Linear Bandits on Uniformly Convex Sets
Linear bandit algorithms yield πͺΜ(nβ(T)) pseudo-regret bounds on compact convex action sets π¦ββ^n and two types of structural assumptions lead to better pseudo-regret bounds. When π¦ is the simplex or an β_p ball with pβ]1,2], there exist bandits algorithms with πͺΜ(β(nT)) pseudo-regret bounds. Here, we derive bandit algorithms for some strongly convex sets beyond β_p balls that enjoy pseudo-regret bounds of πͺΜ(β(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 πͺ(β(n)). However, this comes at the expense of asymptotic rates in T varying between πͺΜ(β(T)) and πͺΜ(T).
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