Adaptive Sequential Experiments with Unknown Information Arrival Processes

Abstract

Sequential experiments are often characterized by an exploration-exploitation tradeoff that is captured by the multi-armed bandit (MAB) framework. This framework has been studied and applied, typically when at each time period feedback is received only on the action that was selected at that period. However, in many practical settings additional data may become available between decision epochs. We introduce a generalized MAB formulation, which considers a broad class of distributions that are informative about mean rewards, and allows observations from these distributions to arrive according to an arbitrary and a priori unknown arrival process. When it is known how to map auxiliary data to reward estimates, by obtaining matching lower and upper bounds we characterize a spectrum of minimax complexities for this class of problems as a function of the information arrival process, which captures how salient characteristics of this process impact achievable performance. In terms of achieving optimal performance, we establish that upper confidence bound and posterior sampling policies possess natural robustness with respect to the information arrival process without any adjustments, which uncovers a novel property of these popular policies and further lends credence to their appeal. When the mappings connecting auxiliary data and rewards are a priori unknown, we characterize necessary and sufficient conditions under which auxiliary information allows performance improvement. We devise a new policy that is based on two different upper confidence bounds (one that accounts for auxiliary observation and one that does not) and establish the near-optimality of this policy. We use data from a large media site to analyze the value that may be captured in practice by leveraging auxiliary data for designing content recommendations.