First and zeroth-order implementations of the regularized Newton method with lazy approximated Hessians

TL;DR

This paper introduces first- and zeroth-order implementations of the regularized Newton method with adaptive parameter fitting and lazy Hessian updates, achieving improved complexity bounds for non-convex optimization.

Contribution

It presents novel Hessian-free and derivative-free algorithms with adaptive regularization and lazy Hessian updates, providing better complexity bounds for non-convex problems.

Findings

Global complexity bound of O(n^{1/2} ε^{-3/2}) for Hessian-free method

Global complexity bound of O(n^{3/2} ε^{-3/2}) for derivative-free method

Algorithms do not require knowledge of Lipschitz constants

Abstract

In this work, we develop first-order (Hessian-free) and zero-order (derivative-free) implementations of the Cubically regularized Newton method for solving general non-convex optimization problems. For that, we employ finite difference approximations of the derivatives. We use a special adaptive search procedure in our algorithms, which simultaneously fits both the regularization constant and the parameters of the finite difference approximations. It makes our schemes free from the need to know the actual Lipschitz constants. Additionally, we equip our algorithms with the lazy Hessian update that reuse a previously computed Hessian approximation matrix for several iterations. Specifically, we prove the global complexity bound of $\mathcal{O}( n^{1/2} \epsilon^{-3/2})$ function and gradient evaluations for our new Hessian-free method, and a bound of $\mathcal{O}( n^{3/2} \epsilon^{-3/2} )$ function evaluations for the derivative-free method, where $n$ is the dimension of the problem and $\epsilon$ is the desired accuracy for the gradient norm. These complexity bounds significantly improve the previously known ones in terms of the joint dependence on $n$ and $\epsilon$, for the first-order and zeroth-order non-convex optimization.