Sequential Batch Learning in Finite-Action Linear Contextual Bandits

Abstract

We study the sequential batch learning problem in linear contextual bandits with finite action sets, where the decision maker is constrained to split incoming individuals into (at most) a fixed number of batches and can only observe outcomes for the individuals within a batch at the batch's end. Compared with both standard online contextual-bandit learning and offline policy learning in contextual bandits, this sequential batch learning problem provides a finer-grained formulation of many personalized sequential decision making problems in practical applications, including medical treatment in clinical trials, product recommendation in e-commerce and adaptive experiment design in crowdsourcing. We study two settings of the problem: one where the contexts are arbitrarily generated and the other where the context vectors are mutually independent across actions and time and follow a common Gaussian distribution. In each setting, we establish a regret lower bound and provide an algorithm, whose regret upper bound nearly matches the lower bound. As an important insight revealed therefrom, in the former setting, we show that the number of batches required to achieve the fully online performance is polynomial in the time horizon, while for the latter setting, a pure-exploitation algorithm with a judicious batch partition scheme achieves the fully online performance even when the number of batches is less than logarithmic in the time horizon. In the stochastic context setting, we additionally provide tight margin-based (i.e. instance-dependent) upper and lower regret bounds that delineate performance in terms of how difficult the problem instance is. Together, our results provide a near-complete characterization of sequential decision making in linear contextual bandits when batch constraints are present.

To appear in Operations Research

References in corpus (4)

• Further Optimal Regret Bounds for Thompson Sampling
• Doubly Robust Policy Evaluation and Learning
• Efficient Optimal Learning for Contextual Bandits
• Nonparametric Bandits with Covariates

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• Design of Experiments for Stochastic Contextual Linear Bandits
• Linear Bandits with Limited Adaptivity and Learning Distributional Optimal Design
• Almost Optimal Batch-Regret Tradeoff for Batch Linear Contextual Bandits
• Pareto Optimal Model Selection in Linear Bandits
• Robust Bandit Learning with Imperfect Context
• The Impact of Batch Learning in Stochastic Bandits