  In this paper, we give a sharp analysis for Stochastic Gradient Descent (SGD)
and prove that SGD is able to efficiently escape from saddle points and find an
$(\epsilon, O(\epsilon^{0.5}))$-approximate second-order stationary point in
$\tilde{O}(\epsilon^{-3.5})$ stochastic gradient computations for generic
nonconvex optimization problems, when the objective function satisfies
gradient-Lipschitz, Hessian-Lipschitz, and dispersive noise assumptions. This
result subverts the classical belief that SGD requires at least
$O(\epsilon^{-4})$ stochastic gradient computations for obtaining an
$(\epsilon,O(\epsilon^{0.5}))$-approximate second-order stationary point. Such
SGD rate matches, up to a polylogarithmic factor of problem-dependent
parameters, the rate of most accelerated nonconvex stochastic optimization
algorithms that adopt additional techniques, such as Nesterov's momentum
acceleration, negative curvature search, as well as quadratic and cubic
regularization tricks. Our novel analysis gives new insights into nonconvex SGD
and can be potentially generalized to a broad class of stochastic optimization
algorithms.
