Quantum Algorithms for Non-smooth Non-convex Optimization
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866909357809074176 |
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| author | Liu, Chengchang Guan, Chaowen He, Jianhao Lui, John C. S. |
| author_facet | Liu, Chengchang Guan, Chaowen He, Jianhao Lui, John C. S. |
| contents | This paper considers the problem for finding the $(δ,ε)$-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of $\tilde{\mathcal{O}}(d^{3/2}δ^{-1}ε^{-3})$ on the stochastic function value oracle, where $d$ is the dimension of the problem. We also enhance the query complexity to $\tilde{\mathcal{O}}(d^{3/2}δ^{-1}ε^{-7/3})$ by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of $ε$ by a factor of $ε^{-2/3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Algorithms for Non-smooth Non-convex Optimization Liu, Chengchang Guan, Chaowen He, Jianhao Lui, John C. S. Quantum Physics Optimization and Control This paper considers the problem for finding the $(δ,ε)$-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of $\tilde{\mathcal{O}}(d^{3/2}δ^{-1}ε^{-3})$ on the stochastic function value oracle, where $d$ is the dimension of the problem. We also enhance the query complexity to $\tilde{\mathcal{O}}(d^{3/2}δ^{-1}ε^{-7/3})$ by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of $ε$ by a factor of $ε^{-2/3}$. |
| title | Quantum Algorithms for Non-smooth Non-convex Optimization |
| topic | Quantum Physics Optimization and Control |
| url | https://arxiv.org/abs/2410.16189 |