Optimism Stabilizes Thompson Sampling for Adaptive Inference

Thompson sampling (TS) is widely used for stochastic multi-armed bandits, yet its inferential behavior under adaptive data collection is subtle. Classical asymptotic theory for sample means can fail because arm-specific sample sizes are random, history-dependent, and coupled with the observed rewards through the action-selection rule. A useful sufficient condition for valid asymptotic inference is \emph{stability}, which requires each arm’s pull count to concentrate around a deterministic scale. While stability is now understood for several UCB-type algorithms, vanilla TS can be unstable, leading to nonstandard asymptotics and potentially invalid Wald-type confidence intervals. We identify optimism as a general mechanism for stabilizing Thompson sampling. In the $K$-armed Gaussian bandit with any fixed $K\ge2$, we study two optimistic TS variants. The first is TS with posterior variance inflation; the second keeps the posterior variance unchanged but adds an explicit optimism bonus to the posterior mean. For both variants, we prove stability: optimal arms asymptotically share the horizon uniformly, while each suboptimal arm is sampled on a sharp gap-dependent logarithmic scale. For variance-inflated TS, this resolves the open problem posed by Halder et al. (2025) by extending their two-armed stability theory to general $K$-armed bandits, including instances with multiple optimal arms. For the mean-bonus variant, our result shows that stability can also be achieved through a direct optimistic shift of the posterior center, without inflating the posterior variance. The main technical novelty lies in the treatment of variance-inflated TS with multiple optimal arms. In this regime, stability requires proving that the randomized competition among statistically indistinguishable optimal arms converges to a deterministic allocation. We isolate a limiting pure-noise competition and prove a negative-feedback property: over-sampled optimal arms become less likely to win futur