We introduce novel convergence results for asynchronous iterations that appear in the analysis of parallel and distributed optimization algorithms. The results are simple to apply and give explicit estimates for how the degree of asynchrony impacts the convergence rates of the iterates. Our results shorten, streamline and strengthen existing convergence proofs for several asynchronous optimization methods and allow us to establish convergence guarantees for popular algorithms that were thus far lacking a complete theoretical understanding. Specifically, we use our results to derive better iteration complexity bounds for proximal incremental aggregated gradient methods, to obtain tighter guarantees depending on the average rather than maximum delay for the asynchronous stochastic gradient descent method, to provide less conservative analyses of the speedup conditions for asynchronous block-coordinate implementations of Krasnoselskii-Mann iterations, and to quantify the convergence rates for totally asynchronous iterations under various assumptions on communication delays and update rates.

翻译：我们提出了异步迭代的新型收敛结果，这些结果出现在并行与分布式优化算法的分析中。该结果易于应用，并给出了异步程度对迭代收敛速度影响的明确估计。我们的结果简化、精简并强化了多种异步优化方法的现有收敛性证明，并为迄今缺乏完整理论理解的流行算法建立了收敛性保证。具体而言，我们利用这些结果为邻近增量聚合梯度方法推导出更优的迭代复杂度界；针对异步随机梯度下降方法，获得依赖于平均延迟而非最大延迟的更紧致保证；为Krasnoselskii-Mann迭代的异步分块坐标实现提供更少保守性的加速条件分析；并量化了在通信延迟与更新速率的不同假设下完全异步迭代的收敛速度。