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We give a sharp convergence rate for the asynchronous stochastic gradientdescent (ASGD) algorithms when the loss function is a perturbed quadraticfunction based on the stochastic modified equations introduced in [An et al.Stochastic modified equations for the asynchronous stochastic gradient descent,arXiv:1805.08244]. We prove that when the number of local workers is largerthan the expected staleness, then ASGD is more efficient than stochasticgradient descent. Our theoretical result also suggests that longer delaysresult in slower convergence rate. Besides, the learning rate cannot be smallerthan a threshold inversely proportional to the expected staleness.

A Sharp Convergence Rate for the Asynchronous Stochastic Gradient Descent