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

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 $K=\left(\genfrac{}{}{0ex}{}{d}{m}\right)$ 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 $1-\delta$, 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. We complement this upper bound with a matching high-probability lower bound. For all sufficiently small $\delta$, every randomized policy admits a deterministic adaptive non-anticipating adversary for which, with probability at least $\delta$, \[ R_T = Ω\left(\sqrt{dT\log(K/δ)}\right). \] Thus, the rate is minimax optimal up to universal constants in this regime. In particular, setting $m=1$ proves that the $\log K$ for ordinary $K$-armed bandits against adaptive non-anticipating adversaries is unavoidable, closing the remaining $\sqrt{\log K}$ gap between confidence-tuned upper and lower bounds left by Gerchinovitz and Lattimore.