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[2405.11454] Gradient Testing and Estimation by Comparisons
Summary
The paper presents algorithms for gradient testing and estimation using a comparison oracle, optimizing query efficiency for smooth functions in both classical and quantum settings.
Why It Matters
This research addresses the challenge of efficiently estimating gradients in machine learning, which is crucial for optimization tasks. By utilizing comparison oracles, the proposed methods enhance the performance of gradient-based algorithms, potentially impacting various applications in machine learning and quantum computing.
Key Takeaways
- Introduces a gradient testing algorithm that requires O(1) queries.
- Develops a gradient estimation algorithm with O(n log(1/ε)) queries, proven to be optimal.
- Explores quantum comparison oracles, achieving O(log(n/ε)) query complexity.
Computer Science > Machine Learning arXiv:2405.11454 (cs) [Submitted on 19 May 2024 (v1), last revised 19 Feb 2026 (this version, v2)] Title:Gradient Testing and Estimation by Comparisons Authors:Xiwen Tao, Chenyi Zhang, Helin Wang, Yexin Zhang, Tongyang Li View a PDF of the paper titled Gradient Testing and Estimation by Comparisons, by Xiwen Tao and 4 other authors View PDF HTML (experimental) 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. Comments: Subjects: Machine Learning (cs.LG); Data Structures and Algorithms (cs.DS); Optimization and Control (math.OC) Cite as: arXiv:2405.11454 [cs.LG] (or arXiv:2405.1...
