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

2026-02-03

AI Summary

This work addresses the limitations of existing accelerated noisy power methods in decentralized principal component analysis (PCA) and other settings where only inexact matrix-vector products are available. Prior analyses impose overly stringent constraints on perturbation magnitudes, hindering practical applicability. We present an improved convergence analysis of the accelerated noisy power method that guarantees accelerated convergence rates under significantly milder perturbation conditions. Building on this, we propose the first decentralized PCA algorithm with provable accelerated convergence guarantees. The algorithm achieves optimal convergence rates and noise tolerance while maintaining communication costs comparable to non-accelerated methods, and we establish a matching lower bound showing that its worst-case performance cannot be further improved.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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.

Problem

• Accelerated Noisy Power Method
• Decentralized PCA
• Inexact matrix-vector products
• Perturbation bounds
• Convergence guarantees

Innovation

• Accelerated Noisy Power Method
• Decentralized PCA
• Perturbation Analysis
• Optimal Convergence Rate
• Matrix Approximation