Complexity guarantees for nonconvex Newton-MR under inexact Hessian information

Abstract We consider an extension of the Newton-MR algorithm for nonconvex unconstrained optimization to the settings where Hessian information is approximated. Under a particular noise model on the Hessian matrix, we investigate the iteration and operation complexities of this variant to achieve appropriate sub-optimality criteria in several nonconvex settings. We do this by first considering functions that satisfy the (generalized) Polyak–Łojasiewicz condition, a special sub-class of nonconvex functions. We show that, under certain conditions, our algorithm achieves global linear convergence rate. We then consider more general nonconvex settings where the rate to obtain first-order sub-optimality is shown to be sub-linear. In all these settings we show that our algorithm converges regardless of the degree of approximation of the Hessian as well as the accuracy of the solution to the sub-problem. Finally, we compare the performance of our algorithm with several alternatives on a few machine learning problems.