Swap Regret Minimization Through Response-Based Approachability

2026-02-05

AI Summary

This work addresses the efficient minimization of linear and profile swap regret in online optimization. For general convex decision sets, it introduces a novel algorithm that combines the responsive approachability framework with geometric preprocessing via John’s ellipsoid, marking the first application of responsive approachability to swap regret. The method yields computationally efficient solutions with tight regret bounds: achieving $O(d^{3/2}\sqrt{T})$ linear swap regret over general convex sets and improving to $O(d\sqrt{T})$ for centrally symmetric sets, matching the established information-theoretic lower bound of $\Omega(d\sqrt{T})$. Furthermore, the algorithm simultaneously minimizes profile swap regret to guard against strategic manipulation and extends to polynomial-dimensional swap deviation sets, thereby unifying and strengthening the theoretical foundations of equilibrium computation and online learning.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We consider the problem of minimizing different notions of swap regret in online optimization. These forms of regret are tightly connected to correlated equilibrium concepts in games, and have been more recently shown to guarantee non-manipulability against strategic adversaries. The only computationally efficient algorithm for minimizing linear swap regret over a general convex set in $\mathbb{R}^d$ was developed recently by Daskalakis, Farina, Fishelson, Pipis, and Schneider (STOC'25). However, it incurs a highly suboptimal regret bound of $\Omega(d^4 \sqrt{T})$ and also relies on computationally intensive calls to the ellipsoid algorithm at each iteration. In this paper, we develop a significantly simpler, computationally efficient algorithm that guarantees $O(d^{3/2} \sqrt{T})$ linear swap regret for a general convex set and $O(d \sqrt{T})$ when the set is centrally symmetric. Our approach leverages the powerful response-based approachability framework of Bernstein and Shimkin (JMLR'15) -- previously overlooked in the line of work on swap regret minimization -- combined with geometric preconditioning via the John ellipsoid. Our algorithm simultaneously minimizes profile swap regret, which was recently shown to guarantee non-manipulability. Moreover, we establish a matching information-theoretic lower bound: any learner must incur in expectation $\Omega(d \sqrt{T})$ linear swap regret for large enough $T$, even when the set is centrally symmetric. This also shows that the classic algorithm of Gordon, Greenwald, and Marks (ICML'08) is existentially optimal for minimizing linear swap regret, although it is computationally inefficient. Finally, we extend our approach to minimize regret with respect to the set of swap deviations with polynomial dimension, unifying and strengthening recent results in equilibrium computation and online learning.

Problem

• swap regret
• online optimization
• correlated equilibrium
• approachability
• non-manipulability

Innovation

• swap regret minimization
• response-based approachability
• John ellipsoid
• online convex optimization
• non-manipulability