Adaptive Newton-CG methods with global and local analysis for unconstrained optimization with H\"older continuous Hessian

TL;DR

This paper introduces adaptive Newton-CG algorithms for nonconvex optimization with Hölder continuous Hessians, improving efficiency and convergence over existing methods by eliminating nested line searches.

Contribution

The paper proposes two novel adaptive Newton-CG algorithms that leverage auto-conditioning to improve iteration complexity and achieve local superlinear convergence.

Findings

Achieve optimal iteration complexity ${\\mathcal O}(H_f^{1/(1+\nu)}\epsilon^{-(2+\nu)/(1+\nu)})$.

Eliminate the nested line search to enhance computational efficiency.

Demonstrate practical advantages through numerical experiments.

Abstract

In this paper, we study Newton-conjugate gradient (Newton-CG) methods for minimizing a nonconvex function $f$ whose Hessian is $(H_f,\nu)$-H\"older continuous with modulus $H_f>0$ and exponent $\nu\in(0,1]$. Recently proposed Newton-CG methods for this problem adopt (i) non-adaptive regularization and (ii) a nested line-search procedure, where (i) often leads to inefficient early progress and the loss of local superlinear convergence, and (ii) may incur high computational cost due to multiple solves of the Newton system per iteration. To address these limitations, we propose two novel Newton-CG algorithms, depending on the availability of $\nu$, that adaptively regularize the Newton system by leveraging the auto-conditioning technique to eliminate the nested line search. The proposed algorithms achieve the best-known iteration complexity ${\mathcal O}\big(H_f^{1/(1+\nu)}\epsilon^{-(2+\nu)/(1+\nu)}\big)$ for finding an $\epsilon$-stationary point and, simultaneously, enjoy local superlinear convergence near nondegenerate local minimizers. Numerical experiments further demonstrate the practical advantages of our algorithms over existing approaches.