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

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.