Hybrid Block Successive Approximation for One-Sided Non-Convex Min-Max Problems: Algorithms and Applications

Abstract

The min-max problem, also known as the saddle point problem, is a class of optimization problems in which we minimize and maximize two subsets of variables simultaneously. This class of problems can be used to formulate a wide range of signal processing and communication (SPCOM) problems.Despite its popularity, existing theory for this class has been mainly developed for problems with certain special convex-concave structure. Therefore, it cannot be used to guide the algorithm design for many interesting problems in SPCOM, where some kind of non-convexity often arises.In this work, we consider a general block-wise one-sided non-convex min-max problem, in which the minimization problem consists of multiple blocks and is non-convex, while the maximization problem is (strongly) concave. We propose a class of simple algorithms named Hybrid Block Successive Approximation (HiBSA), which alternatingly performs gradient descent-type steps for the minimization blocks and one gradient ascent-type step for the maximization problem. A key element in the proposed algorithm is the introduction of certain properly designed regularization and penalty terms, which are used to stabilize the algorithm and ensure convergence. For the first time, we show that such simple alternating min-max algorithms converge to first-order stationary solutions, with quantifiable global rates. To validate the efficiency of the proposed algorithms, we conduct numerical tests on a number of information processing and wireless communication problems, including the robust learning problem, the non-convex min-utility maximization problems, and certain wireless jamming problem arising in interfering channels. * The first two authors contributed equally to this work, and the names are listed alphabetically. S. Lu, I. Tsaknakis and M. Hong are with the DRAFT Problem (1) is quite generic, and it arises in a wide range of signal processing and communication (SPCOM) applications. We list of few of these applications below. A. Motivating Examples in SPCOMDistributed non-convex optimization: Consider a network of K agents defined by a connected graph G = {V, E} with |V| = K, where each agent i can communicate with its neighbors. A generic problem formulation that captures many problems that appear in distributed machine learning and signal processing (e.g., [2]-[5]) can be formulated as follows:

Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity

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Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems

“…For example, multiple-loop algorithms have been studied in [29,30,31,32,33]. Our work in this paper is closely related to the recent literature on studying single-loop algorithm [34,35,36,37,38]. While these works study discretetime algorithms, we consider continuous-time counterpart.…”

Solving Non-Convex Non-Differentiable Min-Max Games Using Proximal Gradient Method

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“…Under this assumption, different algorithms such as optimistic mirror descent [13,14,15,16], Frank-Wolfe algorithm [17,18] and Primal-Dual method [19] have been studied.In the general non-convex settings, [20] considers the weakly convex-concave case and proposes a primal-dual based approach for finding approximate stationary solutions. More recently, the research works [21,22,23,24] * This arXiv submission includes the details of the proofs for the paper accepted for publication in the proceeding of the 45 th International Conference on Acoustics, Speech, and Signal Processing (ICASSP).…”