p-Mean Regret for Stochastic Bandits

TL;DR

This paper introduces a flexible $p$-mean regret framework for stochastic bandits, providing a unified UCB-based algorithm with new bounds that balance fairness and efficiency across different $p$ values.

Contribution

The work extends $p$-mean welfare to bandit regret, proposing a simple unified algorithm with novel bounds applicable to a range of $p$ values, including Nash regret.

Findings

Achieves $p$-mean regret bounds of $ ilde{O}( oot{T^{1/2|p|}})$ for $p eq 0$

Matches lower bounds for $0< p extless 1$ up to logarithmic factors

Unifies analysis for average and Nash regret with a single algorithm.

Abstract

In this work, we extend the concept of the $p$-mean welfare objective from social choice theory (Moulin 2004) to study $p$-mean regret in stochastic multi-armed bandit problems. The $p$-mean regret, defined as the difference between the optimal mean among the arms and the $p$-mean of the expected rewards, offers a flexible framework for evaluating bandit algorithms, enabling algorithm designers to balance fairness and efficiency by adjusting the parameter $p$. Our framework encompasses both average cumulative regret and Nash regret as special cases. We introduce a simple, unified UCB-based algorithm (Explore-Then-UCB) that achieves novel $p$-mean regret bounds. Our algorithm consists of two phases: a carefully calibrated uniform exploration phase to initialize sample means, followed by the UCB1 algorithm of Auer, Cesa-Bianchi, and Fischer (2002). Under mild assumptions, we prove that our algorithm achieves a $p$-mean regret bound of $\tilde{O}\left(\sqrt{\frac{k}{T^{\frac{1}{2|p|}}}}\right)$ for all $p \leq -1$, where $k$ represents the number of arms and $T$ the time horizon. When $-1<p<0$, we achieve a regret bound of $\tilde{O}\left(\sqrt{\frac{k^{1.5}}{T^{\frac{1}{2}}}}\right)$. For the range $0< p \leq 1$, we achieve a $p$-mean regret scaling as $\tilde{O}\left(\sqrt{\frac{k}{T}}\right)$, which matches the previously established lower bound up to logarithmic factors (Auer et al. 1995). This result stems from the fact that the $p$-mean regret of any algorithm is at least its average cumulative regret for $p \leq 1$. In the case of Nash regret (the limit as $p$ approaches zero), our unified approach differs from prior work (Barman et al. 2023), which requires a new Nash Confidence Bound algorithm. Notably, we achieve the same regret bound up to constant factors using our more general method.