Nearly Minimax-Optimal Regret for Linearly Parameterized Bandits

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.

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