Smoothness-Adaptive Contextual Bandits

• 成果类型: Article
• 署名作者: Gur, Yonatan; Momeni, Ahmadreza; Wager, Stefan
• 署名单位: Stanford University; Stanford University
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2021.2215
• 发表日期: 2022
• 页码: 3198-3216
• 关键词: contextual multi-armed bandits; Holder smoothness; Self-similarity; nonparametric confidence intervals; Nonparametric Estimation; Experiment design

摘要

We study a nonparametric multiarmed 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 nonadaptive policies. We establish that for problem settings with either differentiable or nondifferentiable 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.