Revisiting Normalized Gradient Descent: Fast Evasion of Saddle Points

Abstract: The note considers normalized gradient descent (NGD), a natural modification of classical gradient descent (GD) in optimization problems. A serious shortcoming of GD in non-convex problems is that GD may take arbitrarily long to escape from the neighborhood of a saddle point. This issue can make the convergence of GD arbitrarily slow, particularly in high-dimensional non-convex problems where the relative number of saddle points is often large. The paper focuses on continuous-time descent. It is shown that, contrary to standard GD, NGD escapes saddle points quickly.' In particular, it is shown that (i) NGDalmost never' converges to saddle points and (ii) 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 an application of this result, a global convergence-time bound is established for NGD under mild assumptions.