A stochastic cubic regularisation method with inexact function evaluations and random derivatives for finite sum minimisation

This paper focuses on an Adaptive Cubic Regularisation (ARC) method for
approximating a second-order critical point of a finite sum minimisation problem.
The variant presented belongs to the framework of Bellavia, Gurioli, Morinin
and Toint (2020): it employs random models with accuracy guaranteed with a
sufficiently large prefixed probability and deterministic inexact function
evaluations within a prescribed level of accuracy. Without assuming unbiased
estimators, the expected number of iterations is O( epsilon_1^{-3/2} ) or O( \max[ epsilon_1^{-3/2},epsilon_2^{-3} ] ) when searching for a first- or second-order critical point, respectively, where epsilon_j is the jth-order tolerance. These results match the worst-case optimal complexity for the deterministic counterpart of the method.