Finding Stationary Points by Comparisons
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Computer Science > Machine Learning
Title:Finding Stationary Points by Comparisons
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 $\epsilon$-stationary point using $\widetilde O(n^2/\epsilon^{1.5})$ queries. Our approach uses a subroutine that estimates the normalized Hessian to accuracy $\delta$ using $\widetilde O(n^2\log(1/\delta))$ 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 $\epsilon$-stationary point, which takes $\widetilde O(n/\epsilon^{1.5})$ queries.
| Comments: | 41 pages, 4 figures. To appear in the Forty-Third International Conference on Machine Learning (ICML 2026) |
| Subjects: | Machine Learning (cs.LG); Data Structures and Algorithms (cs.DS); Optimization and Control (math.OC); Quantum Physics (quant-ph) |
| Cite as: | arXiv:2606.27082 [cs.LG] |
| (or arXiv:2606.27082v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.27082
arXiv-issued DOI via DataCite (pending registration)
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