  Escaping saddle points is a central research topic in nonconvex optimization.
In this paper, we propose a simple gradient-based algorithm such that for a
smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an
$\epsilon$-approximate second-order stationary point in $\tilde{O}(\log
n/\epsilon^{1.75})$ iterations. Compared to the previous state-of-the-art
algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/\epsilon^{2})$ or
$\tilde{O}((\log n)^{6}/\epsilon^{1.75})$ iterations, our algorithm is
polynomially better in terms of $\log n$ and matches their complexities in
terms of $1/\epsilon$. For the stochastic setting, our algorithm outputs an
$\epsilon$-approximate second-order stationary point in $\tilde{O}((\log
n)^{2}/\epsilon^{4})$ iterations. Technically, our main contribution is an idea
of implementing a robust Hessian power method using only gradients, which can
find negative curvature near saddle points and achieve the polynomial speedup
in $\log n$ compared to the perturbed gradient descent methods. Finally, we
also perform numerical experiments that support our results.
