Distributed Stochastic Gradient Tracking Algorithm With Variance Reduction for Non-Convex Optimization
Xia Jiang, Xianlin Zeng, Jian Sun, Jie Chen
Abstract
This article proposes a distributed stochastic algorithm with variance reduction for general smooth non-convex finite-sum optimization, which has wide applications in signal processing and machine learning communities. In distributed setting, a large number of samples are allocated to multiple agents in the network. Each agent computes local stochastic gradient and communicates with its neighbors to seek for the global optimum. In this article, we develop a modified variance reduction technique to deal with the variance introduced by stochastic gradients. Combining gradient tracking and variance reduction techniques, this article proposes a distributed stochastic algorithm, gradient tracking algorithm with variance reduction (GT-VR), to solve large-scale non-convex finite-sum optimization over multiagent networks. A complete and rigorous proof shows that the GT-VR algorithm converges to the first-order stationary points with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O({1}/{k})$ </tex-math></inline-formula> convergence rate. In addition, we provide the complexity analysis of the proposed algorithm. Compared with some existing first-order methods, the proposed algorithm has a lower <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(PM\epsilon ^{-1})$ </tex-math></inline-formula> gradient complexity under some mild condition. By comparing state-of-the-art algorithms and GT-VR in numerical simulations, we verify the efficiency of the proposed algorithm.