Improved Analysis of the Accelerated Noisy Power Method with Applications to Decentralized PCA

TL;DR

This paper improves the theoretical understanding of the Accelerated Noisy Power Method, showing it maintains fast convergence under milder noise conditions and introduces the first provably accelerated decentralized PCA algorithm.

Contribution

It provides a worst-case optimal analysis of the Accelerated Noisy Power Method under realistic noise conditions and develops the first accelerated decentralized PCA algorithm.

Findings

Accelerated method retains convergence with less restrictive noise bounds.

Proposed decentralized PCA algorithm has similar communication costs to non-accelerated methods.

First provably accelerated decentralized PCA algorithm.

Abstract

We analyze the Accelerated Noisy Power Method, an algorithm for Principal Component Analysis in the setting where only inexact matrix-vector products are available, which can arise for instance in decentralized PCA. While previous works have established that acceleration can improve convergence rates compared to the standard Noisy Power Method, these guarantees require overly restrictive upper bounds on the magnitude of the perturbations, limiting their practical applicability. We provide an improved analysis of this algorithm, which preserves the accelerated convergence rate under much milder conditions on the perturbations. We show that our new analysis is worst-case optimal, in the sense that the convergence rate cannot be improved, and that the noise conditions we derive cannot be relaxed without sacrificing convergence guarantees. We demonstrate the practical relevance of our results by deriving an accelerated algorithm for decentralized PCA, which has similar communication costs to non-accelerated methods. To our knowledge, this is the first decentralized algorithm for PCA with provably accelerated convergence.