  We study the convergence properties of the VR-PCA algorithm introduced by
\cite{shamir2015stochastic} for fast computation of leading singular vectors.
We prove several new results, including a formal analysis of a block version of
the algorithm, and convergence from random initialization. We also make a few
observations of independent interest, such as how pre-initializing with just a
single exact power iteration can significantly improve the runtime of
stochastic methods, and what are the convexity and non-convexity properties of
the underlying optimization problem.
