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# Analysis of Thompson Sampling for the Multi-armed Bandit Problem

; Proceedings of the 25th Annual Conference on Learning Theory, JMLR Workshop and Conference Proceedings 23:39.1-39.26, 2012.

#### Abstract

The multi-armed bandit problem is a popular model for studying exploration/exploitation trade-off in sequential decision problems. Many algorithms are now available for this well-studied problem. One of the earliest algorithms, given by W. R. Thompson, dates back to 1933. This algorithm, referred to as Thompson Sampling, is a natural Bayesian algorithm. The basic idea is to choose an arm to play according to its probability of being the best arm. Thompson Sampling algorithm has experimentally been shown to be close to optimal. In addition, it is efficient to implement and exhibits several desirable properties such as small regret for delayed feedback. However, theoretical understanding of this algorithm was quite limited. In this paper, for the first time, we show that Thompson Sampling algorithm achieves logarithmic expected regret for the stochastic multi-armed bandit problem. More precisely, for the stochastic two-armed bandit problem, the expected regret in time T is O(\frac\ln T∆ + \frac1∆^3). And, for the stochastic N-armed bandit problem, the expected regret in time T is O(\left[\left(\sum_i=2^N \frac1\Delta_i^2\right)^2\right] \ln T). Our bounds are optimal but for the dependence on \Delta_i and the constant factors in big-Oh.