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In this paper, we consider the convergence of a very generalasynchronous-parallel algorithm called ARock, that takes many well-knownasynchronous algorithms as special cases (gradient descent, proximal gradient,Douglas Rachford, ADMM, etc.). In asynchronous-parallel algorithms, thecomputing nodes simply use the most recent information that they have accessto, instead of waiting for a full update from all nodes in the system. Thismeans that nodes do not have to waste time waiting for information, which canbe a major bottleneck, especially in distributed systems. When the system has$p$ nodes, asynchronous algorithms may complete $\Theta(\ln(p))$ moreiterations than synchronous algorithms in a given time period ("more iterationsper second").  Although asynchronous algorithms may compute more iterations per second,there is error associated with using outdated information. How many moreiterations in total are needed to compensate for this error is still an openquestion. The main results of this paper aim to answer this question. We prove,loosely, that as the size of the problem becomes large, the number ofadditional iterations that asynchronous algorithms need becomes negligiblecompared to the total number ("same quality" of the iterations). Taking thesefacts together, our results provide solid evidence of the potential ofasynchronous algorithms to vastly speed up certain distributed computations.

More Iterations per Second, Same Quality -- Why Asynchronous Algorithms may Drastically Outperform Traditional Ones