Online Min-Max Optimization: From Individual Regrets to Cumulative Saddle Points

TL;DR

This paper introduces an online min-max optimization framework that extends beyond convex-concave functions, proposing new algorithms and bounds for saddle point measures and regret, applicable to complex two-player problems.

Contribution

It develops a novel online min-max optimization approach with bounds on saddle point measures, including a dynamic saddle point regret, under various function conditions.

Findings

Bounds for SDual-Gap and DSP-Reg achieved under strong convexity and min-max EC.

Introduces a class of functions satisfying min-max EC relevant to portfolio selection.

Provides regret bounds under a two-sided Polyak-c{}Lojasiewicz condition.

Abstract

We propose and study an online version of min-max optimization based on cumulative saddle points under a variety of performance measures beyond convex-concave settings. After first observing the incompatibility of (static) Nash equilibrium (SNE-Reg$_T$) with individual regrets even for strongly convex-strongly concave functions, we propose an alternate \emph{static} duality gap (SDual-Gap$_T$) inspired by the online convex optimization (OCO) framework. We provide algorithms that, using a reduction to classic OCO problems, achieve bounds for SDual-Gap$_T$~and a novel \emph{dynamic} saddle point regret (DSP-Reg$_T$), which we suggest naturally represents a min-max version of the dynamic regret in OCO. We derive our bounds for SDual-Gap$_T$~and DSP-Reg$_T$~under strong convexity-strong concavity and a min-max notion of exponential concavity (min-max EC), and in addition we establish a class of functions satisfying min-max EC~that captures a two-player variant of the classic portfolio selection problem. Finally, for a dynamic notion of regret compatible with individual regrets, we derive bounds under a two-sided Polyak-\L{}ojasiewicz (PL) condition.