A Randomized Algorithm for Nonconvex Minimization With Inexact Evaluations and Complexity Guarantees

• Published in: Journal of optimization theory and applications Vol. 207; no. 3; p. 66
• Main Authors: Li, Shuyao, Wright, Stephen J.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2025; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Complexity; Engineering; Folklore; Martingales; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Probability; Regularization methods; Theory of Computation; 68W20; Nonconvex optimization; Second-order methods; 90C17; 90C26; Second-order stationarity; Inexact Hessian; Inexact gradient
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-025-02817-y

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 sign to be positive or negative with equal probability. We allow gradients to be inexact (with a bound on their error relative to the size of the true quantity) 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.