Arya Akhavan

Arya Akhavan, Alexandre B. Tsybakov

We address the problem of zero-order optimization from noisy observations for an objective function satisfying the Polyak-Łojasiewicz or the strong convexity condition. Additionally, we assume that the objective function has an additive structure and satisfies a higher-order smoothness property, characterized by the Hölder family of functions. The additive model for Hölder classes of functions is well-studied in the literature on nonparametric function estimation, where it is shown that such a model benefits from a substantial improvement of the estimation accuracy compared to the Hölder model without additive structure. We study this established framework in the context of gradient-free optimization. We propose a randomized gradient estimator that, when plugged into a gradient descent algorithm, allows one to achieve minimax optimal optimization error of the order $dT^{-(β-1)/β}$, where $d$ is the dimension of the problem, $T$ is the number of queries and $β\ge 2$ is the Hölder degree of smoothness. We conclude that, in contrast to nonparametric estimation problems, no substantial gain of accuracy can be achieved when using additive models in gradient-free optimization.

Arya Akhavan, Massimiliano Pontil, Alexandre B. Tsybakov

We study the problem of zero-order optimization of a strongly convex function. The goal is to find the minimizer of the function by a sequential exploration of its values, under measurement noise. We study the impact of higher order smoothness properties of the function on the optimization error and on the cumulative regret. To solve this problem we consider a randomized approximation of the projected gradient descent algorithm. The gradient is estimated by a randomized procedure involving two function evaluations and a smoothing kernel. We derive upper bounds for this algorithm both in the constrained and unconstrained settings and prove minimax lower bounds for any sequential search method. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters. Based on this algorithm, we also propose an estimator of the minimum value of the function achieving almost sharp oracle behavior. We compare our results with the state-of-the-art, highlighting a number of key improvements.