Towards Distribution-Free Multi-Armed Bandits with Combinatorial Strategies

We consider the following linearly combinatorial multiarmed bandits (MABs) problem. In a discrete time system, there are K unknown random variables (RVs), i.e., arms, each evolving as an i.i.d stochastic process over time. At each time slot, we select a set of N (N ≤ K) RVs, i.e., strategy, subject to an arbitrarily constraint. We then gain a reward that is a linear combination of observations on selected RVs. Our goal is to minimize the regret, defined as the difference between the summed reward obtained by an optimal static policy that knew the mean of each RV, and that obtained by a specified learning policy that does not know. A prior result for this problem has achieved zero regret (the expect of regret over time approaches zero when time goes to infinity), but dependent on probability distribution of strategies generated by the learning policy. The regret becomes arbitrarily large if the difference between the reward of the best and second best strategy approaches zero. Meanwhile, when there are exponential number of combinations, naive extension of a prior distribution-free policy would cause poor performance in terms of regret, computation and space complexity. We propose an efficient Distribution-Free Learning (DFL) policy that achieves zero regret without dependence on probability distribution of strategies. Our learning policy only requires time and space complexity O(K). When the linear combination is involved with NP-hard problems, our policy provides a flexible scheme to choose possible approximation algorithms to solve the problem efficiently while retaining zero regret.