  The power method is a classical algorithm with broad applications in machine
learning tasks, including streaming PCA, spectral clustering, and low-rank
matrix approximation. The distilled purpose of the vanilla power method is to
determine the largest eigenvalue (in absolute modulus) and its eigenvector of a
matrix. A momentum-based scheme can be used to accelerate the power method, but
achieving an optimal convergence rate with existing algorithms critically
relies on additional spectral information that is unavailable at run-time, and
sub-optimal initializations can result in divergence. In this paper, we provide
a pair of novel momentum-based power methods, which we call the delayed
momentum power method (DMPower) and a streaming variant, the delayed momentum
streaming method (DMStream). Our methods leverage inexact deflation and are
capable of achieving near-optimal convergence with far less restrictive
hyperparameter requirements. We provide convergence analyses for both
algorithms through the lens of perturbation theory. Further, we experimentally
demonstrate that DMPower routinely outperforms the vanilla power method and
that both algorithms match the convergence speed of an oracle running existing
accelerated methods with perfect spectral knowledge.
