Stochastic Compositional Minimax Optimization with Provable Convergence Guarantees

TL;DR

This paper introduces a new algorithm, CODA, for stochastic compositional minimax problems, providing convergence guarantees across various nonconvex and nonconcave settings, with a variance-reduced variant called CODA+.

Contribution

It formalizes stochastic compositional minimax problems, proposes the CODA algorithm with convergence analysis, and introduces CODA+ for improved rates in complex settings.

Findings

CODA converges to stationary points in nonconvex-strongly-concave and nonconvex-concave settings.

CODA+ achieves the best known convergence rates for nonconvex-strongly-concave problems.

The work extends theoretical understanding of stochastic compositional minimax optimization.

Abstract

Stochastic compositional minimax problems are prevalent in machine learning, yet there are only limited established on the convergence of this class of problems. In this paper, we propose a formal definition of the stochastic compositional minimax problem, which involves optimizing a minimax loss with a compositional structure either in primal , dual, or both primal and dual variables. We introduce a simple yet effective algorithm, stochastically Corrected stOchastic gradient Descent Ascent (CODA), which is a descent ascent type algorithm with compositional correction steps, and establish its convergence rate in aforementioned three settings. In the presence of the compositional structure in primal, the objective function typically becomes nonconvex in primal due to function composition. Thus, we consider the nonconvex-strongly-concave and nonconvex-concave settings and show that CODA can efficiently converge to a stationary point. In the case of composition on the dual, the objective function becomes nonconcave in the dual variable, and we demonstrate convergence in the strongly-convex-nonconcave and convex-nonconcave setting. In the case of composition on both variables, the primal and dual variables may lose convexity and concavity, respectively. Therefore, we anaylze the convergence in weakly-convex-weakly-concave setting. We also give a variance reduction version algorithm, CODA+, which achieves the best known rate on nonconvex-strongly-concave and nonconvex-concave compositional minimax problem. This work initiates the theoretical study of the stochastic compositional minimax problem on various settings and may inform modern machine learning scenarios such as domain adaptation or robust model-agnostic meta-learning.