Stochastic Optimization for Nonconvex Problem With Inexact Hessian Matrix, Gradient, and Function

Trust region (TR) and adaptive regularization using cubics (ARC) have proven to have some very appealing theoretical properties for nonconvex optimization by concurrently computing function value, gradient, and Hessian matrix to obtain the next search direction and the adjusted parameters. Although stochastic approximations help largely reduce the computational cost, it is challenging to theoretically guarantee the convergence rate. In this article, we explore a family of stochastic TR (STR) and stochastic ARC (SARC) methods that can simultaneously provide inexact computations of the Hessian matrix, gradient, and function values. Our algorithms require much fewer propagations overhead per iteration than TR and ARC. We prove that the iteration complexity to achieve
$\epsilon$
-approximate second-order optimality is of the same order as the exact computations demonstrated in previous studies. In addition, the mild conditions on inexactness can be met by leveraging a random sampling technology in the finite-sum minimization problem. Numerical experiments with a nonconvex problem support these findings and demonstrate that, with the same or a similar number of iterations, our algorithms require less computational overhead per iteration than current second-order methods.