Batched Bandits with Heavy-Tailed Rewards

Heavy-tailed distributions arise naturally in sequential decision-making problems such as financial investment, online advertising, and network management, where rare but extreme outcomes can dominate performance. Heavy-tailed bandits model online decision-making in these settings by assuming only that rewards $X$ satisfy $\mathbb{E}[|X|^{1+ε}]\leq u$, for some tail exponent $ε\in(0,1]$ and moment bound $u<+\infty$. However, most existing regret minimization algorithms require these parameters to be known. This assumption is particularly restrictive in practice: $ε$ and $u$ govern the frequency and magnitude of rare events and are therefore precisely the quantities that are hardest to infer reliably from limited observations. Motivated by an open problem posed by Genalti and Metelli at COLT 2025, we resolve the assumption-free adaptation problem for heavy-tailed bandits and characterize the price in the regret of not knowing the tail parameters. We first study adaptation to the moment bound $u$ for a fixed tail exponent $ε$. We prove that every algorithm unaware of $u$, or of any upper bound on it, must obey a sharp trade-off between its distribution-dependent and distribution-free regret guarantees. We then introduce a scheduled-exploration algorithm that requires no knowledge of $u$ and matches the resulting adaptation frontier up to logarithmic factors. Finally, we show that the same algorithm can be instanced without knowing $ε$ by calibrating its exploration schedule to the endpoint $ε=1$. It achieves sublinear regret for every fixed $ε>0$, while no algorithm can guarantee sublinear regret uniformly over all $ε\in(0,1]$. Altogether, our results resolve the COLT open problem without additional distributional assumptions and provide a sharp characterization of the statistical cost of adapting to unknown heavy tails.

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.

The multi-armed bandit problem is a central framework in sequential decision-making, extensively studied under sub-Gaussian reward assumptions. However, real-world applications often involve heavy-tailed reward distributions and decentralized, information-asymmetric interactions. We study multi-agent multi-armed bandits with heavy-tailed rewards under three information-asymmetry regimes: unobserved actions with common rewards, observed actions with independent rewards, and unobserved actions with independent rewards. We develop robust decentralized algorithms for each setting and derive regret guarantees that nearly match centralized heavy-tailed rates. Experiments on a Pareto-distributed reward environment validate our theoretical findings and illustrate the trade-offs between synchronization, coordination, and exploration across the three regimes.