Existing asynchronous distributed optimization algorithms often use
diminishing step-sizes that cause slow practical convergence, or use fixed
step-sizes that depend on and decrease with an upper bound of the delays. Not
only are such delay bounds hard to obtain in advance, but they also tend to be
large and rarely attained, resulting in unnecessarily slow convergence. This
paper develops asynchronous versions of two distributed algorithms, Prox-DGD
and DGD-ATC, for solving consensus optimization problems over undirected
networks. In contrast to alternatives, our algorithms can converge to the fixed
point set of their synchronous counterparts using step-sizes that are
independent of the delays. We establish convergence guarantees for strongly and
weakly convex problems under both partial and total asynchrony. We also show
that the convergence speed of the two asynchronous methods adapts to the actual
level of asynchrony rather than being constrained by the worst-case. Numerical
experiments demonstrate a strong practical performance of our asynchronous
algorithms.

该研究开发了两种分布式算法（Prox-DGD 和 DGD-ATC）的异步版本，用于解决无向网络上的共识优化问题，并且与其他算法相比，我们的算法可以使用与延迟无关的步长来收敛到它们的同步对应算法的固定点集。该研究还在部分异步和完全异步情况下，为强凸和弱凸问题建立了收敛保证，并展示了两种异步方法的收敛速度能够适应实际异步程度而不受最坏情况的限制。数值实验表明我们的异步算法具有很强的实际性能。