Improved Analysis of the Accelerated Noisy Power Method with Applications to Decentralized PCA
This work addresses the practical limitations of previous accelerated PCA methods in decentralized environments, offering the first decentralized PCA algorithm with provably accelerated convergence.
The paper tackles the problem of Principal Component Analysis (PCA) when only inexact matrix-vector products are available, such as in decentralized settings, by providing an improved analysis of the Accelerated Noisy Power Method that preserves accelerated convergence rates under milder noise conditions and is shown to be worst-case optimal.
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.