Gradient Testing and Estimation by Comparisons
Fuente:
arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866908840826503168 |
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| author | Tao, Xiwen Zhang, Chenyi Wang, Helin Zhang, Yexin Li, Tongyang |
| author_facet | Tao, Xiwen Zhang, Chenyi Wang, Helin Zhang, Yexin Li, Tongyang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11454 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient Testing and Estimation by Comparisons Tao, Xiwen Zhang, Chenyi Wang, Helin Zhang, Yexin Li, Tongyang Machine Learning Data Structures and Algorithms Optimization and Control 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. |
| title | Gradient Testing and Estimation by Comparisons |
| topic | Machine Learning Data Structures and Algorithms Optimization and Control |
| url | https://arxiv.org/abs/2405.11454 |