Nearly MinimaxOptimal Regret for Linearly Parameterized Bandits
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Proceedings of the ThirtySecond Conference on Learning Theory, PMLR 99:21732174, 2019.
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 VariableConfidenceLevel (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 informationtheoretical lower bound also improves previous results by one $\sqrt{\log T}$ factor, revealing a regret scaling quite different from classical multiarmed 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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