Communication-efficient Algorithms for Distributed Stochastic Principal Component Analysis

TL;DR

This paper develops communication-efficient distributed algorithms for Principal Component Analysis that achieve near-centralized accuracy, introducing correction and iterative methods to improve efficiency and consistency across different data regimes.

Contribution

It introduces a simple correction step and an iterative algorithm for distributed PCA that improve communication efficiency and estimation accuracy compared to existing methods.

Findings

Correction step ensures consistency with centralized ERM for large n.

Iterative algorithm accelerates convergence over previous methods.

Algorithms perform well across various data regimes.

Abstract

We study the fundamental problem of Principal Component Analysis in a statistical distributed setting in which each machine out of $m$ stores a sample of $n$ points sampled i.i.d. from a single unknown distribution. We study algorithms for estimating the leading principal component of the population covariance matrix that are both communication-efficient and achieve estimation error of the order of the centralized ERM solution that uses all $mn$ samples. On the negative side, we show that in contrast to results obtained for distributed estimation under convexity assumptions, for the PCA objective, simply averaging the local ERM solutions cannot guarantee error that is consistent with the centralized ERM. We show that this unfortunate phenomena can be remedied by performing a simple correction step which correlates between the individual solutions, and provides an estimator that is consistent with the centralized ERM for sufficiently-large $n$. We also introduce an iterative distributed algorithm that is applicable in any regime of $n$, which is based on distributed matrix-vector products. The algorithm gives significant acceleration in terms of communication rounds over previous distributed algorithms, in a wide regime of parameters.