  Many modern machine learning applications - from online principal component
analysis to covariance matrix identification and dictionary learning - can be
formulated as minimization problems on Riemannian manifolds, and are typically
solved with a Riemannian stochastic gradient method (or some variant thereof).
However, in many cases of interest, the resulting minimization problem is not
geodesically convex, so the convergence of the chosen solver to a desirable
solution - i.e., a local minimizer - is by no means guaranteed. In this paper,
we study precisely this question, that is, whether stochastic Riemannian
optimization algorithms are guaranteed to avoid saddle points with probability
1. For generality, we study a family of retraction-based methods which, in
addition to having a potentially much lower per-iteration cost relative to
Riemannian gradient descent, include other widely used algorithms, such as
natural policy gradient methods and mirror descent in ordinary convex spaces.
In this general setting, we show that, under mild assumptions for the ambient
manifold and the oracle providing gradient information, the policies under
study avoid strict saddle points / submanifolds with probability 1, from any
initial condition. This result provides an important sanity check for the use
of gradient methods on manifolds as it shows that, almost always, the limit
state of a stochastic Riemannian algorithm can only be a local minimizer.
