In this paper, we present a unified analysis of methods for such a wide class of problems as variational inequalities, which includes minimization problems and saddle point problems. We develop our analysis on the modified Extra-Gradient method (the classic algorithm for variational inequalities) and consider the strongly monotone and monotone cases, which corresponds to strongly-convex-strongly-concave and convex-concave saddle point problems. The theoretical analysis is based on parametric assumptions about Extra-Gradient iterations. Therefore, it can serve as a strong basis for combining the already existing type methods and also for creating new algorithms. In particular, to show this we develop new robust methods, which include methods with quantization, coordinate methods, distributed randomized local methods, and others. Most of these approaches have never been considered in the generality of variational inequalities and have previously been used only for minimization problems. The robustness of the new methods is also confirmed by numerical experiments with GANs.
In this article we propose a descent method for equality and inequality constrained multiobjective optimization problems (MOPs) which generalizes the steepest descent method for unconstrained MOPs by Fliege and Svaiter to constrained problems by using two active set strategies. Under some regularity assumptions on the problem, we show that accumulation points of our descent method satisfy a necessary condition for local Pareto optimality. Finally, we show the typical behavior of our method in a numerical example.
We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective function; convex or strongly convex objective and constraint; deterministic or randomized information about the objective and constraint. We hope that it is convenient for a reader to have all the methods for different settings in one place. Described methods are based on Mirror Descent algorithm and switching subgradient scheme. One of our focus is to propose, fo…
We propose a first-order method for solving inequality constrained optimization problems. The method is derived from our previous work [12], a modified search direction method (MSDM) that applies the singular-value decomposition of normalized gradients. In this work, we simplify its computational framework to a "gradient descent akin" method, i.e., the search direction is computed using a linear combination of the negative and normalized objective and constraint gradient. The main focus of this work is to provide a mathematical aspect to the method. We analyze the global behavior and convergen…
In this paper, we study the D-gap function associated with a nonsmooth and nonmonotone variational inequality problem. We present some exact formulas for the subderivative, the regular subdifferential set, and the limiting subdifferential set of the D-gap function. By virtue of these formulas, we provide some sufficient and necessary conditions for the Kurdyka-Łojasiewicz inequality property and the error bound property for the D-gap functions. As an application of our Kurdyka-Łojasiewicz inequality result and the abstract convergence result in [Attouch, et al., Convergence of descent methods …
Consensus on nonlinear spaces is of use in many control applications. This paper proposes a gradient descent flow algorithm for consensus on hypersurfaces. We show that if an inequality holds, then the system converges for almost all initial conditions and all connected graphs. The inequality involves the hypersurface Gauss map and the gradient and Hessian of the implicit equation. Moreover, for the inequality to hold, it is necessary that the manifold is the boundary of a convex set. The literature already contains an algorithm for consensus on hypersurfaces. That algorithm on any ellipsoid i…
This paper deals with the solving of variational inequality problem where the constrained set is given as the intersection of a number of fixed-point sets. To this end, we present an extrapolated sequential constraint method. At each iteration, the proposed method is updated based on the ideas of a hybrid conjugate gradient method used to accelerate the well-known hybrid steepest descent method, and an extrapolated cyclic cutter method for solving a common fixed point problem. We prove strong convergence of the method under some suitable assumptions of step-size sequences. We finally show the …
Minimax problems, such as generative adversarial network, adversarial training, and fair training, are widely solved by a multi-step gradient descent ascent (MGDA) method in practice. However, its convergence guarantee is limited. In this paper, inspired by the primal-dual hybrid gradient method, we propose a new semi-anchoring (SA) technique for the MGDA method. This makes the MGDA method find a stationary point of a structured nonconvex-nonconcave composite minimax problem; its saddle-subdifferential operator satisfies the weak Minty variational inequality condition. The resulting method, na…
In this paper, we consider a distributed constrained optimization problem with delayed subgradient information over the time-varying communication network, where each agent can only communicate with its neighbors and the communication channel has a limited data rate. We propose an adaptive quantization method to address this problem. A mirror descent algorithm with delayed subgradient information is established based on the theory of Bregman divergence. With non-Euclidean Bregman projection-based scheme, the proposed method essentially generalizes many previous classical Euclidean projection-b…
Microsoft Research（微软研究院）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Stein Variational Gradient Descent (SVGD) is an algorithm for sampling from a target density which is known up to a multiplicative constant. Although SVGD is a popular algorithm in practice, its theoretical study is limited to a few recent works. We study the convergence of SVGD in the population limit, (i.e., with an infinite number of particles) to sample from a non-logconcave target distribution satisfying Talagrand's inequality T1. We first establish the convergence of the algorithm. Then, we establish a dimension-dependent complexity bound in terms of the Kernelized Stein Discrepancy (KSD…
Black-box functions are broadly used to model complex problems that provide no explicit information but the input and output. Despite existing studies of black-box function optimization, the solution set satisfying an inequality with a black-box function plays a more significant role than only one optimum in many practical situations. Covering as much as possible of the solution set through limited evaluations to the black-box objective function is defined as the Black-Box Coverage (BBC) problem in this paper. We formalized this problem in a sample-based search paradigm and constructed a cover…
Existing implementations of gradient-based optimisation methods typically assume that the problem is posed in Euclidean space. When solving optimality problems on function spaces, the functional derivative is then inaccurately represented with respect to $\ell^2$ instead of the inner product induced by the function space. This error manifests as a mesh dependence in the number of iterations required to solve the optimisation problem. In this paper, an analytic estimate is derived for this iteration count in the case of a simple and generic discretised optimisation problem. The system analysed …
An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to possibly non-smooth and non-convex lower semi-continuous functions that satisfy the Kurdyka--Lojasiewicz (KL) inequality, which comprises a huge class of problems. Most of the recent algorithms that explicitly prove convergence using the KL inequality can cast into the abstract framework in this paper and, therefore, the generated sequence converges to a stat…
In this work, we reformulate the problem of existence of maximal elements for preference relations as a variational inequality problem in the sense of Stampacchia. Similarly, we establish the uniqueness of maximal elements using a variational inequality problem in the sense of Minty. In both of these approaches, we use the normal cone operator to find existence and uniqueness results, under mild assumptions. In addition, we provide an algorithm for finding such maximal elements, which is inspired by the steepest descent method for minimization. Under certain conditions, we prove that the seque…
We consider stochastic gradient descent algorithms for minimizing a non-smooth, strongly-convex function. Several forms of this algorithm, including suffix averaging, are known to achieve the optimal $O(1/T)$ convergence rate in expectation. We consider a simple, non-uniform averaging strategy of Lacoste-Julien et al. (2011) and prove that it achieves the optimal $O(1/T)$ convergence rate with high probability. Our proof uses a recently developed generalization of Freedman's inequality. Finally, we compare several of these algorithms experimentally and show that this non-uniform averaging stra…
We consider the problem of recovering a real-valued $n$-dimensional signal from $m$ phaseless, linear measurements and analyze the amplitude-based non-smooth least squares objective. We establish local convergence of subgradient descent with optimal sample complexity based on the uniform concentration of a random, discontinuous matrix-valued operator arising from the objective's gradient dynamics. While common techniques to establish uniform concentration of random functions exploit Lipschitz continuity, we prove that the discontinuous matrix-valued operator satisfies a uniform matrix concentr…
School of Science, Sun Yat-sen University（中山大学理学院）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统工程及工程管理学系）；School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
Linear convergence of proximal descent schemes on the Wasserstein space
Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan（日本东京RIKEN先进智能研究中心）；School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK（赫瑞-瓦特大学数学与计算机科学学院）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（爱丁堡数学科学麦克斯韦研究所）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；School of Mathematics, University of Edinburgh, UK, The Alan Turing Institute, UK（爱丁堡大学数学学院、英国艾伦·图灵研究所）
In 1963, Polyak proposed a simple condition that is sufficient to show a global linear convergence rate for gradient descent. This condition is a special case of the Łojasiewicz inequality proposed in the same year, and it does not require strong convexity (or even convexity). In this work, we show that this much-older Polyak-Łojasiewicz (PL) inequality is actually weaker than the main conditions that have been explored to show linear convergence rates without strong convexity over the last 25 years. We also use the PL inequality to give new analyses of randomized and greedy coordinate descent…
Provably solving stochastic convex optimization problems with constraints is essential for various problems in science, business, and statistics. Recently proposed XOR-Stochastic Gradient Descent (XOR-SGD) provides a convergence rate guarantee solving the constraints-free version of the problem by leveraging XOR-Sampling. However, the task becomes more difficult when additional equality and inequality constraints are needed to be satisfied. Here we propose XOR-PGD, a novel algorithm based on Projected Gradient Descent (PGD) coupled with the XOR sampler, which is guaranteed to solve the constra…
As the size and complexity of models and datasets grow, so does the need for communication-efficient variants of stochastic gradient descent that can be deployed to perform parallel model training. One popular communication-compression method for data-parallel SGD is QSGD (Alistarh et al., 2017), which quantizes and encodes gradients to reduce communication costs. The baseline variant of QSGD provides strong theoretical guarantees, however, for practical purposes, the authors proposed a heuristic variant which we call QSGDinf, which demonstrated impressive empirical gains for distributed train…
We provide a theoretical algorithm for checking local optimality and escaping saddles at nondifferentiable points of empirical risks of two-layer ReLU networks. Our algorithm receives any parameter value and returns: local minimum, second-order stationary point, or a strict descent direction. The presence of $M$ data points on the nondifferentiability of the ReLU divides the parameter space into at most $2^M$ regions, which makes analysis difficult. By exploiting polyhedral geometry, we reduce the total computation down to one convex quadratic program (QP) for each hidden node, $O(M)$ (in)equa…
University of Waterloo（滑铁卢大学）；Department of Electrical and Computer Engineering（电气与计算机工程系）；Department of Statistics and Actuarial Science（统计与精算科学系）；David R. Cheriton School of Computer Science（大卫·R·谢里顿计算机科学学院）；Centre for Pattern Analysis and Machine Intelligence（模式分析与机器智能中心）；Machine Learning Laboratory（机器学习实验室）；Data Analytics Laboratory（数据分析实验室）
This is a tutorial and survey paper on Karush-Kuhn-Tucker (KKT) conditions, first-order and second-order numerical optimization, and distributed optimization. After a brief review of history of optimization, we start with some preliminaries on properties of sets, norms, functions, and concepts of optimization. Then, we introduce the optimization problem, standard optimization problems (including linear programming, quadratic programming, and semidefinite programming), and convex problems. We also introduce some techniques such as eliminating inequality, equality, and set constraints, adding sl…
Jiangxi Normal University（江西师范大学）；The Hong Kong University of Science and Technology（香港科技大学）；Northwestern University（西北大学）；City University of Hong Kong（香港城市大学）
Deep learning has aroused extensive attention due to its great empirical success. The efficiency of the block coordinate descent (BCD) methods has been recently demonstrated in deep neural network (DNN) training. However, theoretical studies on their convergence properties are limited due to the highly nonconvex nature of DNN training. In this paper, we aim at providing a general methodology for provable convergence guarantees for this type of methods. In particular, for most of the commonly used DNN training models involving both two- and three-splitting schemes, we establish the global conve…
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Institute for Infocomm Research, A*STAR（新加坡科技研究局资讯通信研究院）；Nanyang Technological University（南洋理工大学）；Singapore University of Technology and Design（新加坡科技设计大学）
Owing to their connection with generative adversarial networks (GANs), saddle-point problems have recently attracted considerable interest in machine learning and beyond. By necessity, most theoretical guarantees revolve around convex-concave (or even linear) problems; however, making theoretical inroads towards efficient GAN training depends crucially on moving beyond this classic framework. To make piecemeal progress along these lines, we analyze the behavior of mirror descent (MD) in a class of non-monotone problems whose solutions coincide with those of a naturally associated variational i…
The monotone variational inequality is a central problem in mathematical programming that unifies and generalizes many important settings such as smooth convex optimization, two-player zero-sum games, convex-concave saddle point problems, etc. The extragradient algorithm by Korpelevich [1976] and the optimistic gradient descent-ascent algorithm by Popov [1980] are arguably the two most classical and popular methods for solving monotone variational inequalities. Despite their long histories, the following major problem remains open. What is the last-iterate convergence rate of the extragradient…
Conditional Value-at-Risk ($\mathrm{CV@R}$) is one of the most popular measures of risk, which has been recently considered as a performance criterion in supervised statistical learning, as it is related to desirable operational features in modern applications, such as safety, fairness, distributional robustness, and prediction error stability. However, due to its variational definition, $\mathrm{CV@R}$ is commonly believed to result in difficult optimization problems, even for smooth and strongly convex loss functions. We disprove this statement by establishing noisy (i.e., fixed-accuracy) li…
