A randomized algorithm for nonconvex minimization with inexact evaluations and complexity guarantees
This work addresses optimization challenges in machine learning for nonconvex settings, offering incremental improvements in efficiency for empirical risk minimization.
The paper tackles the problem of minimizing smooth nonconvex functions with inexact gradient and Hessian evaluations, achieving approximate second-order optimality by using a randomized step direction for negative curvature. It results in improved gradient sample complexity for empirical risk minimization problems.
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.