Efficient Swap Regret Minimization in Combinatorial Bandits

TL;DR

This paper introduces a new no-swap regret algorithm for combinatorial bandits that achieves polylogarithmic dependence on the number of actions, making it computationally efficient for large-scale problems.

Contribution

The paper presents the first efficient no-swap regret algorithm with polylogarithmic dependence on the action space size in combinatorial bandits, and demonstrates its practical implementation.

Findings

Achieves polylogarithmic swap regret in combinatorial bandits.

Provides an efficient implementation with polylogarithmic per-iteration complexity.

Proves tightness of the regret bounds for the proposed algorithm.

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.