  Loss functions with a large number of saddle points are one of the major
obstacles for training modern machine learning models efficiently. First-order
methods such as gradient descent are usually the methods of choice for training
machine learning models. However, these methods converge to saddle points for
certain choices of initial guesses. In this paper, we propose a modification of
the recently proposed Laplacian smoothing gradient descent [Osher et al.,
arXiv:1806.06317], called modified Laplacian smoothing gradient descent
(mLSGD), and demonstrate its potential to avoid saddle points without
sacrificing the convergence rate. Our analysis is based on the attraction
region, formed by all starting points for which the considered numerical scheme
converges to a saddle point. We investigate the attraction region's dimension
both analytically and numerically. For a canonical class of quadratic
functions, we show that the dimension of the attraction region for mLSGD is
floor((n-1)/2), and hence it is significantly smaller than that of the gradient
descent whose dimension is n-1.
