Before revision:   The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is  not a saddle point, and the convergence rate is the same as that of Newton's method  .
After revision:  The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is not a saddle point, and the convergence rate is the same as that of Newton's method . At the end of the paper, we present some issues (saddle points and convergence) one faces when implementing Newton's method and modifications into Deep Neural Networks. In the appendix, we test the good performance of New Q-Newton's method on various benchmark test functions such as Rastrigin, Askley, Rosenbroch and many other, against algorithms such as Newton's method, BFGS, Adaptive Cubic Regularization, Random damping Newton's method and Inertial Newton's method, as well as Unbounded Two-way Backtracking Gradient Descent. The experiments demonstrate in particular that the assumption that f is C^3 is necessary for some conclusions in the main theoretical results .