On the Optimality of Averaging in Distributed Statistical Learning

TL;DR

This paper analyzes the statistical error of averaging in distributed empirical risk minimization, showing it is asymptotically optimal in fixed dimensions but incurs a linear accuracy loss in high-dimensional regimes, guiding practical machine learning deployment.

Contribution

It provides asymptotically exact error expressions for distributed averaging in both fixed and high-dimensional settings, revealing when averaging is optimal or incurs a loss.

Findings

Averaging is asymptotically as accurate as centralized solutions in fixed dimensions.

In high-dimensional regimes, averaging causes a linear accuracy loss with the number of machines.

The results inform optimal choices for the number of machines in distributed learning.

Abstract

A common approach to statistical learning with big-data is to randomly split it among $m$ machines and learn the parameter of interest by averaging the $m$ individual estimates. In this paper, focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy. We consider two large-sample settings: First, a classical setting where the number of parameters $p$ is fixed, and the number of samples per machine $n\to\infty$. Second, a high-dimensional regime where both $p,n\to\infty$ with $p/n \to \kappa \in (0,1)$. For both regimes and under suitable assumptions, we present asymptotically exact expressions for this estimation error. In the fixed-$p$ setting, under suitable assumptions, we prove that to leading order averaging is as accurate as the centralized solution. We also derive the second order error terms, and show that these can be non-negligible, notably for non-linear models. The high-dimensional setting, in contrast, exhibits a qualitatively different behavior: data splitting incurs a first-order accuracy loss, which to leading order increases linearly with the number of machines. The dependence of our error approximations on the number of machines traces an interesting accuracy-complexity tradeoff, allowing the practitioner an informed choice on the number of machines to deploy. Finally, we confirm our theoretical analysis with several simulations.