uniINF: Best-of-Both-Worlds Algorithm for Parameter-Free Heavy-Tailed MABs

TL;DR

uniINF is a novel parameter-free algorithm for heavy-tailed multi-armed bandits that performs optimally in both stochastic and adversarial settings without prior knowledge of environment parameters.

Contribution

It introduces the first parameter-free, Best-of-Both-Worlds algorithm for heavy-tailed MABs, extending robustness to adversarial environments and eliminating the need for known tail parameters.

Findings

Achieves nearly-optimal regret in stochastic and adversarial environments.

First parameter-free algorithm with BoBW guarantees for heavy-tailed MABs.

Develops new techniques like auto-balancing learning rates and adaptive loss tuning.

Abstract

In this paper, we present a novel algorithm, uniINF, for the Heavy-Tailed Multi-Armed Bandits (HTMAB) problem, demonstrating robustness and adaptability in both stochastic and adversarial environments. Unlike the stochastic MAB setting where loss distributions are stationary with time, our study extends to the adversarial setup, where losses are generated from heavy-tailed distributions that depend on both arms and time. Our novel algorithm `uniINF` enjoys the so-called Best-of-Both-Worlds (BoBW) property, performing optimally in both stochastic and adversarial environments without knowing the exact environment type. Moreover, our algorithm also possesses a Parameter-Free feature, i.e., it operates without the need of knowing the heavy-tail parameters $(\sigma, \alpha)$ a-priori. To be precise, uniINF ensures nearly-optimal regret in both stochastic and adversarial environments, matching the corresponding lower bounds when $(\sigma, \alpha)$ is known (up to logarithmic factors). To our knowledge, uniINF is the first parameter-free algorithm to achieve the BoBW property for the heavy-tailed MAB problem. Technically, we develop innovative techniques to achieve BoBW guarantees for Parameter-Free HTMABs, including a refined analysis for the dynamics of log-barrier, an auto-balancing learning rate scheduling scheme, an adaptive skipping-clipping loss tuning technique, and a stopping-time analysis for logarithmic regret.