  The alternating gradient descent (AGD) is a simple but popular algorithm
which has been applied to problems in optimization, machine learning, data
ming, and signal processing, etc. The algorithm updates two blocks of variables
in an alternating manner, in which a gradient step is taken on one block, while
keeping the remaining block fixed. When the objective function is nonconvex, it
is well-known the AGD converges to the first-order stationary solution with a
global sublinear rate.
  In this paper, we show that a variant of AGD-type algorithms will not be
trapped by "bad" stationary solutions such as saddle points and local maximum
points. In particular, we consider a smooth unconstrained optimization problem,
and propose a perturbed AGD (PA-GD) which converges (with high probability) to
the set of second-order stationary solutions (SS2) with a global sublinear
rate. To the best of our knowledge, this is the first alternating type
algorithm which takes $\mathcal{O}(\text{polylog}(d)/\epsilon^{7/3})$
iterations to achieve SS2 with high probability [where polylog$(d)$ is
polynomial of the logarithm of dimension $d$ of the problem].
