Gradient-free stochastic optimization of derivatives under strong convexity

Abstract

We consider the problem of minimizing the $k$-th order partial derivative $f=\partial_j^k g$ of an unknown function $g$ along a fixed coordinate direction $j$, based on noisy queries of $g$. Assuming that $g$ has Hölder regularity ${β+k}$ for some $β\ge 2$, that $f$ is strongly convex on a compact convex set $Θ\subset\mathbb{R}^d$ and that $g$ and $f$ satisfy mild boundedness and Lipschitz regularity conditions on $Θ$, we propose a kernel-based estimator of $\nabla f$ and analyze the projected stochastic gradient algorithm driven by this estimator. We obtain a non-asymptotic upper bound on the optimization error of the order $d^{(2β+k-1)/(β+k)}\,N^{-(β-1)/(β+k)}$, where $N$ is the total number of queries. We also establish a minimax lower bound of the order $N^{-(β-1)/(β+k)}$ showing that this rate is optimal in $N$ over all sequential algorithms.