  We analyze stochastic gradient descent for optimizing non-convex functions.
In many cases for non-convex functions the goal is to find a reasonable local
minimum, and the main concern is that gradient updates are trapped in saddle
points. In this paper we identify strict saddle property for non-convex problem
that allows for efficient optimization. Using this property we show that
stochastic gradient descent converges to a local minimum in a polynomial number
of iterations. To the best of our knowledge this is the first work that gives
global convergence guarantees for stochastic gradient descent on non-convex
functions with exponentially many local minima and saddle points. Our analysis
can be applied to orthogonal tensor decomposition, which is widely used in
learning a rich class of latent variable models. We propose a new optimization
formulation for the tensor decomposition problem that has strict saddle
property. As a result we get the first online algorithm for orthogonal tensor
decomposition with global convergence guarantee.
