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

2026-02-11

AI Summary

This work addresses the incompatibility between traditional static Nash equilibria and individual regret in online dynamic games. To bridge this gap, the authors introduce two new performance measures: the Static Duality Gap (SDual-Gap) and the Dynamic Saddle-Point Regret (DSP-Reg). They develop a unified theoretical framework applicable to strongly convex–strongly concave functions, min-max exponentially concave functions, and those satisfying the two-sided Polyak–Łojasiewicz condition. By reducing the problem to classical online convex optimization (OCO), they design corresponding algorithms and establish tight theoretical bounds for the proposed metrics. The framework not only unifies the analysis across diverse function classes but also demonstrates practical efficacy through applications such as two-player portfolio selection, confirming its generality and real-world relevance.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• online min-max optimization
• cumulative saddle points
• individual regrets
• dynamic regret
• duality gap

Innovation

• cumulative saddle points
• dynamic saddle point regret
• static duality gap
• min-max exponential concavity
• online min-max optimization