Identifying near-optimal decisions in linear-in-parameter bandit models with continuous decision sets

Sanjay P. Bhat, Chaitanya Amballa

Keywords: best arm identification, linear bandits

TL;DR: We design an algorithm and provide theoretical bounds for the best arm identification problem in the fixed confidence setting with continuous decision sets.

Abstract: We consider an online optimization problem in a bandit setting in which a learner chooses decisions from a continuous decision set at discrete decision epochs, and receives noisy rewards from the environment in response. While the noise samples are assumed to be independent and sub-Gaussian, the mean reward at each epoch is a fixed but unknown linear function of a feature vector, which depends on the decision through a known (and possibly nonlinear) feature map. We study the problem within the framework of best-arm identification with fixed confidence, and provide a template algorithm for approximately learning the optimal decision in a probably approximately correct (PAC) setting. More precisely, the template algorithm samples the decision space till a stopping condition is met, and returns a subset of decisions such that, with the required confidence, every element of the subset is approximately optimal for the unknown mean reward function. We provide a sample complexity bound for the template algorithm and then specialize it to the case where the mean-reward function is a univariate polynomial of a single decision variable. We provide an implementable algorithm for this case by explicitly instantiating all the steps in the template algorithm. Finally, we provide experimental results to demonstrate the efficacy of our algorithms.