Cubic regularized subspace Newton for non-convex optimization

• URL: http://arxiv.org/abs/2406.16666v1
• Date: Mon, 24 Jun 2024 14:20:02 GMT
• Title: Cubic regularized subspace Newton for non-convex optimization
• Authors: Jim Zhao, Aurelien Lucchi, Nikita Doikov,
• Abstract summary: We analyze a coordinate second-order SSCN which can be interpreted as applying stationary regularization in random subspaces. We demonstrate substantial speed-ups compared to conventional first-order methods.
• Score: 3.481985817302898
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\mathcal{O}(\epsilon^{-3/2})$ of the cubic regularization (CR) rate. Additionally, we propose an adaptive sampling scheme ensuring exact convergence rate of $\mathcal{O}(\epsilon^{-3/2}, \epsilon^{-3})$ 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.

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