A Stochastic Objective-Function-Free Adaptive Regularization Method with Optimal Complexity

TL;DR

This paper introduces a stochastic second-order adaptive-regularization method that avoids computing the objective function, yet guarantees optimal complexity for finding critical points, with applications to inexact derivatives and finite-sum minimization.

Contribution

It presents a novel objective-function-free adaptive regularization method with optimal complexity bounds for nonconvex optimization, tolerant to noise and inexact evaluations.

Findings

Achieves $ ilde{O}( ext{epsilon}^{-3/2})$ complexity for critical points.

Demonstrates effectiveness on large binary classification problems.

Handles inexact derivatives and finite-sum minimization through sampling.

Abstract

A fully stochastic second-order adaptive-regularization method for unconstrained nonconvex optimization is presented which never computes the objective-function value, but yet achieves the optimal $\mathcal{O}(\epsilon^{-3/2})$ complexity bound for finding first-order critical points. The method is noise-tolerant and the inexactness conditions required for convergence depend on the history of past steps. Applications to cases where derivative evaluation is inexact and to minimization of finite sums by sampling are discussed. Numerical experiments on large binary classification problems illustrate the potential of the new method.