An Improved Analysis of Gradient Tracking for Decentralized Machine Learning

• URL: http://arxiv.org/abs/2202.03836v1
• Date: Tue, 8 Feb 2022 12:58:14 GMT
• Title: An Improved Analysis of Gradient Tracking for Decentralized Machine Learning
• Authors: Anastasia Koloskova, Tao Lin, Sebastian U. Stich
• Abstract summary: We consider decentralized machine learning over a network where the training data is distributed across $n$ agents. The agent's common goal is to find a model that minimizes the average of all local loss functions. We improve the dependency on $p$ from $mathcalO(p-1)$ to $mathcalO(p-1)$ in the noiseless case.
• Score: 34.144764431505486
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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