Alternating and Parallel Proximal Gradient Methods for Nonsmooth, Nonconvex Minimax: A Unified Convergence Analysis

Eyal Cohen, Marc Teboulle*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

There is growing interest in nonconvex minimax problems that is driven by an abundance of applications. Our focus is on nonsmooth, nonconvex-strongly concave minimax, thus departing from the more common weakly convex and smooth models assumed in the recent literature. We present proximal gradient schemes with either parallel or alternating steps. We show that both methods can be analyzed through a single scheme within a unified analysis that relies on expanding a general convergence mechanism used for analyzing nonconvex, nonsmooth optimization problems. In contrast to the current literature, which focuses on the complexity of obtaining nearly approximate stationary solutions, we prove subsequence convergence to a critical point of the primal objective and global convergence when the latter is semialgebraic. Furthermore, the complexity results we provide are with respect to approximate stationary solutions. Lastly, we expand the scope of problems that can be addressed by generalizing one of the steps with a Bregman proximal gradient update, and together with a few adjustments to the analysis, this allows us to extend the convergence and complexity results to this broader setting.

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalMathematics of Operations Research
DOIs
StatePublished - 2024

Funding

FundersFunder number
Israel Sci-ence Foundation
Deutsche Forschungsgemeinschaft800240
Israel Science Foundation2619-20

    Keywords

    • Bregman distance
    • Kurdyka–Lojasiewicz property
    • convergence rate
    • global convergence
    • nonconvex nonsmooth minimax
    • nonsmooth minimization-maximization
    • proximal gradient method

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