Linear Bandits on Uniformly Convex Sets

03/10/2021

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by   Thomas Kerdreux, et al.

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Linear bandit algorithms yield 𝒪̃(n√(T)) pseudo-regret bounds on compact convex action sets 𝒦⊂ℝ^n and two types of structural assumptions lead to better pseudo-regret bounds. When 𝒦 is the simplex or an ℓ_p ball with p∈]1,2], there exist bandits algorithms with 𝒪̃(√(nT)) pseudo-regret bounds. Here, we derive bandit algorithms for some strongly convex sets beyond ℓ_p balls that enjoy pseudo-regret bounds of 𝒪̃(√(nT)), which answers an open question from [BCB12, 5.5.]. Interestingly, when the action set is uniformly convex but not necessarily strongly convex, we obtain pseudo-regret bounds with a dimension dependency smaller than 𝒪(√(n)). However, this comes at the expense of asymptotic rates in T varying between 𝒪̃(√(T)) and 𝒪̃(T).