Cubic regularized subspace Newton for non-convex optimization
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909513089548288 |
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| author | Zhao, Jim Lucchi, Aurelien Doikov, Nikita |
| author_facet | Zhao, Jim Lucchi, Aurelien Doikov, Nikita |
| contents | This paper addresses the optimization problem of minimizing non-convex continuous functions, which is relevant in the context of high-dimensional machine learning applications characterized by over-parametrization. We analyze a randomized coordinate second-order method named SSCN which can be interpreted as applying cubic regularization in random subspaces. This approach effectively reduces the computational complexity associated with utilizing second-order information, rendering it applicable in higher-dimensional scenarios. Theoretically, we establish convergence guarantees for non-convex functions, with interpolating rates for arbitrary subspace sizes and allowing inexact curvature estimation. When increasing subspace size, our complexity matches $\mathcal{O}(ε^{-3/2})$ of the cubic regularization (CR) rate. Additionally, we propose an adaptive sampling scheme ensuring exact convergence rate of $\mathcal{O}(ε^{-3/2}, ε^{-3})$ to a second-order stationary point, even without sampling all coordinates. Experimental results demonstrate substantial speed-ups achieved by SSCN compared to conventional first-order methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16666 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cubic regularized subspace Newton for non-convex optimization Zhao, Jim Lucchi, Aurelien Doikov, Nikita Machine Learning Numerical Analysis Optimization and Control This paper addresses the optimization problem of minimizing non-convex continuous functions, which is relevant in the context of high-dimensional machine learning applications characterized by over-parametrization. We analyze a randomized coordinate second-order method named SSCN which can be interpreted as applying cubic regularization in random subspaces. This approach effectively reduces the computational complexity associated with utilizing second-order information, rendering it applicable in higher-dimensional scenarios. Theoretically, we establish convergence guarantees for non-convex functions, with interpolating rates for arbitrary subspace sizes and allowing inexact curvature estimation. When increasing subspace size, our complexity matches $\mathcal{O}(ε^{-3/2})$ of the cubic regularization (CR) rate. Additionally, we propose an adaptive sampling scheme ensuring exact convergence rate of $\mathcal{O}(ε^{-3/2}, ε^{-3})$ to a second-order stationary point, even without sampling all coordinates. Experimental results demonstrate substantial speed-ups achieved by SSCN compared to conventional first-order methods. |
| title | Cubic regularized subspace Newton for non-convex optimization |
| topic | Machine Learning Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2406.16666 |