Accelerated Stochastic Power Iteration

Principal component analysis (PCA) is one of the most powerful tools in machine learning. The simplest method for PCA, the power iteration, requires mathcal O(1/Delta) full-data passes to recover the principal component of a matrix with eigen-gap Delta. Lanczos, a significantly more complex method, achieves an accelerated rate of mathcal O(1/√Delta) passes. Modern applications, however, motivate methods that only ingest a subset of available data, known as the stochastic setting. In the online stochastic setting, simple algorithms like Oja's iteration achieve the optimal sample complexity mathcal O(σ2/Delta2). Unfortunately, they are fully sequential, and also require mathcal O(σ2/Delta2) iterations, far from the mathcal O(1/√Delta) rate of Lanczos. We propose a simple variant of the power iteration with an added momentum term, that achieves both the optimal sample and iteration complexity. In the full-pass setting, standard analysis shows that momentum achieves the accelerated rate, mathcal O(1/√Delta). We demonstrate empirically that naively applying momentum to a stochastic method, does not result in acceleration. We perform a novel, tight variance analysis that reveals the "breaking-point variance" beyond which this acceleration does not occur. By combining this insight with modern variance reduction techniques, we construct stochastic PCA algorithms, for the online and offline setting, that achieve an accelerated iteration complexity mathcal O(1/√Delta). Due to the embarassingly parallel nature of our methods, this acceleration translates directly to wall-clock time if deployed in a parallel environment. Our approach is very general, and applies to many non-convex optimization problems that can now be accelerated using the same technique.