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

arXiv:2608.12231v3 Announce Type: replace 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. 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 action in hindsight of $R_T =O(\sqrt{dT\log(K/\delta)}). $ This matches the high-probability regret bound of the finite-action EXP3 with Kiefer-Wolfowitz exploration 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. This resolves an open problem posed by Maiti et al. We complement this upper bound with a matching high-probability lower bound. For all sufficiently small $\delta$ and all sufficiently large horizons, every randomized algorithm admits an adaptive non-anticipating adversary for which, with probability at least $\delta$, $R_T=\Omega( \sqrt{dT\log(K/\delta)}). $ Thus, the rate is minimax optimal up to universal constants in this regime. In particular, setting $m=1$ shows that the $\log K$ factor in the high-probability regret bound for ordinary $K$-armed bandits against adaptive non-anticipating adversaries is unavoidable. This strengthens the lower bound of Gerchinovitz and Lattimore, closing the remaining $\sqrt{\log K}$ gap between high-probability upper and lower regret bounds.