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]^d. The reward functions r:[0,1]^d -> R are assumed to intrinsically depend on at most k coordinate variables implying r(x_1,..,x_d) = g(x_{i_1},..,x_{i_k}) for distinct and unknown i_1,..,i_k from {1,..,d} and some locally Holder continuous g:[0,1]^k -> R with exponent 0 < alpha <= 1. Firstly we consider the setting where (i_1,..,i_k) is fixed across time. We propose a simple modification of the CAB1 algorithm where we construct the discrete set of points to obtain a bound of O(n^((alpha+k)/(2*alpha+k)) (log n)^((alpha)/(2*alpha+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} and is probabilistic, hence our result holds with high probability. Secondly we show that in case (i_1,..,i_k) is allowed to vary at each time step, then the low dimensional structure of the reward functions is useless in the sense that the worst-case regret incurred by any algorithm will be Omega(2^d).