On the Suboptimality of Thompson Sampling in High Dimensions

In this paper we consider Thompson Sampling for combinatorial semi-bandits. We demonstrate that, perhaps surprisingly, Thompson Sampling is sub-optimal for this problem in the sense that its regret scales exponentially in the ambient dimension, and its minimax regret scales almost linearly. This phenomenon occurs under a wide variety of assumptions including both non-linear and linear reward functions in the Bernoulli distribution setting. We also show that including a fixed amount of forced exploration to Thompson Sampling does not alleviate the problem. We complement our theoretical results with numerical results and show that in practice Thompson Sampling indeed can perform very poorly in some high dimension situations.