We propose adaptive, line search-free second-order methods with optimal rate of convergence for solving convex-concave min-max problems. By means of an adaptive step size, our algorithms feature a simple update rule that requires solving only one linear system per iteration, eliminating the need for line search or backtracking mechanisms. Specifically, we base our algorithms on the optimistic method and appropriately combine it with second-order information. Moreover, distinct from common adaptive schemes, we define the step size recursively as a function of the gradient norm and the prediction error in the optimistic update. We first analyze a variant where the step size requires knowledge of the Lipschitz constant of the Hessian. Under the additional assumption of Lipschitz continuous gradients, we further design a parameter-free version by tracking the Hessian Lipschitz constant locally and ensuring the iterates remain bounded. We also evaluate the practical performance of our algorithm by comparing it to existing second-order algorithms for minimax optimization.
We investigate the challenge of multi-output learning, where the goal is to learn a vector-valued function based on a supervised data set. This includes a range of important problems in Machine Learning including multi-target regression, multi-class classification and multi-label classification. We begin our analysis by introducing the self-bounding Lipschitz condition for multi-output loss functions, which interpolates continuously between a classical Lipschitz condition and a multi-dimensional analogue of a smoothness condition. We then show that the self-bounding Lipschitz condition gives r…
Adaptive algorithms like AdaGrad and AMSGrad are successful in nonconvex optimization owing to their parameter-agnostic ability -- requiring no a priori knowledge about problem-specific parameters nor tuning of learning rates. However, when it comes to nonconvex minimax optimization, direct extensions of such adaptive optimizers without proper time-scale separation may fail to work in practice. We provide such an example proving that the simple combination of Gradient Descent Ascent (GDA) with adaptive stepsizes can diverge if the primal-dual stepsize ratio is not carefully chosen; hence, a fo…
China University of Petroleum-Beijing（中国石油大学（北京））；College of Information Science and Engineering（信息科学与工程学院）
In this paper, we study the optimistic online convex optimization problem in dynamic environments. Existing works have shown that Ader enjoys an $O\left(\sqrt{\left(1+P_T\right)T}\right)$ dynamic regret upper bound, where $T$ is the number of rounds, and $P_T$ is the path length of the reference strategy sequence. However, Ader is not environment-adaptive. Based on the fact that optimism provides a framework for implementing environment-adaptive, we replace Greedy Projection (GP) and Normalized Exponentiated Subgradient (NES) in Ader with Optimistic-GP and Optimistic-NES respectively, and name…
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…
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…
UC Berkeley（加州大学伯克利分校）；University of Washington（华盛顿大学）
This paper establishes that optimistic algorithms attain gap-dependent and non-asymptotic logarithmic regret for episodic MDPs. In contrast to prior work, our bounds do not suffer a dependence on diameter-like quantities or ergodicity, and smoothly interpolate between the gap dependent logarithmic-regret, and the $\widetilde{\mathcal{O}}(\sqrt{HSAT})$-minimax rate. The key technique in our analysis is a novel "clipped" regret decomposition which applies to a broad family of recent optimistic algorithms for episodic MDPs.
The Pennsylvania State University（宾夕法尼亚州立大学）；Massachusetts Institute of Technology（麻省理工学院）
Despite the established convergence theory of Optimistic Gradient Descent Ascent (OGDA) and Extragradient (EG) methods for the convex-concave minimax problems, little is known about the theoretical guarantees of these methods in nonconvex settings. To bridge this gap, for the first time, this paper establishes the convergence of OGDA and EG methods under the nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings by providing a unified analysis through the lens of single-call extra-gradient methods. We further establish lower bounds on the convergence of GDA/OGDA/EG, shedding …
In online learning an algorithm plays against an environment with losses possibly picked by an adversary at each round. The generality of this framework includes problems that are not adversarial, for example offline optimization, or saddle point problems (i.e. min max optimization). However, online algorithms are typically not designed to leverage additional structure present in non-adversarial problems. Recently, slight modifications to well-known online algorithms such as optimism and adaptive step sizes have been used in several domains to accelerate online learning -- recovering optimal r…
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 …
k Nearest Neighbor (kNN) method is a simple and popular statistical method for classification and regression. For both classification and regression problems, existing works have shown that, if the distribution of the feature vector has bounded support and the probability density function is bounded away from zero in its support, the convergence rate of the standard kNN method, in which k is the same for all test samples, is minimax optimal. On the contrary, if the distribution has unbounded support, we show that there is a gap between the convergence rate achieved by the standard kNN method a…
We study online convex optimization under stochastic sub-gradient observation faults, where we introduce adaptive algorithms with minimax optimal regret guarantees. We specifically study scenarios where our sub-gradient observations can be noisy or even completely missing in a stochastic manner. To this end, we propose algorithms based on sub-gradient descent method, which achieve tight minimax optimal regret bounds. When necessary, these algorithms utilize properties of the underlying stochastic settings to optimize their learning rates (step sizes). These optimizations are the main factor in…
Enhanced Adaptive Gradient Algorithms for Nonconvex-PL Minimax Optimization
College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics（南京航空航天大学计算机科学与技术学院）；MIIT Key Laboratory of Pattern Analysis and Machine Intelligence（信息产业部模式分析与机器智能重点实验室）；School of Automation Science and Engineering, Xi’an Jiaotong University（西安交通大学自动化科学与工程学院）
Online Convex Optimization plays a key role in large scale machine learning. Early approaches to this problem were conservative, in which the main focus was protection against the worst case scenario. But recently several algorithms have been developed for tightening the regret bounds in easy data instances such as sparsity, predictable sequences, and curved losses. In this work we unify some of these existing techniques to obtain new update rules for the cases when these easy instances occur together. First we analyse an adaptive and optimistic update rule which achieves tighter regret bound …
Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret
Facebook AI Research（Facebook人工智能研究院）；Inria Lille（法国国家信息与自动化研究所里尔中心）；Tsinghua University（清华大学）；University of Washington（华盛顿大学）；Facebook AI Research Paris（Facebook人工智能研究院巴黎）；DeepMind Paris（DeepMind巴黎）
INRIA Lille - Nord Europe（法国里尔 - 北欧 INRIA）；DeepMind Paris（巴黎深度思考公司（DeepMind Paris））；SequeL team（SequeL团队（非独立机构，按要求不提取））
We address the problem of optimizing a Brownian motion. We consider a (random) realization $W$ of a Brownian motion with input space in $[0,1]$. Given $W$, our goal is to return an $ε$-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order $\log^2(1/ε)$. This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive---each query depends on previous values---and is an instan…
Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Department of Statistics and Data Science, Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）
We consider the problem of global optimization of an unknown non-convex smooth function with zeroth-order feedback. In this setup, an algorithm is allowed to adaptively query the underlying function at different locations and receives noisy evaluations of function values at the queried points (i.e. the algorithm has access to zeroth-order information). Optimization performance is evaluated by the expected difference of function values at the estimated optimum and the true optimum. In contrast to the classical optimization setup, first-order information like gradients are not directly accessibl…
For linear time-invariant systems with uncertain parameters belonging to a finite set, we present a purely deterministic approach to multiple-model estimation and propose an algorithm based on the minimax criterion using constrained quadratic programming. The estimator tends to learn the dynamics of the system, and once the uncertain parameters have been sufficiently estimated, the estimator behaves like a standard Kalman filter.
For linear time-invariant systems having a state matrix with uncertain sign, we formulate a minimax adaptive control problem as a zero sum dynamic game. Explicit expressions for the optimal value function and the optimal control law are given in terms of a Riccati equation. The optimal control law is adaptive in the sense the past data is used to estimate the uncertain sign for prediction of future dynamics. Once the sign has been estimated, the controller behaves like standard H-infinity optimal state feedback.
Multi-agent interactions are increasingly important in the context of reinforcement learning, and the theoretical foundations of policy gradient methods have attracted surging research interest. We investigate the global convergence of natural policy gradient (NPG) algorithms in multi-agent learning. We first show that vanilla NPG may not have parameter convergence, i.e., the convergence of the vector that parameterizes the policy, even when the costs are regularized (which enabled strong convergence guarantees in the policy space in the literature). This non-convergence of parameters leads to…
This paper presents local asymptotic minimax regret lower bounds for adaptive Linear Quadratic Regulators (LQR). We consider affinely parametrized $B$-matrices and known $A$-matrices and aim to understand when logarithmic regret is impossible even in the presence of structural side information. After defining the intrinsic notion of an uninformative optimal policy in terms of a singularity condition for Fisher information we obtain local minimax regret lower bounds for such uninformative instances of LQR by appealing to van Trees' inequality (Bayesian Cramér-Rao) and a representation of regret…
EXOTIC: An Exact, Optimistic, Tree-Based Algorithm for Min-Max Optimization
Department of Electrical and Computer Engineering, and Data Science and AI Institute at Johns Hopkins University（约翰霍普金斯大学电气与计算机工程系及数据科学与人工智能研究院）；Institute for Computational and Mathematical Engineering at Stanford University（斯坦福大学计算与数学工程研究所）；Center for Systems and Control, IIT Bombay（印度理工学院班加罗尔系统与控制中心）
This paper studies the adaptive optimal stationary control of continuous-time linear stochastic systems with both additive and multiplicative noises, using reinforcement learning techniques. Based on policy iteration, a novel off-policy reinforcement learning algorithm, named optimistic least-squares-based policy iteration, is proposed which is able to find iteratively near-optimal policies of the adaptive optimal stationary control problem directly from input/state data without explicitly identifying any system matrices, starting from an initial admissible control policy. The solutions given …
In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and investigate a novel algorithm under the name of OGAProx, consisting of an optimistic gradient ascent step in the smooth variable coupled with a proximal step of the regulariser, and which is alternated with a {proximal step} in the nonsmooth component of the coupling function. We consider the situations…
The problem of estimating the $\mathcal{H}_\infty$-norm of an LTI system from noisy input/output measurements has attracted recent attention as an alternative to parameter identification for bounding unmodeled dynamics in robust control. In this paper, we study lower bounds for $\mathcal{H}_\infty$-norm estimation under a query model where at each iteration the algorithm chooses a bounded input signal and receives the response of the chosen signal corrupted by white noise. We prove that when the underlying system is an FIR filter, $\mathcal{H}_\infty$-norm estimation is no more efficient than …
The University of Iowa（爱荷华大学）；IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Adaptive gradient algorithms perform gradient-based updates using the history of gradients and are ubiquitous in training deep neural networks. While adaptive gradient methods theory is well understood for minimization problems, the underlying factors driving their empirical success in min-max problems such as GANs remain unclear. In this paper, we aim at bridging this gap from both theoretical and empirical perspectives. First, we analyze a variant of Optimistic Stochastic Gradient (OSG) proposed in~\citep{daskalakis2017training} for solving a class of non-convex non-concave min-max problem a…
JD Explore Academy, JD.com Inc（京东集团京东探索研究院）；University of Chicago（芝加哥大学）
Large scale convex-concave minimax problems arise in numerous applications, including game theory, robust training, and training of generative adversarial networks. Despite their wide applicability, solving such problems efficiently and effectively is challenging in the presence of large amounts of data using existing stochastic minimax methods. We study a class of stochastic minimax methods and develop a communication-efficient distributed stochastic extragradient algorithm, LocalAdaSEG, with an adaptive learning rate suitable for solving convex-concave minimax problems in the Parameter-Serve…
Motivated by the training of Generative Adversarial Networks (GANs), we study methods for solving minimax problems with additional nonsmooth regularizers. We do so by employing \emph{monotone operator} theory, in particular the \emph{Forward-Backward-Forward (FBF)} method, which avoids the known issue of limit cycling by correcting each update by a second gradient evaluation. Furthermore, we propose a seemingly new scheme which recycles old gradients to mitigate the additional computational cost. In doing so we rediscover a known method, related to \emph{Optimistic Gradient Descent Ascent (OGD…
