Thompson Sampling and Approximate Inference

We study the effects of approximate inference on the performance of Thompson sampling in the k -armed bandit problems. Thompson sampling is a successful algorithm for online decision-making but requires posterior inference, which often must be approximated in practice. We show that even small constant inference error (in \alpha -divergence) can lead to poor performance (linear regret) due to under-exploration (for \alpha 1) or over-exploration (for \alpha 0) by the approximation. While for \alpha 0 this is unavoidable, for \alpha \leq 0 the regret can be improved by adding a small amount of forced exploration even when the inference error is a large constant.