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A central challenge to many fields of science and engineering involvesminimizing non-convex error functions over continuous, high dimensional spaces.Gradient descent or quasi-Newton methods are almost ubiquitously used toperform such minimizations, and it is often thought that a main source ofdifficulty for these local methods to find the global minimum is theproliferation 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 profounddifficulty originates from the proliferation of saddle points, not localminima, especially in high dimensional problems of practical interest. Suchsaddle points are surrounded by high error plateaus that can dramatically slowdown learning, and give the illusory impression of the existence of a localminimum. Motivated by these arguments, we propose a new approach tosecond-order optimization, the saddle-free Newton method, that can rapidlyescape high dimensional saddle points, unlike gradient descent and quasi-Newtonmethods. We apply this algorithm to deep or recurrent neural network training,and provide numerical evidence for its superior optimization performance.

Identifying and attacking the saddle point problem in high-dimensional non-convex optimization