  Asynchronous algorithms have attracted much attention recently due to the
crucial demands on solving large-scale optimization problems. However, the
accelerated versions of asynchronous algorithms are rarely studied. In this
paper, we propose the "momentum compensation" technique to accelerate
asynchronous algorithms for convex problems. Specifically, we first accelerate
the plain Asynchronous Gradient Descent, which achieves a faster
$O(1/\sqrt{\epsilon})$ (v.s. $O(1/\epsilon)$) convergence rate for non-strongly
convex functions, and $O(\sqrt{\kappa}\log(1/\epsilon))$ (v.s. $O(\kappa
\log(1/\epsilon))$) for strongly convex functions to reach an $\epsilon$-
approximate minimizer with the condition number $\kappa$. We further apply the
technique to accelerate modern stochastic asynchronous algorithms such as
Asynchronous Stochastic Coordinate Descent and Asynchronous Stochastic Gradient
Descent. Both of the resultant practical algorithms are faster than existing
ones by order. To the best of our knowledge, we are the first to consider
accelerated algorithms that allow updating by delayed gradients and are the
first to propose truly accelerated asynchronous algorithms. Finally, the
experimental results on a shared memory system show that acceleration can lead
to significant performance gains on ill-conditioned problems.
