Conditioning the complexity of random landscapes on marginal optima
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912140178227200 |
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| author | Kent-Dobias, Jaron |
| author_facet | Kent-Dobias, Jaron |
| contents | Marginal optima are minima or maxima of a function with many nearly flat directions. In settings with many competing optima, marginal ones tend to attract algorithms and physical dynamics. Often, the important family of marginal attractors are a vanishing minority compared with nonmarginal optima and other unstable stationary points. We introduce a generic technique for conditioning the statistics of stationary points in random landscapes on their marginality, and apply it in three isotropic settings with qualitatively different structure: in the spherical spin-glasses, where the energy is Gaussian and its Hessian is GOE; in multispherical spin glasses, which are Gaussian but non-GOE; and in sums of squared spherical random functions, which are non-Gaussian. In these problems we are able to fully characterize the distribution of marginal optima in the landscape, including when they are in the minority. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02082 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conditioning the complexity of random landscapes on marginal optima Kent-Dobias, Jaron Disordered Systems and Neural Networks Statistical Mechanics Marginal optima are minima or maxima of a function with many nearly flat directions. In settings with many competing optima, marginal ones tend to attract algorithms and physical dynamics. Often, the important family of marginal attractors are a vanishing minority compared with nonmarginal optima and other unstable stationary points. We introduce a generic technique for conditioning the statistics of stationary points in random landscapes on their marginality, and apply it in three isotropic settings with qualitatively different structure: in the spherical spin-glasses, where the energy is Gaussian and its Hessian is GOE; in multispherical spin glasses, which are Gaussian but non-GOE; and in sums of squared spherical random functions, which are non-Gaussian. In these problems we are able to fully characterize the distribution of marginal optima in the landscape, including when they are in the minority. |
| title | Conditioning the complexity of random landscapes on marginal optima |
| topic | Disordered Systems and Neural Networks Statistical Mechanics |
| url | https://arxiv.org/abs/2407.02082 |