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# Accelerated Stochastic Power Iteration

*Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics*, PMLR 84:58-67, 2018.

#### Abstract

Principal component analysis (PCA) is one of the most powerful tools for analyzing matrices in machine learning. In this paper, we study methods to accelerate power iteration in the stochastic setting by adding a momentum term. While in the deterministic setting, power iteration with momentum has optimal iteration complexity, we show that naively adding momentum to a stochastic method does not always result in acceleration. We perform a novel, tight variance analysis that reveals a "breaking-point variance" beyond which this acceleration does not occur. Combining this insight with modern variance reduction techniques yields a simple version of power iteration with momentum that achieves the optimal iteration complexities in both the online and offline setting. Our methods are embarrassingly parallel and can produce wall-clock-time speedups. Our approach is very general and applies to many non-convex optimization problems that can now be accelerated using the same technique.