The Typical Behavior of Bandit Algorithms

We establish strong laws of large numbers and central limit theorems for the regret of two of the most popular bandit algorithms: Thompson sampling and UCB. Here, our characterizations of the regret distribution complement the characterizations of the tail of the regret distribution recently developed by Fan and Glynn (2021) (arXiv:2109.13595). The tail characterizations there are associated with atypical bandit behavior on trajectories where the optimal arm mean is under-estimated, leading to mis-identification of the optimal arm and large regret. In contrast, our SLLN's and CLT's here describe the typical behavior and fluctuation of regret on trajectories where the optimal arm mean is properly estimated. We find that Thompson sampling and UCB satisfy the same SLLN and CLT, with the asymptotics of both the SLLN and the (mean) centering sequence in the CLT matching the asymptotics of expected regret. Both the mean and variance in the CLT grow at $\log(T)$ rates with the time horizon $T$. Asymptotically as $T \to \infty$, the variability in the number of plays of each sub-optimal arm depends only on the rewards received for that arm, which indicates that each sub-optimal arm contributes independently to the overall CLT variance.