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We study the problem of zero-order optimization of a strongly convexfunction. The goal is to find the minimizer of the function by a sequentialexploration of its values, under measurement noise. We study the impact ofhigher order smoothness properties of the function on the optimization errorand on the cumulative regret. To solve this problem we consider a randomizedapproximation of the projected gradient descent algorithm. The gradient isestimated by a randomized procedure involving two function evaluations and asmoothing kernel. We derive upper bounds for this algorithm both in theconstrained and unconstrained settings and prove minimax lower bounds for anysequential search method. Our results imply that the zero-order algorithm isnearly optimal in terms of sample complexity and the problem parameters. Basedon this algorithm, we also propose an estimator of the minimum value of thefunction achieving almost sharp oracle behavior. We compare our results withthe state-of-the-art, highlighting a number of key improvements.

Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits