On Optimal Regularization Parameters via Bilevel Learning
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914646467805184 |
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| author | Ehrhardt, Matthias J. Gazzola, Silvia Scott, Sebastian J. |
| author_facet | Ehrhardt, Matthias J. Gazzola, Silvia Scott, Sebastian J. |
| contents | Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter. Selection of an appropriate regularization parameter is critical, with various choices leading to very different reconstructions. Classical strategies used to determine a suitable parameter value include the discrepancy principle and the L-curve criterion, and in recent years a supervised machine learning approach called bilevel learning has been employed. Bilevel learning is a powerful framework to determine optimal parameters and involves solving a nested optimization problem. While previous strategies enjoy various theoretical results, the well-posedness of bilevel learning in this setting is still an open question. In particular, a necessary property is positivity of the determined regularization parameter. In this work, we provide a new condition that better characterizes positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and high-dimensional problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_18394 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Optimal Regularization Parameters via Bilevel Learning Ehrhardt, Matthias J. Gazzola, Silvia Scott, Sebastian J. Optimization and Control Machine Learning 65K10 (Primary) 65F22 (Secondary) Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter. Selection of an appropriate regularization parameter is critical, with various choices leading to very different reconstructions. Classical strategies used to determine a suitable parameter value include the discrepancy principle and the L-curve criterion, and in recent years a supervised machine learning approach called bilevel learning has been employed. Bilevel learning is a powerful framework to determine optimal parameters and involves solving a nested optimization problem. While previous strategies enjoy various theoretical results, the well-posedness of bilevel learning in this setting is still an open question. In particular, a necessary property is positivity of the determined regularization parameter. In this work, we provide a new condition that better characterizes positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and high-dimensional problems. |
| title | On Optimal Regularization Parameters via Bilevel Learning |
| topic | Optimization and Control Machine Learning 65K10 (Primary) 65F22 (Secondary) |
| url | https://arxiv.org/abs/2305.18394 |