  We consider in this work small random perturbations (of multiplicative noise
type) of the gradient flow. We prove that under mild conditions, when the
potential function is a Morse function with additional strong saddle condition,
the perturbed gradient flow converges to the neighborhood of local minimizers
in $O(\ln (\varepsilon^{-1}))$ time on the average, where $\varepsilon$ is the
scale of the random perturbation. Under a change of time scale, this indicates
that for the diffusion process that approximates the stochastic gradient
method, it takes (up to logarithmic factor) only a linear time of inverse
stepsize to evade from all saddle points. This can be regarded as a
manifestation of fast convergence of the discrete-time stochastic gradient
method, the latter being used heavily in modern statistical machine learning.
