Convergence analysis of a proximal stochastic gradient algorithm with adaptive sampling for non-convex and non-smooth composite optimization problems

This paper examines the convergence and computational complexity of a proximal stochastic gradient algorithm that adaptively incorporates sampling techniques for solving large-scale, non-convex, and non-smooth problems, with a particular emphasis on problems that involve the combination of two non-convex functions.This an area that has been scarcely explored by current methods.By adjusting adaptively the sampling size (or mini-batch size) throughout the algorithm's iterations, this method aims to balance the trade-off between stochastic gradient noise and convergence stability.It maintains a convergence rate similar to that of the proximal gradient method.Moreover, when the objective function is a Kurdyka-ojasiewicz (KL) function, we demonstrate the convergence rate of the expected function value on a case-by-case basis, achieving linear convergence under optimal conditions.Finally, some preliminary numerical results validate the effectiveness and robustness of the proposed method.