  We provide larger step-size restrictions for which gradient descent based
algorithms (almost surely) avoid strict saddle points. In particular, consider
a twice differentiable (non-convex) objective function whose gradient has
Lipschitz constant L and whose Hessian is well-behaved. We prove that the
probability of initial conditions for gradient descent with step-size up to 2/L
converging to a strict saddle point, given one uniformly random initialization,
is zero. This extends previous results up to the sharp limit imposed by the
convex case. In addition, the arguments hold in the case when a learning rate
schedule is given, with either a continuous decaying rate or a piece-wise
constant schedule.
