Revisiting Normalized Gradient Descent: Fast Evasion of Saddle Points.

The paper considers normalized gradient descent (NGD), a natural modification of classical gradient descent (GD) in optimization problems. It is shown that, contrary to GD, NGD escapes saddle points “quickly.” A serious shortcoming of GD in nonconvex problems is that it can take arbitrarily long to escape from the neighborhood of a saddle point. In practice, this issue can significantly slow the convergence of GD, particularly in high-dimensional nonconvex problems. The paper focuses on continuous-time dynamics. It is shown that 1) NGD “almost never” converges to saddle points and 2) the time required for NGD to escape from a ball of radius $r$ about a saddle point $x^*$ is at most $5\sqrt{\kappa }r$ , where $\kappa$ is the condition number of the Hessian of $f$ at $x^*$. As a simple application of these results, a global convergence-time bound is established for NGD under mild assumptions. [ABSTRACT FROM AUTHOR]