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Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) isone of the most popular algorithms in distributed machine learning. However,its convergence properties for these complicated nonconvex problems is stilllargely unknown, because of the current technical limit. Therefore, in thispaper, we propose to analyze the algorithm through a simpler but nontrivialnonconvex problem - streaming PCA, which helps us to understand Aync-MSGDbetter even for more general problems. Specifically, we establish theasymptotic rate of convergence of Async-MSGD for streaming PCA by diffusionapproximation. Our results indicate a fundamental tradeoff between asynchronyand momentum: To ensure convergence and acceleration through asynchrony, wehave to reduce the momentum (compared with Sync-MSGD). To the best of ourknowledge, this is the first theoretical attempt on understanding Async-MSGDfor distributed nonconvex stochastic optimization. Numerical experiments onboth streaming PCA and training deep neural networks are provided to supportour findings for Async-MSGD.

Towards Understanding Acceleration Tradeoff between Momentum and Asynchrony in Nonconvex Stochastic Optimization