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

• URL: http://arxiv.org/abs/2310.18841v2
• Date: Tue, 26 Mar 2024 17:39:30 GMT
• Title: A randomized algorithm for nonconvex minimization with inexact evaluations and complexity guarantees
• Authors: Shuyao Li, Stephen J. Wright,
• Abstract summary: We consider minimization of a smooth non oracle function with inexact to gradient Hessian. A novel feature of our method is that if an approximate direction of negative curvature is chosen, we choose a sense relax to be negative with equal gradients.
• Score: 7.08249229857925
• License: http://creativecommons.org/licenses/by-sa/4.0/
• 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.

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