Stochastic Analysis of an Adaptive Cubic Regularisation Method under Inexact Gradient Evaluations and Dynamic Hessian Accuracy

TL;DR

This paper extends an adaptive cubic regularisation method to stochastic nonconvex optimization, allowing inexact gradient and Hessian computations, and proves it maintains optimal iteration complexity with practical numerical benefits.

Contribution

It introduces a stochastic variant of an adaptive cubic regularisation method with inexact derivatives, preserving optimal complexity bounds.

Findings

Expected iteration complexity remains O(epsilon^(-3/2)).

Inexact derivatives can reduce computational costs.

Numerical tests confirm practical efficiency gains.

Abstract

We here adapt an extended version of the adaptive cubic regularisation method with dynamic inexact Hessian information for nonconvex optimisation in [3] to the stochastic optimisation setting. While exact function evaluations are still considered, this novel variant inherits the innovative use of adaptive accuracy requirements for Hessian approximations introduced in [3] and additionally employs inexact computations of the gradient. Without restrictions on the variance of the errors, we assume that these approximations are available within a sufficiently large, but fixed, probability and we extend, in the spirit of [18], the deterministic analysis of the framework to its stochastic counterpart, showing that the expected number of iterations to reach a first-order stationary point matches the well known worst-case optimal complexity. This is, in fact, still given by O(epsilon^(-3/2)), with respect to the first-order epsilon tolerance. Finally, numerical tests on nonconvex finite-sum minimisation confirm that using inexact first and second-order derivatives can be beneficial in terms of the computational savings.