This paper introduces a novel algorithm, the Perturbed Proximal Preconditioned SPIDER algorithm (3P-SPIDER), designed to solve finite sum non-convex composite optimization. It is a stochastic Variable Metric Forward-Backward algorithm, which allows approximate preconditioned forward operator and uses a variable metric proximity operator as the backward operator; it also proposes a mini-batch strategy with variance reduction to address the finite sum setting. We show that 3P-SPIDER extends some Stochastic preconditioned Gradient Descent-based algorithms and some Incremental Expectation Maximization algorithms to composite optimization and to the case the forward operator can not be computed in closed form. We also provide an explicit control of convergence in expectation of 3P-SPIDER, and study its complexity in order to satisfy the epsilon-approximate stationary condition. Our results are the first to combine the composite non-convex optimization setting, a variance reduction technique to tackle the finite sum setting by using a minibatch strategy and, to allow deterministic or random approximations of the preconditioned forward operator. Finally, through an application to inference in a logistic regression model with random effects, we numerically compare 3P-SPIDER to other stochastic forward-backward algorithms and discuss the role of some design parameters of 3P-SPIDER.

翻译：本文提出了一种新颖算法——扰动近端预条件SPIDER算法（3P-SPIDER），旨在解决有限和形式的非凸复合优化问题。该算法是一种随机变度量前向-后向算法，允许采用近似预条件前向算子，并使用变度量邻近算子作为后向算子；同时，它提出了一种结合方差缩减的小批量策略以应对有限和设置。我们证明，3P-SPIDER将若干基于随机预条件梯度下降的算法和增量期望最大化算法扩展至复合优化场景，并适用于前向算子无法以闭式求解的情形。本文还给出了3P-SPIDER期望收敛性的显式控制，并分析了其满足ε-近似稳定条件的复杂度。我们的成果首次将复合非凸优化设置、利用小批量策略处理有限和问题的方差缩减技术，以及允许对预条件前向算子进行确定性或随机近似相结合。最后，通过随机效应逻辑回归模型中的推断应用，我们将3P-SPIDER与其他随机前向-后向算法进行数值比较，并讨论了3P-SPIDER若干设计参数的作用。