Tight Regret Bounds for Stochastic Combinatorial Semi-Bandits

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 / Δ) n) and O (K L n 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 Δ 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.