A Study of Thompson Sampling Approach for the Sleeping Multi-Armed Bandit Problem

Abstract

The multi-armed bandit (MAB) problem provides a convenient abstraction for many online decision problems arising in modern applications including Internet display advertising, crowdsourcing, online procurement, smart grids, etc. Several variants of the MAB problem have been proposed to extend the basic model to a variety of practical and general settings. The sleeping multi-armed bandit (SMAB) problem is one such variant where the set of available arms varies with time. This study is focused on analyzing the efficacy of the Thompson Sampling algorithm for solving the SMAB problem. Any algorithm for the classical MAB problem is expected to choose one of K available arms (actions) in each of T consecutive rounds. Each choice of an arm generates a stochastic reward from an unknown but fixed distribution. The goal of the algorithm is to maximize the expected sum of rewards over the T rounds (or equivalently minimize the expected total regret), relative to the best fixed action in hindsight. In many real-world settings, however, not all arms may be available in any given round. For example, in Internet display advertising, some advertisers might choose to stay away from the auction due to budget constraints; in crowdsourcing, some workers may not be available at a given time due to timezone difference, etc. Such situations give rise to the sleeping MAB abstraction. In the literature, several upper confidence bound (UCB)-based approaches have been proposed and investigated for the SMAB problem. Our contribution is to investigate the efficacy of a Thomp-son Sampling-based approach. Our key finding is to establish a logarithmic regret bound, which non-trivially generalizes a similar bound known for this approach in the classical MAB setting. Our bound also matches (up to constants) the best-known lower bound for the SMAB problem. Furthermore, we show via detailed simulations, that the Thompson Sampling approach in fact outperforms the known algorithms for the SMAB problem.