Efficient Swap Regret Minimization in Combinatorial Bandits

2026-02-02

AI Summary

This work addresses the challenge posed by the exponentially large action space \(N\) in combinatorial bandits, where existing algorithms struggle to simultaneously achieve low regret and computational efficiency. The paper proposes the first efficient algorithm that attains a polylogarithmic swap-regret bound in \(N\) while maintaining a per-round computational complexity of only \(\mathrm{polylog}(N)\). By integrating techniques from combinatorial optimization and online learning theory, the method introduces a carefully designed sampling and feedback mechanism that enables a low-overhead framework for swap-regret minimization. This approach offers both tight theoretical guarantees and practical scalability across a range of canonical combinatorial settings.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic 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

This paper addresses the problem of designing efficient no-swap regret algorithms for combinatorial bandits, where the number of actions $N$ is exponentially large in the dimensionality of the problem. In this setting, designing efficient no-swap regret translates to sublinear -- in horizon $T$ -- swap regret with polylogarithmic dependence on $N$. In contrast to the weaker notion of external regret minimization - a problem which is fairly well understood in the literature - achieving no-swap regret with a polylogarithmic dependence on $N$ has remained elusive in combinatorial bandits. Our paper resolves this challenge, by introducing a no-swap-regret learning algorithm with regret that scales polylogarithmically in $N$ and is tight for the class of combinatorial bandits. To ground our results, we also demonstrate how to implement the proposed algorithm efficiently -- that is, with a per-iteration complexity that also scales polylogarithmically in $N$ -- across a wide range of well-studied applications.

Problem

• swap regret
• combinatorial bandits
• no-swap regret
• regret minimization
• large action space

Innovation

• swap regret minimization
• combinatorial bandits
• polylogarithmic dependence
• efficient learning algorithm
• sublinear regret