$, (Lin et al., 2020) proved the convergence of GDA with a stepsize ratio $η_{\mathbf{y}}/η_{\mathbf{x}}=Θ(κ^2)$ where $η_{\mathbf{x}}$ and $η_{\mathbf{y}}$ are the stepsizes for $\mathbf{x}$ and $\mathbf{y}$ and…
Mirror Descent-Ascent for mean-field min-max problems
Center for Advanced Intelligence Project, RIKEN（日本东京RIKEN先进智能项目中心）；School of Mathematical and Computer Sciences, Heriot-Watt University（赫里奥特-瓦特大学数学与计算机科学学院）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（爱丁堡数学科学研究所）；School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）；The Alan Turing Institute（艾伦·图灵研究所）
A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems
Shenzhen Research Institute of Big Data（深圳大数据研究院）；School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）；Department of ISE, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校工业与系统工程系）
University of Utah（犹他大学）；The Ohio State University（俄亥俄州立大学）
The gradient descent-ascent (GDA) algorithm has been widely applied to solve minimax optimization problems. In order to achieve convergent policy parameters for minimax optimization, it is important that GDA generates convergent variable sequences rather than convergent sequences of function values or gradient norms. However, the variable convergence of GDA has been proved only under convexity geometries, and there lacks understanding for general nonconvex minimax optimization. This paper fills such a gap by studying the convergence of a more general proximal-GDA for regularized nonconvex-stro…
Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis
Department of ACMS University of Notre Dame（ACMS系，诺特大学）；Department of Mathematics University of California（数学系，加州大学）；Department of Mathematics University of Southern California（数学系，南加州大学）
We study the role that a finite timescale separation parameter $τ$ has on gradient descent-ascent in two-player non-convex, non-concave zero-sum games where the learning rate of player 1 is denoted by $γ_1$ and the learning rate of player 2 is defined to be $γ_2=τγ_1$. Existing work analyzing the role of timescale separation in gradient descent-ascent has primarily focused on the edge cases of players sharing a learning rate ($τ=1$) and the maximizing player approximately converging between each update of the minimizing player ($τ\rightarrow \infty$). For the parameter choice of $τ=1$, it is k…
ETH Zurich（苏黎世联邦理工学院）；UC Berkeley（加州大学伯克利分校）；Lehigh University（理海大学）
Communication remains the most significant bottleneck in the performance of distributed optimization algorithms for large-scale machine learning. In this paper, we propose a communication-efficient framework, CoCoA, that uses local computation in a primal-dual setting to dramatically reduce the amount of necessary communication. We provide a strong convergence rate analysis for this class of algorithms, as well as experiments on real-world distributed datasets with implementations in Spark. In our experiments, we find that as compared to state-of-the-art mini-batch versions of SGD and SDCA alg…
Georgia Institute of Technology（佐治亚理工学院）；Virginia Tech（弗吉尼亚理工大学）
We study the problem of finding the Nash equilibrium in a two-player zero-sum Markov game. Due to its formulation as a minimax optimization program, a natural approach to solve the problem is to perform gradient descent/ascent with respect to each player in an alternating fashion. However, due to the non-convexity/non-concavity of the underlying objective function, theoretical understandings of this method are limited. In our paper, we consider solving an entropy-regularized variant of the Markov game. The regularization introduces structure into the optimization landscape that make the soluti…
MIT-IBM Watson AI Lab, IBM Research（IBM研究中心MIT-IBM沃森人工智能实验室）；Thomas J. Watson Research Center, IBM Research（IBM研究中心托马斯·J·沃森研究中心）；University of Minnesota（明尼苏达大学）；Tsinghua University（清华大学）；Northeastern University（东北大学）；CSAIL, MIT（麻省理工学院计算机科学与人工智能实验室）；MIT（麻省理工学院）
In this paper, we study the problem of constrained robust (min-max) optimization ina black-box setting, where the desired optimizer cannot access the gradients of the objective function but may query its values. We present a principled optimization framework, integrating a zeroth-order (ZO) gradient estimator with an alternating projected stochastic gradient descent-ascent method, where the former only requires a small number of function queries and the later needs just one-step descent/ascent update. We show that the proposed framework, referred to as ZO-Min-Max, has a sub-linear convergence …
Innopolis University（Innopolis 大学）；HSE University（俄罗斯国立高等经济大学）；Yandex（Yandex 公司）；MBZUAI（穆罕默德·本·扎耶德人工智能大学）；Mila（米拉研究所）；DIRO, Université de Montréal（蒙特利尔大学 DIRO 学院）；AMS and MINDS, Johns Hopkins University（约翰斯·霍普金斯大学 AMS 与 MINDS 研究中心）
Stochastic Gradient Descent-Ascent (SGDA) is one of the most prominent algorithms for solving min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. The success of the method led to several advanced extensions of the classical SGDA, including variants with arbitrary sampling, variance reduction, coordinate randomization, and distributed variants with compression, which were extensively studied in the literature, especially during the last few years. In this paper, we propose a unified convergence analysis that covers a large variety of sto…
A.I. SQUARE CONNECT；Universite du Littoral - Cote d’Opale（滨海大学）；Université Paris Dauphine（巴黎多芬大学）
In machine learning, Feature Selection (FS) is a major part of efficient algorithm. It fuels the algorithm and is the starting block for our prediction. In this paper, we present a new method, called Optimal Coordinate Ascent (OCA) that allows us selecting features among block and individual features. OCA relies on coordinate ascent to find an optimal solution for gradient boosting methods score (number of correctly classified samples). OCA takes into account the notion of dependencies between variables forming blocks in our optimization. The coordinate ascent optimization solves the issue of …
Mila and DIRO, Université de Montréal（蒙特利尔大学 Mila 与 DIRO）
Two of the most prominent algorithms for solving unconstrained smooth games are the classical stochastic gradient descent-ascent (SGDA) and the recently introduced stochastic consensus optimization (SCO) [Mescheder et al., 2017]. SGDA is known to converge to a stationary point for specific classes of games, but current convergence analyses require a bounded variance assumption. SCO is used successfully for solving large-scale adversarial problems, but its convergence guarantees are limited to its deterministic variant. In this work, we introduce the expected co-coercivity condition, explain it…
Sogang University（西江大学）；University of Pennsylvania（宾夕法尼亚大学）；Princeton University（普林斯顿大学）
Minimizing the inclusive Kullback-Leibler (KL) divergence with stochastic gradient descent (SGD) is challenging since its gradient is defined as an integral over the posterior. Recently, multiple methods have been proposed to run SGD with biased gradient estimates obtained from a Markov chain. This paper provides the first non-asymptotic convergence analysis of these methods by establishing their mixing rate and gradient variance. To do this, we demonstrate that these methods-which we collectively refer to as Markov chain score ascent (MCSA) methods-can be cast as special cases of the Markov c…
Howard Hughes Medical Institute（霍华德·休斯医学研究所）；MIT（麻省理工学院）；Princeton University（普林斯顿大学）
A dynamical system is defined in terms of the gradient of a payoff function. Dynamical variables are of two types, ascent and descent. The ascent variables move in the direction of the gradient, while the descent variables move in the opposite direction. Dynamical systems of this form or very similar forms have been studied in diverse fields such as game theory, optimization, neural networks, and population biology. Gradient descent-ascent is approximated as a Newtonian dynamical system that conserves total energy, defined as the sum of the kinetic energy and a potential energy that is proport…
University of Southern California（南加州大学）；Princeton University（普林斯顿大学）
Recent applications that arise in machine learning have surged significant interest in solving min-max saddle point games. This problem has been extensively studied in the convex-concave regime for which a global equilibrium solution can be computed efficiently. In this paper, we study the problem in the non-convex regime and show that an \varepsilon--first order stationary point of the game can be computed when one of the player's objective can be optimized to global optimality efficiently. In particular, we first consider the case where the objective of one of the players satisfies the Polya…
Penn State Behrend（宾夕法尼亚州立大学贝伦德分校）；University of California, Davis（加州大学戴维斯分校）；University of Iowa（爱荷华大学）
Due to the big size of data and limited data storage volume of a single computer or a single server, data are often stored in a distributed manner. Thus, performing large-scale machine learning operations with the distributed datasets through communication networks is often required. In this paper, we study the convergence rate of the distributed dual coordinate ascent for distributed machine learning problems in a general tree-structured network. Since a tree network model can be understood as the generalization of a star network model, our algorithm can be thought of as the generalization of…
Stochastic gradient descent-ascent (SGDA) is one of the main workhorses for solving finite-sum minimax optimization problems. Most practical implementations of SGDA randomly reshuffle components and sequentially use them (i.e., without-replacement sampling); however, there are few theoretical results on this approach for minimax algorithms, especially outside the easier-to-analyze (strongly-)monotone setups. To narrow this gap, we study the convergence bounds of SGDA with random reshuffling (SGDA-RR) for smooth nonconvex-nonconcave objectives with Polyak-Łojasiewicz (PŁ) geometry. We analyze b…
NEC Labs America（美国NEC实验室）；Michigan State University（密歇根州立大学）
In \citep{Yangnips13}, the author presented distributed stochastic dual coordinate ascent (DisDCA) algorithms for solving large-scale regularized loss minimization. Extraordinary performances have been observed and reported for the well-motivated updates, as referred to the practical updates, compared to the naive updates. However, no serious analysis has been provided to understand the updates and therefore the convergence rates. In the paper, we bridge the gap by providing a theoretical analysis of the convergence rates of the practical DisDCA algorithm. Our analysis helped by empirical stud…
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 con…
Cornell University（康奈尔大学）；Google Research（谷歌研究院）；University of Chicago（芝加哥大学）
Unlike nonconvex optimization, where gradient descent is guaranteed to converge to a local optimizer, algorithms for nonconvex-nonconcave minimax optimization can have topologically different solution paths: sometimes converging to a solution, sometimes never converging and instead following a limit cycle, and sometimes diverging. In this paper, we study the limiting behaviors of three classic minimax algorithms: gradient descent ascent (GDA), alternating gradient descent ascent (AGDA), and the extragradient method (EGM). Numerically, we observe that all of these limiting behaviors can arise i…
University of Texas at Austin（得克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；Massachusetts Institute of Technology（麻省理工学院）
Generative adversarial networks (GANs) are a widely used framework for learning generative models. Wasserstein GANs (WGANs), one of the most successful variants of GANs, require solving a minmax optimization problem to global optimality, but are in practice successfully trained using stochastic gradient descent-ascent. In this paper, we show that, when the generator is a one-layer network, stochastic gradient descent-ascent converges to a global solution with polynomial time and sample complexity.
Institut für Neuroinformatik, Ruhr-Universität Bochum（波鸿鲁尔大学神经信息学研究所）
We present a dual subspace ascent algorithm for support vector machine training that respects a budget constraint limiting the number of support vectors. Budget methods are effective for reducing the training time of kernel SVM while retaining high accuracy. To date, budget training is available only for primal (SGD-based) solvers. Dual subspace ascent methods like sequential minimal optimization are attractive for their good adaptation to the problem structure, their fast convergence rate, and their practical speed. By incorporating a budget constraint into a dual algorithm, our method enjoys…
University of California, Davis（加州大学戴维斯分校）；University of Southern California（南加州大学）
In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first consider a deterministic version of the problem. We design and analyze the Zeroth-Order Gradient Descent Ascent (\texttt{ZO-GDA}) algorithm, and provide improved results compared to existing works, in terms of oracle complexity. We also propose the Zeroth-Order Gradient Descent Multi-Step A…
University of Edinburgh（爱丁堡大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent (SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss.
