Continuum Armed Bandit Problem of Few Variables in High Dimensions

We consider the stochastic and adversarial settings of continuum armed bandits where the arms are indexed by [0, 1]. The reward functions r : [0, 1] → R are assumed to intrinsically depend on at most k coordinate variables implying r(x1, . . . , xd) = g(xi1 , . . . , xik ) for distinct and unknown i1, . . . , ik ∈ {1, . . . , d} and some locally Hölder continuous g : [0, 1] → R with exponent α ∈ (0, 1]. Firstly, assuming (i1, . . . , ik) to be fixed across time, we propose a simple modification of the CAB1 algorithm where we construct the discrete set of sampling points to obtain a bound of O(n α+k 2α+k (logn) α 2α+k C(k, d)) on the regret, with C(k, d) depending at most polynomially in k and sub-logarithmically in d. The construction is based on creating partitions of {1, . . . , d} into k disjoint subsets and is probabilistic, hence our result holds with high probability. Secondly we extend our results to also handle the more general case where (i1, . . . , ik) can change over time and derive regret bounds for the same.