This paper targets solving distributed machine learning problems such as federated learning in a communication-efficient fashion. A class of new stochastic gradient descent (SGD) approaches have been developed, which can be viewed as the stochastic generalization to the recently developed lazily aggregated gradient (LAG) method --- justifying the name LASG. LAG adaptively predicts the contribution of each round of communication and chooses only the significant ones to perform. It saves communication while also maintains the rate of convergence. However, LAG only works with deterministic gradients, and applying it to stochastic gradients yields poor performance. The key components of LASG are a set of new rules tailored for stochastic gradients that can be implemented either to save download, upload, or both. The new algorithms adaptively choose between fresh and stale stochastic gradients and have convergence rates comparable to the original SGD. LASG achieves impressive empirical performance --- it typically saves total communication by an order of magnitude.
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