Sampling Allocation of LinUCB: Optimal Design Limits in the Small-Gap Regime

In stochastic linear bandits, the canonical Upper Confidence Bound (UCB) algorithm admits a simple frequentist regret analysis but can be computationally demanding, while Thompson Sampling (TS) is computationally attractive yet typically harder to analyze due to its non-optimistic nature. We propose Absolute Thompson Sampling (ATS), a simple modification of TS that ensures optimism in expectation by replacing the signed exploration noise with its absolute value. This preserves the computational efficiency of TS while avoiding the technically involved anti-concentration arguments common in TS analyses, enabling a simple UCB-style regret analysis. We show that ATS achieves $\tilde{O}(d^{3/2}\sqrt{K})$ regret, matching existing bounds for TS in linear bandits. We further introduce Ensemble Absolute Thompson Sampling (EATS), which takes the maximum over multiple absolute perturbations with normalization by the ensemble size. As the ensemble size grows, EATS converges to the UCB objective, recovering UCB behavior in the limit. Experiments show that moderate ensemble sizes already yield strong performance. Our results point to a bridge between randomized exploration and deterministic optimism both in theory and practice.

Allocation stability is often used to justify Gaussian inference from bandit data, but when is it necessary? In this paper, we address this question for a two-armed, fixed-horizon variance-aware UCB policy with bounded reward distributions that may vary with the horizon. We find a sharp criterion in terms of the reward gap and variances that determines whether the optimal-arm count admits a deterministic approximation with vanishing relative error, while the suboptimal-arm count is always stable. Despite the possible instability of the optimal-arm count, we show that the ordinary Wald statistic for a linear combination of the arm means has a standard normal limit for every fixed nonzero coefficient vector, provided the product of the pull count and reward variance diverges in probability for each arm. Under the same condition, however, this Gaussian approximation holds uniformly over deterministic nonzero coefficient vectors if and only if the optimal-arm count is stable. The analysis relies on two main ingredients: (i) a pathwise comparison with an auxiliary policy whose final optimal-arm count is asymptotically equivalent to the original count and independent of the optimal-arm reward sequence; and (ii) joint limits for the rescaled optimal-arm count and the two studentized sample-mean errors under the original policy, which yield nonstandard Wald limits for certain linear combinations of the arm means with coefficients that vary with the horizon.

We study the tail behavior of regret in stochastic multi-armed bandits for algorithms that are asymptotically optimal in expectation. While minimizing expected regret is the classical objective, recent work shows that even such algorithms can exhibit heavy regret tails, incurring large regret with non-negligible probability. Existing sharp characterizations of regret tails are largely restricted to parametric settings, such as single-parameter exponential families. In this work, we extend the $\KLinf$-UCB algorithm of to a broad nonparametric class of reward distributions satisfying mild assumptions, and establish its asymptotic optimality in expectation. We then analyze the tail behavior of its regret and derive a novel upper bound on the regret tail probability. As special cases, our results recover regret-tail guarantees for both bounded-support and heavy-tailed (moment-bounded) bandit models. Moreover, for the special case of finitely-supported reward distributions, our upper bound matches the known lower bound exactly. Our results thus provide a unified and tight characterization of regret tails for asymptotically optimal KL-based UCB algorithms, going beyond parametric models.