  In machine learning, asynchronous parallel stochastic gradient descent
(APSGD) is broadly used to speed up the training process through multi-workers.
Meanwhile, the time delay of stale gradients in asynchronous algorithms is
generally proportional to the total number of workers, which brings additional
deviation from the accurate gradient due to using delayed gradients. This may
have a negative influence on the convergence of the algorithm. One may ask: How
many workers can we use at most to achieve a good convergence and the linear
speedup?
  In this paper, we consider the second-order convergence of asynchronous
algorithms in non-convex optimization. We investigate the behaviors of APSGD
with consistent read near strictly saddle points and provide a theoretical
guarantee that if the total number of workers is bounded by
$\widetilde{O}(K^{1/3}M^{-1/3})$ ($K$ is the total steps and $M$ is the
mini-batch size), APSGD will converge to good stationary points ($||\nabla
f(x)||\leq \epsilon, \nabla^2 f(x)\succeq -\sqrt{\epsilon}\bm{I},
\epsilon^2\leq O(\sqrt{\frac{1}{MK}}) $) and the linear speedup is achieved.
Our works give the first theoretical guarantee on the second-order convergence
for asynchronous algorithms. The technique we provide can be generalized to
analyze other types of asynchronous algorithms to understand the behaviors of
asynchronous algorithms in distributed asynchronous parallel training.
