On the Convergence of A Data-Driven Regularized Stochastic Gradient Descent for Nonlinear Ill-Posed Problems
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917788588703744 |
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| author | Zhou, Zehui |
| author_facet | Zhou, Zehui |
| contents | Stochastic gradient descent (SGD) is a promising method for solving large-scale inverse problems, due to its excellent scalability with respect to data size. In this work, we analyze a new data-driven regularized stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in infinite dimensional Hilbert spaces. At each step of the iteration, the method randomly selects one equation from the nonlinear system combined with a corresponding equation from the learned system based on training data to obtain a stochastic estimate of the gradient and then performs a descent step with the estimated gradient. We prove the regularizing property of this method under the tangential cone condition and a priori parameter choice and then derive the convergence rates under the additional source condition and range invariance conditions. Several numerical experiments are provided to complement the analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11787 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Convergence of A Data-Driven Regularized Stochastic Gradient Descent for Nonlinear Ill-Posed Problems Zhou, Zehui Numerical Analysis Optimization and Control 65J20, 65J22, 47J06 Stochastic gradient descent (SGD) is a promising method for solving large-scale inverse problems, due to its excellent scalability with respect to data size. In this work, we analyze a new data-driven regularized stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in infinite dimensional Hilbert spaces. At each step of the iteration, the method randomly selects one equation from the nonlinear system combined with a corresponding equation from the learned system based on training data to obtain a stochastic estimate of the gradient and then performs a descent step with the estimated gradient. We prove the regularizing property of this method under the tangential cone condition and a priori parameter choice and then derive the convergence rates under the additional source condition and range invariance conditions. Several numerical experiments are provided to complement the analysis. |
| title | On the Convergence of A Data-Driven Regularized Stochastic Gradient Descent for Nonlinear Ill-Posed Problems |
| topic | Numerical Analysis Optimization and Control 65J20, 65J22, 47J06 |
| url | https://arxiv.org/abs/2403.11787 |