Gradient Testing and Estimation by Comparisons

TL;DR

This paper introduces comparison-based algorithms for gradient testing and estimation of smooth functions, achieving optimal query complexity in classical and quantum models.

Contribution

It presents the first comparison oracle algorithms for gradient testing and estimation with proven optimal query bounds, including quantum enhancements.

Findings

Classical gradient testing uses O(1) queries.

Classical gradient estimation uses O(n log(1/ε)) queries, proven optimal.

Quantum gradient estimation uses O(log(n/ε)) queries, showing quantum advantage.

Abstract

We study gradient testing and gradient estimation of smooth functions using only a comparison oracle that, given two points, indicates which one has the larger function value. For any smooth $f\colon\mathbb R^n\to\mathbb R$, $\mathbf{x}\in\mathbb R^n$, and $\varepsilon>0$, we design a gradient testing algorithm that determines whether the normalized gradient $\nabla f(\mathbf{x})/\|\nabla f(\mathbf{x})\|$ is $\varepsilon$-close or $2\varepsilon$-far from a given unit vector $\mathbf{v}$ using $O(1)$ queries, as well as a gradient estimation algorithm that outputs an $\varepsilon$-estimate of $\nabla f(\mathbf{x})/\|\nabla f(\mathbf{x})\|$ using $O(n\log(1/\varepsilon))$ queries which we prove to be optimal. Furthermore, we study gradient estimation in the quantum comparison oracle model where queries can be made in superpositions, and develop a quantum algorithm using $O(\log (n/\varepsilon))$ queries.