Convergence rate of the (1+1)-evolution strategy on locally strongly convex functions with lipschitz continuous gradient
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915471298658304 |
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| author | Morinaga, Daiki Fukuchi, Kazuto Sakuma, Jun Akimoto, Youhei |
| author_facet | Morinaga, Daiki Fukuchi, Kazuto Sakuma, Jun Akimoto, Youhei |
| contents | Evolution strategy (ES) is one of the promising classes of algorithms for black-box continuous optimization. Despite its broad successes in applications, theoretical analysis on the speed of its convergence is limited on convex quadratic functions and their monotonic transformation. In this study, an upper bound and a lower bound of the rate of linear convergence of the (1+1)-ES on locally $L$-strongly convex functions with $U$-Lipschitz continuous gradient are derived as $\exp\left(-Ω_{d\to\infty}\left(\frac{L}{d\cdot U}\right)\right)$ and $\exp\left(-\frac1d\right)$, respectively. Notably, any prior knowledge on the mathematical properties of the objective function, such as Lipschitz constant, is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12467 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Convergence rate of the (1+1)-evolution strategy on locally strongly convex functions with lipschitz continuous gradient Morinaga, Daiki Fukuchi, Kazuto Sakuma, Jun Akimoto, Youhei Optimization and Control Neural and Evolutionary Computing 65K05, 90C25, 90C26, 90C56, 90C59 G.1.6 Evolution strategy (ES) is one of the promising classes of algorithms for black-box continuous optimization. Despite its broad successes in applications, theoretical analysis on the speed of its convergence is limited on convex quadratic functions and their monotonic transformation. In this study, an upper bound and a lower bound of the rate of linear convergence of the (1+1)-ES on locally $L$-strongly convex functions with $U$-Lipschitz continuous gradient are derived as $\exp\left(-Ω_{d\to\infty}\left(\frac{L}{d\cdot U}\right)\right)$ and $\exp\left(-\frac1d\right)$, respectively. Notably, any prior knowledge on the mathematical properties of the objective function, such as Lipschitz constant, is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require it. |
| title | Convergence rate of the (1+1)-evolution strategy on locally strongly convex functions with lipschitz continuous gradient |
| topic | Optimization and Control Neural and Evolutionary Computing 65K05, 90C25, 90C26, 90C56, 90C59 G.1.6 |
| url | https://arxiv.org/abs/2209.12467 |