Gradient Testing and Estimation by Comparisons
Xiwen Tao, Chenyi Zhang, Helin Wang, Yexin Zhang, and Tongyang Li

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 , , and , we design a gradient testing algorithm that determines whether the normalized gradient is -close or -far from a given unit vector using queries, as well as a gradient estimation algorithm that outputs an -estimate of using 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…
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Taxonomy
TopicsStochastic Gradient Optimization Techniques · Advanced Bandit Algorithms Research · Complexity and Algorithms in Graphs
