A randomized algorithm for nonconvex minimization with inexact evaluations and complexity guarantees

TL;DR

This paper introduces a randomized algorithm for nonconvex optimization that handles inexact gradient and Hessian evaluations, providing convergence guarantees and improved sample complexity for empirical risk minimization.

Contribution

It proposes a novel randomized approach that chooses the direction of negative curvature randomly, relaxing inexactness assumptions and offering both expectation and high-probability convergence guarantees.

Findings

Achieves improved gradient sample complexity in empirical risk minimization

Provides convergence guarantees with inexact gradient and Hessian evaluations

Includes both expectation and high-probability bounds in analysis

Abstract

We consider minimization of a smooth nonconvex function with inexact oracle access to gradient and Hessian (without assuming access to the function value) to achieve approximate second-order optimality. A novel feature of our method is that if an approximate direction of negative curvature is chosen as the step, we choose its sense to be positive or negative with equal probability. We allow gradients to be inexact in a relative sense and relax the coupling between inexactness thresholds for the first- and second-order optimality conditions. Our convergence analysis includes both an expectation bound based on martingale analysis and a high-probability bound based on concentration inequalities. We apply our algorithm to empirical risk minimization problems and obtain improved gradient sample complexity over existing works.