Nearly Minimax-Optimal Regret for Linearly Parameterized Bandits

Yingkai Li, Yining Wang, Yuan Zhou

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.