Nearly Minimax-Optimal Regret for Linearly Parameterized Bandits
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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 ≤ 2^d/2 candidate actions per time period, we (1) show that the minimax expected regret is Ω(√(dT T 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 √( T) factors from previous analysis, and our information-theoretical lower bound also improves previous results by one √( 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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