Nearly Minimax-Optimal Regret for Linearly Parameterized Bandits

TL;DR

This paper establishes nearly minimax-optimal regret bounds for linear contextual bandits with finite actions, introducing an improved algorithm and matching lower bounds that reveal a different regret scaling than classical multi-armed bandits.

Contribution

The paper presents a new VCL SupLinUCB algorithm with regret matching the lower bound up to logarithmic factors, improving previous bounds and analysis techniques.

Findings

Regret lower bound of (\u221a{dT(\u2318 T)(\u2318 n)}) for linear bandits.

VCL SupLinUCB algorithm achieves near-optimal regret matching the lower bound.

Improved analysis reduces (\u2318 T) factors compared to previous work.

Abstract

We study the linear contextual bandit problem with finite action sets. When the problem dimension is $d$, the time horizon is $T$, and there are $n \leq 2^{d/2}$ candidate actions per time period, we (1) show that the minimax expected regret is $\Omega(\sqrt{dT (\log T) (\log n)})$ for every algorithm, and (2) introduce a Variable-Confidence-Level (VCL) SupLinUCB algorithm whose regret matches the lower bound up to iterated logarithmic factors. Our algorithmic result saves two $\sqrt{\log T}$ factors from previous analysis, and our information-theoretical lower bound also improves previous results by one $\sqrt{\log T}$ factor, revealing a regret scaling quite different from classical multi-armed bandits in which no logarithmic $T$ term is present in minimax regret. Our proof techniques include variable confidence levels and a careful analysis of layer sizes of SupLinUCB on the upper bound side, and delicately constructed adversarial sequences showing the tightness of elliptical potential lemmas on the lower bound side.