An Efficient Near-Optimal Algorithm for Adversarial $m$-Set Bandits

2026-08-12

AI Summary

This work addresses the exponential blow-up in action space inherent in adversarial combinatorial multi-armed bandits, where at each round an agent selects $m$ items out of $d$ and observes only an aggregated loss. The paper proposes an efficient algorithm that exploits the structural assumption that the loss is determined by a $d$-dimensional item loss vector, thereby avoiding explicit enumeration of all $\binom{d}{m}$ actions. By introducing a dual representation and parameterizing a low-dimensional sampling distribution, the method integrates online learning with combinatorial optimization to achieve, for the first time, a high-probability regret bound of $O(\sqrt{dT \log(K/\delta)})$ with probability at least $1-\delta$ (where $K = \binom{d}{m}$) in polynomial time. This matches the theoretical performance of EXP3-KW while eliminating exponential space complexity, resolving an open problem posed by Maiti et al.

Abstract

We study adversarial combinatorial bandits with $m$-set actions, where at each round the learner selects $m$ out of $d$ items and observes only the aggregate loss of the selected items. The resulting action set contains $K=\binom{d}{m}$ elements and can therefore be exponentially large. Nevertheless, the loss of every action is determined by the same $d$-dimensional vector of item losses. We propose a computationally efficient algorithm that exploits this structure without explicitly enumerating the action set. Against adaptive non-anticipating adversaries, it guarantees, with probability at least $1-δ$, regret against the best fixed action of \[ R_T = O\left(\sqrt{dT\log(K/δ)}\right). \] This matches the high-probability regret bound of the finite-action EXP3-KW algorithm of Zimmert and Lattimore, whose direct implementation may require exponential space. Our algorithm instead represents each sampling distribution with $d$ parameters and runs in polynomial time without enumerating the action set. Thus, it resolves the open problem posed by Maiti et al.

Problem

• adversarial bandits
• combinatorial bandits
• m-set actions
• large action space
• regret minimization

Innovation

• adversarial combinatorial bandits
• m-set actions
• efficient algorithm
• regret bound
• polynomial-time sampling