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 π’ͺΜƒ(n√(T)) pseudo-regret bounds on compact convex action sets π’¦βŠ‚β„^n and two types of structural assumptions lead to better pseudo-regret bounds. When 𝒦 is the simplex or an β„“_p ball with p∈]1,2], there exist bandits algorithms with π’ͺΜƒ(√(nT)) pseudo-regret bounds. Here, we derive bandit algorithms for some strongly convex sets beyond β„“_p balls that enjoy pseudo-regret bounds of π’ͺΜƒ(√(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 π’ͺ(√(n)). However, this comes at the expense of asymptotic rates in T varying between π’ͺΜƒ(√(T)) and π’ͺΜƒ(T).

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