Parameter-Free Heavy-Tailed Bandits

• URL: http://arxiv.org/abs/2607.29460v1
• Date: Fri, 31 Jul 2026 14:27:28 GMT
• Title: Parameter-Free Heavy-Tailed Bandits
• Abstract summary: We study adaptation to the moment bound $u$ for a fixed tail exponent $$.<n>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.
• Score: 36.983245230853434
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Bayesian Best-Arm Identification with Abstention: A Polynomial-to-Exponential Phase Transition [51.253617466579335]
  We study the Bayesian fixed-budget best-arm identification problem in which a learner can abstain from making a terminal recommendation.<n>We analyze the probability of undetected error--the risk of recommending a suboptimal arm without abstaining.
  arXiv  Detail & Related papers  (2026-06-28T05:07:27Z)
• Unified Framework of Distributional Regret in Multi-Armed Bandits and Reinforcement Learning [39.8867004581646]
  We formalize a distributional regret bound as a probabilistic guarantee that holds uniformly over all confidence levels $in (0,1]$.<n>We present a simple UCBVI-style algorithm with exploration bonus $minc_1,k/N, c_2,k/sqrtN$, where $N$ denotes the visit count and $(c_1,k,c_2,k)$ are user-specified parameters.<n>Our bounds achieve optimal trade-offs between expected and distributional regret in both minimax and instance-dependent regimes
  arXiv  Detail & Related papers  (2026-05-06T16:38:30Z)
• No-Regret Linear Bandits under Gap-Adjusted Misspecification [38.592043705502725]
  Existing linear bandits work usually relies on a uniform misspecification parameter $epsilon$ that measures the sup-norm error of the best linear approximation.<n>We propose a more natural model of misspecification which only requires the approximation error at each input $x$ to be proportional to the suboptimality gap at $x$.<n>We show that the classical LinUCB algorithm -- designed for the realizable case -- is automatically robust against such $rho$-gap-adjusted misspecification.
  arXiv  Detail & Related papers  (2025-01-09T16:44:53Z)
• Variance-Dependent Regret Bounds for Non-stationary Linear Bandits [52.872628573907434]
  We propose algorithms that utilize the variance of the reward distribution as well as the $B_K$, and show that they can achieve tighter regret upper bounds. We introduce two novel algorithms: Restarted Weighted$textOFUL+$ and Restarted $textSAVE+$. Notably, when the total variance $V_K$ is much smaller than $K$, our algorithms outperform previous state-of-the-art results on non-stationary linear bandits under different settings.
  arXiv  Detail & Related papers  (2024-03-15T23:36:55Z)
• $(\epsilon, u)$-Adaptive Regret Minimization in Heavy-Tailed Bandits [29.966828248335972]
  We study the regret minimization problem when $epsilon$ and $u$ are unknown to the learner. AdaR-UCB is the first algorithm that enjoys regret guarantees nearly matching those of the non-adaptive heavy-tailed case.
  arXiv  Detail & Related papers  (2023-10-04T17:11:15Z)
• Settling the Sample Complexity of Online Reinforcement Learning [92.02082223856479]
  We show how to achieve minimax-optimal regret without incurring any burn-in cost.<n>We extend our theory to unveil the influences of problem-dependent quantities like the optimal value/cost and certain variances.
  arXiv  Detail & Related papers  (2023-07-25T15:42:11Z)
• Variance-Dependent Regret Bounds for Linear Bandits and Reinforcement Learning: Adaptivity and Computational Efficiency [90.40062452292091]
  We present the first computationally efficient algorithm for linear bandits with heteroscedastic noise. Our algorithm is adaptive to the unknown variance of noise and achieves an $tildeO(d sqrtsum_k = 1K sigma_k2 + d)$ regret. We also propose a variance-adaptive algorithm for linear mixture Markov decision processes (MDPs) in reinforcement learning.
  arXiv  Detail & Related papers  (2023-02-21T00:17:24Z)
• Causal Bandits for Linear Structural Equation Models [58.2875460517691]
  This paper studies the problem of designing an optimal sequence of interventions in a causal graphical model. It is assumed that the graph's structure is known and has $N$ nodes. Two algorithms are proposed for the frequentist (UCB-based) and Bayesian settings.
  arXiv  Detail & Related papers  (2022-08-26T16:21:31Z)
• Minimal Expected Regret in Linear Quadratic Control [79.81807680370677]
  We devise an online learning algorithm and provide guarantees on its expected regret. This regret at time $T$ is upper bounded (i) by $widetildeO((d_u+d_x)sqrtd_xT)$ when $A$ and $B$ are unknown.
  arXiv  Detail & Related papers  (2021-09-29T14:07:21Z)
• Nearly Optimal Regret for Stochastic Linear Bandits with Heavy-Tailed Payoffs [35.988644745703645]
  We analyze the linear bandits with heavy-tailed payoffs, where the payoffs admit finite $1+epsilon$ moments. We propose two novel algorithms which enjoy a sublinear regret bound of $widetildeO(dfrac12Tfrac11+epsilon)$.
  arXiv  Detail & Related papers  (2020-04-28T13:01:38Z)
• Optimal $\delta$-Correct Best-Arm Selection for Heavy-Tailed Distributions [2.2940141855172036]
  We consider the problem of identifying the one with the maximum mean using a $delta$-correct algorithm. Lower bounds for $delta$-correct algorithms are well known. We propose a $delta$-correct algorithm that matches the lower bound as $delta$ reduces to zero.
  arXiv  Detail & Related papers  (2019-08-24T05:31:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.