In this thesis, I study the minimax oracle complexity of distributed stochastic optimization. First, I present the "graph oracle model", an extension of the classic oracle complexity framework that can be applied to study distributed optimization algorithms. Next, I describe a general approach to proving optimization lower bounds for arbitrary randomized algorithms (as opposed to more restricted classes of algorithms, e.g., deterministic or "zero-respecting" algorithms), which is used extensively throughout the thesis. For the remainder of the thesis, I focus on the specific case of the "intermittent communication setting", where multiple computing devices work in parallel with limited communication amongst themselves. In this setting, I analyze the theoretical properties of the popular Local Stochastic Gradient Descent (SGD) algorithm in convex setting, both for homogeneous and heterogeneous objectives. I provide the first guarantees for Local SGD that improve over simple baseline methods, but show that Local SGD is not optimal in general. In pursuit of optimal methods in the intermittent communication setting, I then show matching upper and lower bounds for the intermittent communication setting with homogeneous convex, heterogeneous convex, and homogeneous non-convex objectives. These upper bounds are attained by simple variants of SGD which are therefore optimal. Finally, I discuss several additional assumptions about the objective or more powerful oracles that might be exploitable in order to develop better intermittent communication algorithms with better guarantees than our lower bounds allow.
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…
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…
School of Computer Science（计算机学院）；Wuhan University（武汉大学）；The University of Sydney（悉尼大学）；School of Cyber Science and Technology（网络空间科学与技术学院）；Shenzhen Campus of Sun Yat-sen University（中山大学深圳校区）；Nanyang Technological University（南洋理工大学）
University of Utah（犹他大学）；University of Science and Technology of China（中国科学技术大学）；Ant Group（蚂蚁集团）；JD company（京东公司）
Gradient descent-ascent (GDA) is a widely used algorithm for minimax optimization. However, GDA has been proved to converge to stationary points for nonconvex minimax optimization, which are suboptimal compared with local minimax points. In this work, we develop cubic regularization (CR) type algorithms that globally converge to local minimax points in nonconvex-strongly-concave minimax optimization. We first show that local minimax points are equivalent to second-order stationary points of a certain envelope function. Then, inspired by the classic cubic regularization algorithm, we propose an…
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…
We develop a new theoretical framework, the \emph{envelope complexity}, to analyze the minimax regret with logarithmic loss functions and derive a Bayesian predictor that adaptively achieves the minimax regret over high-dimensional $\ell_1$-balls within a factor of two. The prior is newly derived for achieving the minimax regret and called the \emph{spike-and-tails~(ST) prior} as it looks like. The resulting regret bound is so simple that it is completely determined with the smoothness of the loss function and the radius of the balls except with logarithmic factors, and it has a generalized fo…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Twente（特文特大学）；Pontificia Universidad Católica de Chile（智利天主教大学）；EPFL（洛桑联邦理工学院）；ETH Zürich（苏黎世联邦理工学院）
This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity bounds of $Ω(\sqrtκΔLε^{-2})$ and $Ω(n+\sqrt{nκ}ΔLε^{-2})$ for the two settings, respectively, where $κ$ is the condition number, $L$ is the smoothness constant, and $Δ$ is the initial gap. Our result reveals substantial gaps between these limits and best-known upper bounds in the literature. To close these gaps, we introduce a generic acceleration scheme that …
The Hong Kong University of Science and Technology（香港科技大学）；Peking University（北京大学）
This paper considers stochastic first-order algorithms for convex-concave minimax problems of the form $\min_{\bf x}\max_{\bf y}f(\bf x, \bf y)$, where $f$ can be presented by the average of $n$ individual components which are $L$-average smooth. For $μ_x$-strongly-convex-$μ_y$-strongly-concave setting, we propose a new method which could find a $\varepsilon$-saddle point of the problem in $\tilde{\mathcal O} \big(\sqrt{n(\sqrt{n}+κ_x)(\sqrt{n}+κ_y)}\log(1/\varepsilon)\big)$ stochastic first-order complexity, where $κ_x\triangleq L/μ_x$ and $κ_y\triangleq L/μ_y$. This upper bound is near optim…
In this paper, we consider nonconvex minimax optimization, which is gaining prominence in many modern machine learning applications such as GANs. Large-scale edge-based collection of training data in these applications calls for communication-efficient distributed optimization algorithms, such as those used in federated learning, to process the data. In this paper, we analyze Local stochastic gradient descent ascent (SGDA), the local-update version of the SGDA algorithm. SGDA is the core algorithm used in minimax optimization, but it is not well-understood in a distributed setting. We prove th…
The minimax optimization over Riemannian manifolds (possibly nonconvex constraints) has been actively applied to solve many problems, such as robust dimensionality reduction and deep neural networks with orthogonal weights (Stiefel manifold). Although many optimization algorithms for minimax problems have been developed in the Euclidean setting, it is difficult to convert them into Riemannian cases, and algorithms for nonconvex minimax problems with nonconvex constraints are even rare. On the other hand, to address the big data challenges, decentralized (serverless) training techniques have re…
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 …
Peking University（北京大学）；University of California, Los Angles（加州大学洛杉矶分校）
The empirical success of Multi-agent reinforcement learning is encouraging, while few theoretical guarantees have been revealed. In this work, we prove that the plug-in solver approach, probably the most natural reinforcement learning algorithm, achieves minimax sample complexity for turn-based stochastic game (TBSG). Specifically, we plan in an empirical TBSG by utilizing a `simulator' that allows sampling from arbitrary state-action pair. We show that the empirical Nash equilibrium strategy is an approximate Nash equilibrium strategy in the true TBSG and give both problem-dependent and probl…
The Ohio State University（俄亥俄州立大学）；Princeton University（普林斯顿大学）
Many important machine learning applications amount to solving minimax optimization problems, and in many cases there is no access to the gradient information, but only the function values. In this paper, we focus on such a gradient-free setting, and consider the nonconvex-strongly-concave minimax stochastic optimization problem. In the literature, various zeroth-order (i.e., gradient-free) minimax methods have been proposed, but none of them achieve the potentially feasible computational complexity of $\mathcal{O}(ε^{-3})$ suggested by the stochastic nonconvex minimization theorem. In this pa…
Near-Optimal Algorithms for Making the Gradient Small in Stochastic Minimax Optimization
Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；School of Data Science, Fudan University（复旦大学数据科学学院）；Shanghai Key Laboratory for Contemporary Applied Mathematics（上海市现代应用数学重点实验室）
Stochastic Zeroth-Order Optimization under Strongly Convexity and Lipschitz Hessian: Minimax Sample Complexity
University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）；University of Texas at Dallas（德克萨斯大学达拉斯分校）；University of California, Berkeley（加利福尼亚大学伯克利分校）；New York University（纽约大学）；Princeton University（普林斯顿大学）
We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset, then averages the local solutions into a global predictor. This partitioning leads to a substantial reduction in computation time versus the standard approach of performing kernel ridge regression on all N samples. Our two main theorems establish that despite the computational speed-up, statistical optimal…
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…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）
In this paper, we revisit the smooth and strongly-convex-strongly-concave minimax optimization problem. Zhang et al. (2021) and Ibrahim et al. (2020) established the lower bound $Ω\left(\sqrt{κ_xκ_y} \log \frac{1}ε\right)$ on the number of gradient evaluations required to find an $ε$-accurate solution, where $κ_x$ and $κ_y$ are condition numbers for the strong convexity and strong concavity assumptions. However, the existing state-of-the-art methods do not match this lower bound: algorithms of Lin et al. (2020) and Wang and Li (2020) have gradient evaluation complexity $\mathcal{O}\left( \sqrt…
University of Pittsburgh（匹兹堡大学）；Simon Fraser University（西蒙弗雷泽大学）
In the paper, we propose a class of accelerated zeroth-order and first-order momentum methods for both nonconvex mini-optimization and minimax-optimization. Specifically, we propose a new accelerated zeroth-order momentum (Acc-ZOM) method for black-box mini-optimization where only function values can be obtained. Moreover, we prove that our Acc-ZOM method achieves a lower query complexity of $\tilde{O}(d^{3/4}ε^{-3})$ for finding an $ε$-stationary point, which improves the best known result by a factor of $O(d^{1/4})$ where $d$ denotes the variable dimension. In particular, our Acc-ZOM does no…
We design accelerated algorithms with improved rates for several fundamental classes of optimization problems. Our algorithms all build upon techniques related to the analysis of primal-dual extragradient methods via relative Lipschitzness proposed recently by [CST21]. (1) Separable minimax optimization. We study separable minimax optimization problems $\min_x \max_y f(x) - g(y) + h(x, y)$, where $f$ and $g$ have smoothness and strong convexity parameters $(L^x, μ^x)$, $(L^y, μ^y)$, and $h$ is convex-concave with a $(Λ^{xx}, Λ^{xy}, Λ^{yy})$-blockwise operator norm bounded Hessian. We provide …
This paper resolves a longstanding open question pertaining to the design of near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current state-of-the-art first-order algorithms find an approximate Nash equilibrium using $\tilde{O}(κ_{\mathbf x}+κ_{\mathbf y})$ or $\tilde{O}(\min\{κ_{\mathbf x}\sqrt{κ_{\mathbf y}}, \sqrt{κ_{\mathbf x}}κ_{\mathbf y}\})$ gradient evaluations, where $κ_{\mathbf x}$ and $κ_{\mathbf y}$ are the condition numbers for the strong-convexity and strong-concavity assumptions. A gap still remains between these results and t…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；ETH Zürich（苏黎世联邦理工学院）；University of Washington（华盛顿大学）
We study the bilinearly coupled minimax problem: $\min_{x} \max_{y} f(x) + y^\top A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions and admit first-order gradient oracles. Surprisingly, no known first-order algorithms have hitherto achieved the lower complexity bound of $Ω((\sqrt{\frac{L_x}{μ_x}} + \frac{\|A\|}{\sqrt{μ_x μ_y}} + \sqrt{\frac{L_y}{μ_y}}) \log(\frac1{\varepsilon}))$ for solving this problem up to an $\varepsilon$ primal-dual gap in the general parameter regime, where $L_x, L_y,μ_x,μ_y$ are the corresponding smoothness and strongly convexity constants. We cl…
Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(ε^{-4/7})$ Second-Order Oracle Complexity
Tsinghua University（清华大学）；Shanghai Qizhi Institute（上海期智研究院）；The Chinese University of Hong Kong（香港中文大学）；Fudan University（复旦大学）；Shanghai AI Lab（上海人工智能实验室）
University of Waterloo（滑铁卢大学）；Waterloo AI Institute（滑铁卢人工智能研究所）；Vector Institute（向量研究所）；Borealis AI
Differential games, in particular two-player sequential zero-sum games (a.k.a. minimax optimization), have been an important modeling tool in applied science and received renewed interest in machine learning due to many recent applications, such as adversarial training, generative models and reinforcement learning. However, existing theory mostly focuses on convex-concave functions with few exceptions. In this work, we propose two novel Newton-type algorithms for nonconvex-nonconcave minimax optimization. We prove their local convergence at strict local minimax points, which are surrogates of …
While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic bounds on the clustering accuracy and sample complexity of learning a mixture of two isotropic Gaussians in high dimensions under small mean separation. If there is a sparse subset of relevant dimensions that determine the mean separation, then the sample complexity only depends …
Structured prediction can be considered as a generalization of many standard supervised learning tasks, and is usually thought as a simultaneous prediction of multiple labels. One standard approach is to maximize a score function on the space of labels, which decomposes as a sum of unary and pairwise potentials, each depending on one or two specific labels, respectively. For this approach, several learning and inference algorithms have been proposed over the years, ranging from exact to approximate methods while balancing the computational complexity. However, in contrast to binary and multicl…
Matching Multi-Loop Complexities with a Single Loop: Optimal Optimization Stationarity and Best-Known Game Stationarity in Nonconvex--Concave Minimax Optimization
We note that known methods achieving the optimal oracle complexity for first order convex optimization require quadratic memory, and ask whether this is necessary, and more broadly seek to characterize the minimax number of first order queries required to optimize a convex Lipschitz function subject to a memory constraint.
post.bgu.ac.il（BGU（可能是Ben-Gurion University of the Negev，具体需结合上下文））
We exhibit an efficient procedure for testing, based on a single long state sequence, whether an unknown Markov chain is identical to or $\varepsilon$-far from a given reference chain. We obtain nearly matching (up to logarithmic factors) upper and lower sample complexity bounds for our notion of distance, which is based on total variation. Perhaps surprisingly, we discover that the sample complexity depends solely on the properties of the known reference chain and does not involve the unknown chain at all, which is not even assumed to be ergodic.
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）
MIT（麻省理工学院）；Weizmann Institute of Science（魏茨曼科学研究所）
Multi-modal distributions are commonly used to model clustered data in statistical learning tasks. In this paper, we consider the Mixed Linear Regression (MLR) problem. We propose an optimal transport-based framework for MLR problems, Wasserstein Mixed Linear Regression (WMLR), which minimizes the Wasserstein distance between the learned and target mixture regression models. Through a model-based duality analysis, WMLR reduces the underlying MLR task to a nonconvex-concave minimax optimization problem, which can be provably solved to find a minimax stationary point by the Gradient Descent Asce…
Dictionary learning is the problem of estimating the collection of atomic elements that provide a sparse representation of measured/collected signals or data. This paper finds fundamental limits on the sample complexity of estimating dictionaries for tensor data by proving a lower bound on the minimax risk. This lower bound depends on the dimensions of the tensor and parameters of the generative model. The focus of this paper is on second-order tensor data, with the underlying dictionaries constructed by taking the Kronecker product of two smaller dictionaries and the observed data generated b…
ETH Zurich（苏黎世联邦理工学院）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学习系统中心）；Amazon（亚马逊）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Washington（华盛顿大学）
We study differentially private (DP) algorithms for smooth stochastic minimax optimization, with stochastic minimization as a byproduct. The holy grail of these settings is to guarantee the optimal trade-off between the privacy and the excess population loss, using an algorithm with a linear time-complexity in the number of training samples. We provide a general framework for solving differentially private stochastic minimax optimization (DP-SMO) problems, which enables the practitioners to bring their own base optimization algorithm and use it as a black-box to obtain the near-optimal privacy…
