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# Tight Regret Bounds for Stochastic Combinatorial Semi-Bandits

*Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics*, PMLR 38:535-543, 2015.

#### Abstract

A stochastic combinatorial semi-bandit is an online learning problem where at each step a learning agent chooses a subset of ground items subject to constraints, and then observes stochastic weights of these items and receives their sum as a payoff. In this paper, we close the problem of computationally and sample efficient learning in stochastic combinatorial semi-bandits. In particular, we analyze a UCB-like algorithm for solving the problem, which is known to be computationally efficient; and prove O(K L (1 / ∆) \log n) and O(\sqrtK L n \log n) upper bounds on its n-step regret, where L is the number of ground items, K is the maximum number of chosen items, and ∆is the gap between the expected returns of the optimal and best suboptimal solutions. The gap-dependent bound is tight up to a constant factor and the gap-free bound is tight up to a polylogarithmic factor.