  In this paper we consider Thompson Sampling (TS) for combinatorial
semi-bandits. We demonstrate that, perhaps surprisingly, TS 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, with Bernoulli distributed rewards and uniform priors. We
also show that including a fixed amount of forced exploration to TS does not
alleviate the problem. We complement our theoretical results with numerical
results and show that in practice TS indeed can perform very poorly in some
high dimensional situations.
