Ahmadreza Momeni

Yonatan Gur, Ahmadreza Momeni, Stefan Wager

We study a non-parametric multi-armed bandit problem with stochastic covariates, where a key complexity driver is the smoothness of payoff functions with respect to covariates. Previous studies have focused on deriving minimax-optimal algorithms in cases where it is a priori known how smooth the payoff functions are. In practice, however, the smoothness of payoff functions is typically not known in advance, and misspecification of smoothness may severely deteriorate the performance of existing methods. In this work, we consider a framework where the smoothness of payoff functions is not known, and study when and how algorithms may adapt to unknown smoothness. First, we establish that designing algorithms that adapt to unknown smoothness of payoff functions is, in general, impossible. However, under a self-similarity condition (which does not reduce the minimax complexity of the dynamic optimization problem at hand), we establish that adapting to unknown smoothness is possible, and further devise a general policy for achieving smoothness-adaptive performance. Our policy infers the smoothness of payoffs throughout the decision-making process, while leveraging the structure of off-the-shelf non-adaptive policies. We establish that for problem settings with either differentiable or non-differentiable payoff functions, this policy matches (up to a logarithmic scale) the regret rate that is achievable when the smoothness of payoffs is known a priori.

Yonatan Gur, Ahmadreza Momeni

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.