  A central challenge to many fields of science and engineering involves
minimizing non-convex error functions over continuous, high dimensional spaces.
Gradient descent or quasi-Newton methods are almost ubiquitously used to
perform such minimizations, and it is often thought that a main source of
difficulty for these local methods to find the global minimum is the
proliferation of local minima with much higher error than the global minimum.
Here we argue, based on results from statistical physics, random matrix theory,
neural network theory, and empirical evidence, that a deeper and more profound
difficulty originates from the proliferation of saddle points, not local
minima, especially in high dimensional problems of practical interest. Such
saddle points are surrounded by high error plateaus that can dramatically slow
down learning, and give the illusory impression of the existence of a local
minimum. Motivated by these arguments, we propose a new approach to
second-order optimization, the saddle-free Newton method, that can rapidly
escape high dimensional saddle points, unlike gradient descent and quasi-Newton
methods. We apply this algorithm to deep or recurrent neural network training,
and provide numerical evidence for its superior optimization performance.
