Statistically Efficient, Polynomial-Time Algorithms for Combinatorial Semi-Bandits

We consider combinatorial semi-bandits over a set of arms X \subset \0,1\ ^d where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields smallest known regret bound R(T) = O( d (łn m)^2 T) / Δ_\min ) after T rounds, m \max_x \in 1^\top x. However, it has computational complexity O(|X|), which is typically exponential in d, and cannot be used large dimensions. propose first that both computationally statistically efficient for problem with asymptotic O(δ_T^-1 poly(d)), δ_T function vanishes arbitrarily slowly. Our approach involves carefully designing AESCB, an approximate version same guarantees. show that, whenever budgeted linear maximization can solved up to given approximation ratio, AESCB implementable polynomial time poly(d)) by repeatedly maximizing subject budget constraint, showing how solve these problems efficiently.