UCB for Large-Scale Pure Exploration: Beyond Sub-Gaussianity

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arXiv:2511.22273v2 Announce Type: replace-cross Abstract: Selecting the best alternative from a finite set is the central objective of ranking and selection (R&S) and best arm identification (BAI). Traditional R&S or BAI approaches have predominantly relied on Gaussian or sub-Gaussian assumptions on the performance distributions of all alternatives, which limit their applicability to non-sub-Gaussian---especially heavy-tailed---problems. The need to move beyond sub-Gaussianity may become even more critical in large-scale problems, which tend to be especially sensitive to distributional specifications. In this paper, motivated by the widespread use of upper confidence bound (UCB) algorithms in sequential decision making, we investigate their performance in large-scale, non-sub-Gaussian R&S settings. We consider the simplest category of UCB algorithms, where the UCB value for each alternative is defined as the sample mean plus an exploration bonus that depends only on its own sample size. We abstract this into a meta-UCB algorithm and propose letting it select the alternative with the largest sample size as the best upon stopping. For this meta-UCB algorithm, we first derive a distribution-free lower bound on the probability of correct selection. Building on this bound, we first study the indifference-zone formulation and show that the meta-UCB algorithm---and therefore a broad class of UCB algorithms---achieves sample optimality as long as the variances are uniformly bounded. We then extend sample optimality to a non-indifference-zone configuration with polynomially …

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