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| Autors principals: | , , |
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| Format: | Preprint |
| Publicat: |
2024
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| Matèries: | |
| Accés en línia: | https://arxiv.org/abs/2403.06708 |
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| _version_ | 1866916776320696320 |
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| author | Maulen-Soto, Rodrigo Fadili, Jalal Attouch, Hedy |
| author_facet | Maulen-Soto, Rodrigo Fadili, Jalal Attouch, Hedy |
| contents | To solve convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable. In this context, under Lipschitz's continuity of the differentiable term and a growth condition of the non-smooth term, our first main result shows almost sure weak convergence of the trajectory process towards a minimizer of the objective function. Then, using Tikhonov regularization with a properly tuned vanishing parameter, we can obtain almost sure strong convergence of the trajectory towards the minimum norm solution. We find an explicit tuning of this parameter when our objective function satisfies a local error-bound inequality. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and Lojasiewicz case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06708 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces Maulen-Soto, Rodrigo Fadili, Jalal Attouch, Hedy Optimization and Control 37N40, 46N10, 49M99, 65B99, 65K05, 65K10, 900B50, 90C25, 60H10, 49J52 To solve convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable. In this context, under Lipschitz's continuity of the differentiable term and a growth condition of the non-smooth term, our first main result shows almost sure weak convergence of the trajectory process towards a minimizer of the objective function. Then, using Tikhonov regularization with a properly tuned vanishing parameter, we can obtain almost sure strong convergence of the trajectory towards the minimum norm solution. We find an explicit tuning of this parameter when our objective function satisfies a local error-bound inequality. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and Lojasiewicz case. |
| title | Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces |
| topic | Optimization and Control 37N40, 46N10, 49M99, 65B99, 65K05, 65K10, 900B50, 90C25, 60H10, 49J52 |
| url | https://arxiv.org/abs/2403.06708 |