[Paper Review] Swap Regret Minimization Through Response-Based Approachability

Ioannis Anagnostides, Gabriele Farina+3 more|arXiv (Cornell University)|Feb 5, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

Introduces a computationally efficient algorithm to minimize linear swap regret over general convex sets using a response-based approachability framework, achieving O(d√T) regret and matching lower bounds, while also minimizing profile swap regret.

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 $Ω(d^4 \sqrt{T})$ 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 $Ω(d \sqrt{T})$ 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.

Motivation & Objective

• Motivate the need for stronger regret notions (swap regret) in online optimization and their connections to correlated equilibria and non-manipulability.
• Develop a computationally efficient algorithm to minimize linear swap regret over general convex sets.
• Show that the algorithm also minimizes profile swap regret, ensuring non-manipulability.
• Provide a matching information-theoretic lower bound and discuss extensions to swap deviations with polynomial dimension.

Proposed method

• Reduce linear swap regret to an approachability problem using the best-response, convex hull K and target set S.
• Apply the response-based approachability algorithm of Bernstein and Shimkin (2015) with a preprocessing step that puts the strategy set into John’s position.
• Introduce geometric preconditioning (John’s position) to bound the Frobenius norms of endomorphisms and of the approachability space.
• Prove an O(d√T) bound on linear swap regret under preconditioning (Algorithm 2: Preconditioned response-based approachability).
• Extend the framework to swap deviations with polynomial dimension using mixed strategies (Algorithm 3).
• Establish matching lower bounds showing Ω(d√T) linear swap regret is unavoidable in the worst case (Theorem 5.1).

Experimental results

Research questions

• RQ1 Can linear swap regret be minimized efficiently over general convex sets beyond the simplex?
• RQ2 What is the tight regret rate achievable for linear swap regret with preconditioning, and is it information-theoretically optimal?
• RQ3 Can the approach be extended to swap deviations with polynomial dimension while preserving computational efficiency?
• RQ4 How does minimization of linear and profile swap regret relate to non-manipulability against strategic adversaries?
• RQ5 What are the limitations of existing methods (e.g., ellipsoid-based algorithms) and how does the proposed approach compare?

Key findings

• An efficient algorithm achieves LinearSwapRegT = O(d√T) for general convex sets after John’s preconditioning.
• There is a matching information-theoretic lower bound Ω(d√T) for large T, even for centrally symmetric P, implying optimality (up to constants).
• The classic Gordon et al. algorithm is information-theoretically optimal for linear swap regret but computationally inefficient; the new method improves practical efficiency considerably.
• The approach also minimizes profile swap regret, contributing to non-manipulability against adaptive adversaries.
• The framework extends to swap deviations with polynomial dimension, yielding improved PolyDimSwapRegT bounds over prior work.
• The lower bound construction uses a product set P = B1 × B∞ to demonstrate unavoidable regret growth under adversarial losses.