Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914686057840640 |
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| author | Wang, Hao Zhang, Chenyi Li, Tongyang |
| author_facet | Wang, Hao Zhang, Chenyi Li, Tongyang |
| contents | The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(Nε^{-2/3} + ε^{-8/3})$ queries to a first-order oracle to compute an $ε$-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of $\tilde{O}(\sqrt{N}ε^{-5/3} + ε^{-8/3})$. On the other hand, we prove that quantum algorithms must take $\tildeΩ(\sqrt{N}ε^{-2/3})$ queries to a first order quantum oracle, showing that our dependence on $N$ is optimal up to poly-logarithmic factors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss Wang, Hao Zhang, Chenyi Li, Tongyang Quantum Physics Data Structures and Algorithms Optimization and Control The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(Nε^{-2/3} + ε^{-8/3})$ queries to a first-order oracle to compute an $ε$-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of $\tilde{O}(\sqrt{N}ε^{-5/3} + ε^{-8/3})$. On the other hand, we prove that quantum algorithms must take $\tildeΩ(\sqrt{N}ε^{-2/3})$ queries to a first order quantum oracle, showing that our dependence on $N$ is optimal up to poly-logarithmic factors. |
| title | Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss |
| topic | Quantum Physics Data Structures and Algorithms Optimization and Control |
| url | https://arxiv.org/abs/2402.12745 |