On the Optimality of Batch Policy Optimization Algorithms

Abstract

Batch policy optimization considers leveraging existing data for policy construction before interacting with an environment. Although interest in this problem has grown signi cantly in recent years, its theoretical foundations remain under-developed. To advance the understanding of this problem, we provide three results that characterize the limits and possibilities of batch policy optimization in the nite-armed stochastic bandit setting. First, we introduce a class of con dence-adjusted index algorithms that uni es optimistic and pessimistic principles in a common framework, which enables a general analysis. For this family, we show that any con dence-adjusted index algorithm is minimax optimal, whether it be optimistic, pessimistic or neutral. Our analysis reveals that instance-dependent optimality, commonly used to establish optimality of on-line stochastic bandit algorithms, cannot be achieved by any algorithm in the batch setting. In particular, for any algorithm that performs optimally in some environment, there exists another environment where the same algorithm su ers arbitrarily larger regret. Therefore, to establish a framework for distinguishing algorithms, we introduce a new weighted-minimax criterion that considers the inherent di culty of optimal value prediction. We demonstrate how this criterion can be used to justify commonly used pessimistic principles for batch policy optimization.