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

Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) is one of the most popular algorithms in distributed machine learning. However, its convergence properties for these complicated nonconvex problems is still largely unknown, because of the current technical limit. Therefore, in this paper, we propose to analyze the algorithm through a simpler but nontrivial nonconvex problem - streaming PCA, which helps us to understand Aync-MSGD better even for more general problems. Specifically, we establish the asymptotic rate of convergence of Async-MSGD for streaming PCA by diffusion approximation. Our results indicate a fundamental tradeoff between asynchrony and momentum: To ensure convergence and acceleration through asynchrony, we have to reduce the momentum (compared with Sync-MSGD). To the best of our knowledge, this is the first theoretical attempt on understanding Async-MSGD for distributed nonconvex stochastic optimization. Numerical experiments on both streaming PCA and training deep neural networks are provided to support our findings for Async-MSGD.