Examples of slow convergence for adaptive regularization optimization methods are not isolated
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866916408945803264 |
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| author | Toint, Philippe L. |
| author_facet | Toint, Philippe L. |
| contents | The adaptive regularization algorithm for unconstrained nonconvex optimization was shown in Nesterov and Polyak (2006) and Cartis, Gould and Toint (2011) to require, under standard assumptions, at most $\mathcal{O}(ε^{3/(3-q)})$ evaluations of the objective function and its derivatives of degrees one and two to produce an $ε$-approximate critical point of order $q\in\{1,2\}$. This bound was shown to be sharp for $q \in\{1,2\}$. This note revisits these results and shows that the example for which slow convergence is exhibited is not isolated, but that this behaviour occurs for a subset of univariate functions of nonzero measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16047 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Examples of slow convergence for adaptive regularization optimization methods are not isolated Toint, Philippe L. Optimization and Control Computational Complexity 49M37, 65K05, 68Q17, 68W40, 90C30 F.2.1; G.1.6; I.1.2 The adaptive regularization algorithm for unconstrained nonconvex optimization was shown in Nesterov and Polyak (2006) and Cartis, Gould and Toint (2011) to require, under standard assumptions, at most $\mathcal{O}(ε^{3/(3-q)})$ evaluations of the objective function and its derivatives of degrees one and two to produce an $ε$-approximate critical point of order $q\in\{1,2\}$. This bound was shown to be sharp for $q \in\{1,2\}$. This note revisits these results and shows that the example for which slow convergence is exhibited is not isolated, but that this behaviour occurs for a subset of univariate functions of nonzero measure. |
| title | Examples of slow convergence for adaptive regularization optimization methods are not isolated |
| topic | Optimization and Control Computational Complexity 49M37, 65K05, 68Q17, 68W40, 90C30 F.2.1; G.1.6; I.1.2 |
| url | https://arxiv.org/abs/2409.16047 |