Analysis of Thompson Sampling for the Multi-armed Bandit Problem
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 multi-armed bandit problem. More precisely, for the two-armed bandit problem, the expected regret in time T is O(fracln TDelta + 1/Delta^(3)). And, for the N-armed bandit problem, the expected regret in time T is O([(sum_i = 2^(N) 1/Delta_i^(2))^(2)] ln T). Our bounds are optimal but for the dependence on Delta_i and the constant factors in big-Oh.
Summary¶
Provides one of the first logarithmic expected-regret analyses for Thompson sampling in stochastic multi-armed bandits. The paper helped move Thompson sampling from an empirically attractive Bayesian heuristic toward a theoretically grounded bandit algorithm, though with gap-dependent constants that later work refined.