Quadratic and Cubic Regularisation Methods with Inexact function and Random Derivatives for Finite-Sum Minimisation
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This paper focuses on regularisation methods using models up to the third order to search for up to second-order critical points of a finite-sum minimisation problem. The variant presented belongs to the framework of [3]: 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(ϵ_1^-2) or 𝒪(ϵ_1^-3/2) when searching for a first-order critical point using a second or third order model, respectively, and of 𝒪(max[ϵ_1^-3/2,ϵ_2^-3]) when seeking for second-order critical points with a third order model, in which ϵ_j, j∈{1,2}, is the jth-order tolerance. These results match the worst-case optimal complexity for the deterministic counterpart of the method. Preliminary numerical tests for first-order optimality in the context of nonconvex binary classification in imaging, with and without Artifical Neural Networks (ANNs), are presented and discussed.
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