An Improved Analysis of Gradient Tracking for Decentralized Machine Learning

• Main Authors: Koloskova, Anastasia, Lin, Tao, Stich, Sebastian U
• Format: Journal Article
• Language: English
• Published: 08.02.2022
• Subjects: Computer Science - Distributed, Parallel, and Cluster Computing; Computer Science - Learning; Mathematics - Optimization and Control
• DOI: 10.48550/arxiv.2202.03836

35th Conference on Neural Information Processing Systems (NeurIPS 2021) We consider decentralized machine learning over a network where the training data is distributed across $n$ agents, each of which can compute stochastic model updates on their local data. The agent's common goal is to find a model that minimizes the average of all local loss functions. While gradient tracking (GT) algorithms can overcome a key challenge, namely accounting for differences between workers' local data distributions, the known convergence rates for GT algorithms are not optimal with respect to their dependence on the mixing parameter $p$ (related to the spectral gap of the connectivity matrix). We provide a tighter analysis of the GT method in the stochastic strongly convex, convex and non-convex settings. We improve the dependency on $p$ from $\mathcal{O}(p^{-2})$ to $\mathcal{O}(p^{-1}c^{-1})$ in the noiseless case and from $\mathcal{O}(p^{-3/2})$ to $\mathcal{O}(p^{-1/2}c^{-1})$ in the general stochastic case, where $c \geq p$ is related to the negative eigenvalues of the connectivity matrix (and is a constant in most practical applications). This improvement was possible due to a new proof technique which could be of independent interest.