Convergence properties of an Objective-Function-Free Optimization regularization algorithm, including an $\mathcal{O}(\epsilon^{-3/2})$ complexity bound

TL;DR

This paper introduces an objective-function-free adaptive regularization algorithm for unconstrained nonconvex optimization that achieves optimal worst-case complexity bounds using only derivatives, not function evaluations.

Contribution

The paper presents a novel regularization algorithm that avoids objective function evaluations while maintaining optimal complexity bounds for finding approximate minimizers.

Findings

Achieves $O(oxed{ ext{epsilon}^{-(p+1)/p}})$ complexity for first-order minimizers.

Achieves $O( ext{max}[ ext{epsilon}^{-(p+1)/p}, ext{epsilon}_2^{-(p+1)/(p-1)}])$ complexity for second-order minimizers.

Special case: finds a point with gradient norm less than epsilon in $O( ext{epsilon}^{-3/2})$ iterations.

Abstract

An adaptive regularization algorithm for unconstrained nonconvex optimization is presented in which the objective function is never evaluated, but only derivatives are used. This algorithm belongs to the class of adaptive regularization methods, for which optimal worst-case complexity results are known for the standard framework where the objective function is evaluated. It is shown in this paper that these excellent complexity bounds are also valid for the new algorithm, despite the fact that significantly less information is used. In particular, it is shown that, if derivatives of degree one to $p$ are used, the algorithm will find a $\epsilon_1$-approximate first-order minimizer in at most $O(\epsilon_1^{-(p+1)/p})$ iterations, and an $(\epsilon_1,\epsilon_2)$-approximate second-order minimizer in at most $O(\max[\epsilon^{-(p+1)/p},\epsilon_2^{-(p+1)/(p-1)}])$ iterations. As a special case, the new algorithm using first and second derivatives, when applied to functions with Lipschitz continuous Hessian, will find an iterate $x_k$ at which the gradient's norm is less than $\epsilon_1$ in at most $O(\epsilon_1^{-3/2})$ iterations.