Stochastic Gauss–Seidel type inertial proximal alternating linearized minimization and its application to proximal neural networks

In many optimization problems arising from machine learning, image processing, and statistics communities, the objective functions possess a special form involving huge amounts of data, which encourages the application of stochastic algorithms. In this paper, we study such a broad class of nonconvex nonsmooth minimization problems, whose objective function is the sum of a smooth function of the entire variables and two nonsmooth functions of each variable. We propose to solve this problem with a stochastic Gauss–Seidel type inertial proximal alternating linearized minimization (denoted by SGiPALM) algorithm. We prove that under Kurdyka–Łojasiewicz (KŁ) property and some mild conditions, each bounded sequence generated by SGiPALM with the variance-reduced stochastic gradient estimator globally converges to a critical point after a finite number of iterations, or almost surely satisfies the finite length property. We also apply the SGiPALM algorithm to the proximal neural networks (PNN) with 4 layers for classification tasks on the MNIST dataset and compare it with other deterministic and stochastic optimization algorithms, the results illustrate the effectiveness of the proposed algorithm.