Linear Regression with Distributed Learning: A Generalization Error Perspective

TL;DR

This paper analyzes the generalization error of distributed linear regression, revealing how model partitioning affects performance on unseen data compared to centralized solutions.

Contribution

It provides high-probability bounds on generalization error for distributed linear regression with various data types, highlighting the impact of model partitioning.

Findings

Distributed solutions can have higher generalization error than centralized ones.

Generalization bounds depend on data distribution and model partitioning.

Numerical experiments validate theoretical insights with real and synthetic data.

Abstract

Distributed learning provides an attractive framework for scaling the learning task by sharing the computational load over multiple nodes in a network. Here, we investigate the performance of distributed learning for large-scale linear regression where the model parameters, i.e., the unknowns, are distributed over the network. We adopt a statistical learning approach. In contrast to works that focus on the performance on the training data, we focus on the generalization error, i.e., the performance on unseen data. We provide high-probability bounds on the generalization error for both isotropic and correlated Gaussian data as well as sub-gaussian data. These results reveal the dependence of the generalization performance on the partitioning of the model over the network. In particular, our results show that the generalization error of the distributed solution can be substantially higher than that of the centralized solution even when the error on the training data is at the same level for both the centralized and distributed approaches. Our numerical results illustrate the performance with both real-world image data as well as synthetic data.