A conversion theorem and minimax optimality for continuum contextual bandits

We study the contextual continuum bandits problem, where the learner sequentially receives a side information vector and has to choose an action in a convex set, minimizing a function associated with the context. The goal is to minimize all the underlying functions for the received contexts, leading to the contextual notion of regret, which is stronger than the standard static regret. Assuming that the objective functions are $\gamma$-H\"older with respect to the contexts, $0<\gamma\le 1,$ we demonstrate that any algorithm achieving a sub-linear static regret can be extended to achieve a sub-linear contextual regret. We prove a static-to-contextual regret conversion theorem that provides an upper bound for the contextual regret of the output algorithm as a function of the static regret of the input algorithm. We further study the implications of this general result for three fundamental cases of dependency of the objective function on the action variable: (a) Lipschitz bandits, (b) convex bandits, (c) strongly convex and smooth bandits. For Lipschitz bandits and $\gamma=1,$ combining our results with the lower bound of Slivkins (2014), we prove that the minimax optimal contextual regret for the noise-free adversarial setting is achieved. Then, we prove that in the presence of noise, the contextual regret rate as a function of the number of queries is the same for convex bandits as it is for strongly convex and smooth bandits. Lastly, we present a minimax lower bound, implying two key facts. First, obtaining a sub-linear contextual regret may be impossible over functions that are not continuous with respect to the context. Second, for convex bandits and strongly convex and smooth bandits, the algorithms that we propose achieve, up to a logarithmic factor, the minimax optimal rate of contextual regret as a function of the number of queries.