Communication-efficient Algorithms for Distributed Stochastic Principal Component Analysis
This work addresses the challenge of efficient distributed computation for PCA, which is incremental as it builds on prior distributed optimization methods but focuses on non-convex PCA objectives.
The paper tackles the problem of distributed principal component analysis (PCA) by developing communication-efficient algorithms that achieve estimation error comparable to centralized solutions using all data. It shows that simple averaging fails for PCA, proposes a correction step for consistency, and introduces an iterative algorithm that reduces communication rounds significantly.
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.