Finding Stationary Points by Comparisons

2026-06-25

AI Summary

This work addresses the problem of efficiently finding ε-stationary points in non-convex optimization under the comparison oracle model, where only comparisons between function values are accessible. The authors introduce, for the first time, a Hessian-normalized estimation subroutine based solely on comparison queries, leveraging both Lipschitz continuity of the gradient and structural properties of the Hessian. Building upon this subroutine, they design both classical and quantum algorithms: the classical algorithm requires Õ(n²/ε¹·⁵) comparison queries, while the quantum algorithm exploits quantum superposition to reduce the query complexity to Õ(n/ε¹·⁵). This quantum method constitutes the first algorithm capable of locating ε-stationary points in the comparison oracle setting, achieving a significant improvement in query efficiency over classical approaches.

Abstract

We study the problem of finding stationary points of non-convex functions when access to the objective is provided only through a comparison oracle that, given two points, outputs which has the larger function value. For a twice differentiable $f\colon\mathbb R^n\to\mathbb R$ with Lipschitz gradient and Hessian, we develop an algorithm that visits an $ε$-stationary point using $\widetilde O(n^2/ε^{1.5})$ queries. Our approach uses a subroutine that estimates the normalized Hessian to accuracy $δ$ using $\widetilde O(n^2\log(1/δ))$ queries. We further study this problem with a quantum comparison oracle model where queries can be made in superpositions, and develop the first quantum algorithm that finds an $ε$-stationary point, which takes $\widetilde O(n/ε^{1.5})$ queries.

Problem

• stationary points
• comparison oracle
• non-convex optimization
• query complexity
• quantum algorithm

Innovation

• comparison oracle
• non-convex optimization
• stationary point
• Hessian estimation
• quantum algorithm