Soft quasi-Newton: Guaranteed positive definiteness by relaxing the secant constraint
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913254411862016 |
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| author | Berglund, Erik Zhang, Jiaojiao Johansson, Mikael |
| author_facet | Berglund, Erik Zhang, Jiaojiao Johansson, Mikael |
| contents | We propose a novel algorithm, termed soft quasi-Newton (soft QN), for optimization in the presence of bounded noise. Traditional quasi-Newton algorithms are vulnerable to such perturbations. To develop a more robust quasi-Newton method, we replace the secant condition in the matrix optimization problem for the Hessian update with a penalty term in its objective and derive a closed-form update formula. A key feature of our approach is its ability to maintain positive definiteness of the Hessian inverse approximation. Furthermore, we establish the following properties of soft QN: it recovers the BFGS method under specific limits, it treats positive and negative curvature equally, and it is scale invariant. Collectively, these features enhance the efficacy of soft QN in noisy environments. For strongly convex objective functions and Hessian approximations obtained using soft QN, we develop an algorithm that exhibits linear convergence toward a neighborhood of the optimal solution, even if gradient and function evaluations are subject to bounded perturbations. Through numerical experiments, we demonstrate superior performance of soft QN compared to state-of-the-art methods in various scenarios. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Soft quasi-Newton: Guaranteed positive definiteness by relaxing the secant constraint Berglund, Erik Zhang, Jiaojiao Johansson, Mikael Optimization and Control 90C53 We propose a novel algorithm, termed soft quasi-Newton (soft QN), for optimization in the presence of bounded noise. Traditional quasi-Newton algorithms are vulnerable to such perturbations. To develop a more robust quasi-Newton method, we replace the secant condition in the matrix optimization problem for the Hessian update with a penalty term in its objective and derive a closed-form update formula. A key feature of our approach is its ability to maintain positive definiteness of the Hessian inverse approximation. Furthermore, we establish the following properties of soft QN: it recovers the BFGS method under specific limits, it treats positive and negative curvature equally, and it is scale invariant. Collectively, these features enhance the efficacy of soft QN in noisy environments. For strongly convex objective functions and Hessian approximations obtained using soft QN, we develop an algorithm that exhibits linear convergence toward a neighborhood of the optimal solution, even if gradient and function evaluations are subject to bounded perturbations. Through numerical experiments, we demonstrate superior performance of soft QN compared to state-of-the-art methods in various scenarios. |
| title | Soft quasi-Newton: Guaranteed positive definiteness by relaxing the secant constraint |
| topic | Optimization and Control 90C53 |
| url | https://arxiv.org/abs/2403.02448 |