Minimax optimality for sequential gradient-free minimization of smooth functions and their derivatives
We consider the problem of noisy gradient-free minimization of the k-th order partial derivative of a $\beta$-H{\"o}lder function supported on a d-dimensional cube. We show that T ^{($\beta$+d+k)/(2$\beta$+d)} log(T )^{(\beta-k)/(2\beta+d)} is a non-asymptotic minimax rate of the T step cumulative regret for all $\beta$ \ge 0. In the special case k = 0, our results cover the problem of noisy gradient-free minimization of $\beta$-H{\"o}lder functions, closing the existing gap between the known upper and lower bounds. We show that a minimizer of a suitably chosen local polynomial estimator is rate-optimal. The minimax optimal upper bound is achieved under the passive design, that is, when the query points are i.i.d. Thus, there is no advantage in considering sequential designs when it is only known that f is a $\beta$-H{\"o}lder function with no additional property. We propose an algorithm feasible in polynomial time that constructs a proxy of the minimizer of the local polynomial estimator. The procedure requires computing the estimator on auxiliary random points. The resulting polynomial time algorithm matches the lower bound.