A Regularized Newton Method for Nonconvex Optimization with Global and Local Complexity Guarantees

• URL: http://arxiv.org/abs/2502.04799v3
• Date: Fri, 31 Oct 2025 15:26:14 GMT
• Title: A Regularized Newton Method for Nonconvex Optimization with Global and Local Complexity Guarantees
• Authors: Yuhao Zhou, Jintao Xu, Bingrui Li, Chenglong Bao, Chao Ding, Jun Zhu,
• Abstract summary: We propose a new class of regularizers constructed from the current and previous gradients to solve regularized Newton equation.<n>The proposed algorithm is adaptive, requiring no prior knowledge of the Hessian Lipschitz constant.
• Score: 34.85903827323451
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Finding an $\epsilon$-stationary point of a nonconvex function with a Lipschitz continuous Hessian is a central problem in optimization. Regularized Newton methods are a classical tool and have been studied extensively, yet they still face a trade-off between global and local convergence. Whether a parameter-free algorithm of this type can simultaneously achieve optimal global complexity and quadratic local convergence remains an open question. To bridge this long-standing gap, we propose a new class of regularizers constructed from the current and previous gradients, and leverage the conjugate gradient approach with a negative curvature monitor to solve the regularized Newton equation. The proposed algorithm is adaptive, requiring no prior knowledge of the Hessian Lipschitz constant, and achieves a global complexity of $O(\epsilon^{-3/2})$ in terms of the second-order oracle calls, and $\tilde{O}(\epsilon^{-7/4})$ for Hessian-vector products, respectively. When the iterates converge to a point where the Hessian is positive definite, the method exhibits quadratic local convergence. Preliminary numerical results, including training the physics-informed neural networks, illustrate the competitiveness of our algorithm.

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