  We consider the problem of principal component analysis (PCA) in a streaming
stochastic setting, where our goal is to find a direction of approximate
maximal variance, based on a stream of i.i.d. data points in $\reals^d$. A
simple and computationally cheap algorithm for this is stochastic gradient
descent (SGD), which incrementally updates its estimate based on each new data
point. However, due to the non-convex nature of the problem, analyzing its
performance has been a challenge. In particular, existing guarantees rely on a
non-trivial eigengap assumption on the covariance matrix, which is intuitively
unnecessary. In this paper, we provide (to the best of our knowledge) the first
eigengap-free convergence guarantees for SGD in the context of PCA. This also
partially resolves an open problem posed in \cite{hardt2014noisy}. Moreover,
under an eigengap assumption, we show that the same techniques lead to new SGD
convergence guarantees with better dependence on the eigengap.
