Combinatorial Network Optimization with Unknown Variables: Multi-Armed Bandits with Linear Rewards

Abstract

In the classic multi-armed bandits problem, the goal is to have a policy for dynamically operating arms that each yield stochastic rewards with unknown means. The key metric of interest is regret, defined as the gap between the expected total reward accumulated by an omniscient player that knows the reward means for each arm, and the expected total reward accumulated by the given policy. The policies presented in prior work have storage, computation and regret all growing linearly with the number of arms, which is not scalable when the number of arms is large. We consider in this work a broad class of multi-armed bandits with dependent arms that yield rewards as a linear combination of a set of unknown parameters. For this general framework, we present efficient policies that are shown to achieve regret that grows logarithmically with time, and polynomially in the number of unknown parameters (even though the number of dependent arms may grow exponentially). Furthermore, these policies only require storage that grows linearly in the number of unknown parameters. We show that this generalization is broadly applicable and useful for many interesting tasks in networks that can be formulated as tractable combinatorial optimization problems with linear objective functions, such as maximum weight matching, shortest path, and minimum spanning tree computations.