Escaping From Saddle Points — Online Stochastic Gradient for Tensor Decomposition
Proceedings of The 28th Conference on Learning Theory, PMLR 40:797-842, 2015.
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 \em saddle points. In this paper we identify \em strict saddle property for non-convex problem that allows for efficient optimization. Using this property we show that from an \em arbitrary starting point, 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 \em 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.