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| Natura: | Preprint |
| Pubblicazione: |
2023
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| Accesso online: | https://arxiv.org/abs/2307.03820 |
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| _version_ | 1866912512115474432 |
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| author | Brooks, S. J. |
| author_facet | Brooks, S. J. |
| contents | The Newton, Gauss--Newton and Levenberg--Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some directions than others and the space of possible improvement in sum of squares becomes a long narrow ellipsoid in the linear model. This means that even a small amount of nonlinearity in the problem parameters can cause a proposed point far down the long axis of the ellipsoid to fall outside of the actual curved valley of improved values, even though it is quite nearby. This paper presents a differential equation that `follows' these valleys, based on the technique of geodesic acceleration, which itself provides a 2$^\mathrm{nd}$ order improvement to the Levenberg--Marquardt iteration step. Higher derivatives of this equation are computed that allow $n^\mathrm{th}$ order improvements to the optimisation methods to be derived. These higher-order accelerated methods up to 4$^\mathrm{th}$ order are tested numerically and shown to provide substantial reduction of both number of steps and computation time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03820 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher-Order Corrections to Optimisers based on Newton's Method Brooks, S. J. Numerical Analysis The Newton, Gauss--Newton and Levenberg--Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some directions than others and the space of possible improvement in sum of squares becomes a long narrow ellipsoid in the linear model. This means that even a small amount of nonlinearity in the problem parameters can cause a proposed point far down the long axis of the ellipsoid to fall outside of the actual curved valley of improved values, even though it is quite nearby. This paper presents a differential equation that `follows' these valleys, based on the technique of geodesic acceleration, which itself provides a 2$^\mathrm{nd}$ order improvement to the Levenberg--Marquardt iteration step. Higher derivatives of this equation are computed that allow $n^\mathrm{th}$ order improvements to the optimisation methods to be derived. These higher-order accelerated methods up to 4$^\mathrm{th}$ order are tested numerically and shown to provide substantial reduction of both number of steps and computation time. |
| title | Higher-Order Corrections to Optimisers based on Newton's Method |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2307.03820 |