Nash Regret Guarantees for Linear Bandits

TL;DR

This paper introduces tight bounds for Nash regret in stochastic linear bandits, linking fairness and collective welfare, with algorithms tailored for finite and infinite arm sets under sub-Poisson reward assumptions.

Contribution

It develops a novel algorithm achieving near-optimal Nash regret bounds for linear bandits, incorporating fairness via Nash social welfare and advanced technical tools.

Findings

Achieves Nash regret of O(√(dν/T) log(T|X|)) for finite arm sets.

Provides a bound of O(d^{5/4} ν^{1/2} / √T log(T)) for infinite arm sets.

Results apply to bounded, positive rewards, ensuring broad applicability.

Abstract

We obtain essentially tight upper bounds for a strengthened notion of regret in the stochastic linear bandits framework. The strengthening -- referred to as Nash regret -- is defined as the difference between the (a priori unknown) optimum and the geometric mean of expected rewards accumulated by the linear bandit algorithm. Since the geometric mean corresponds to the well-studied Nash social welfare (NSW) function, this formulation quantifies the performance of a bandit algorithm as the collective welfare it generates across rounds. NSW is known to satisfy fairness axioms and, hence, an upper bound on Nash regret provides a principled fairness guarantee. We consider the stochastic linear bandits problem over a horizon of $T$ rounds and with set of arms ${X}$ in ambient dimension $d$. Furthermore, we focus on settings in which the stochastic reward -- associated with each arm in ${X}$ -- is a non-negative, $\nu$-sub-Poisson random variable. For this setting, we develop an algorithm that achieves a Nash regret of $O\left( \sqrt{\frac{d\nu}{T}} \log( T |X|)\right)$. In addition, addressing linear bandit instances in which the set of arms ${X}$ is not necessarily finite, we obtain a Nash regret upper bound of $O\left( \frac{d^\frac{5}{4}\nu^{\frac{1}{2}}}{\sqrt{T}} \log(T)\right)$. Since bounded random variables are sub-Poisson, these results hold for bounded, positive rewards. Our linear bandit algorithm is built upon the successive elimination method with novel technical insights, including tailored concentration bounds and the use of sampling via John ellipsoid in conjunction with the Kiefer-Wolfowitz optimal design.