Some Unified Theory for Variance Reduced Prox-Linear Methods
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908593077354496 |
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| author | Wu, Yue Grimmer, Benjamin |
| author_facet | Wu, Yue Grimmer, Benjamin |
| contents | This work considers the nonconvex, nonsmooth problem of minimizing a composite objective of the form $f(g(x))+h(x)$ where the inner mapping $g$ is a smooth finite summation or expectation amenable to variance reduction. In such settings, prox-linear methods can enjoy variance-reduced speed-ups despite the existence of nonsmoothness. We provide a unified convergence theory applicable to a wide range of common variance-reduced vector and Jacobian constructions. All the technical conditions we required for variance-reduced methods can be summarized in a single unified assumption. Our theory (i) only requires operator norm bounds on Jacobians (whereas prior works used potentially much larger Frobenius norms), (ii) provides state-of-the-art high probability guarantees, and (iii) allows inexactness in proximal computations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15008 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Unified Theory for Variance Reduced Prox-Linear Methods Wu, Yue Grimmer, Benjamin Optimization and Control This work considers the nonconvex, nonsmooth problem of minimizing a composite objective of the form $f(g(x))+h(x)$ where the inner mapping $g$ is a smooth finite summation or expectation amenable to variance reduction. In such settings, prox-linear methods can enjoy variance-reduced speed-ups despite the existence of nonsmoothness. We provide a unified convergence theory applicable to a wide range of common variance-reduced vector and Jacobian constructions. All the technical conditions we required for variance-reduced methods can be summarized in a single unified assumption. Our theory (i) only requires operator norm bounds on Jacobians (whereas prior works used potentially much larger Frobenius norms), (ii) provides state-of-the-art high probability guarantees, and (iii) allows inexactness in proximal computations. |
| title | Some Unified Theory for Variance Reduced Prox-Linear Methods |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2412.15008 |