  Convergence detection of iterative stochastic optimization methods is of
great practical interest. This paper considers stochastic gradient descent
(SGD) with a constant learning rate and momentum. We show that there exists a
transient phase in which iterates move towards a region of interest, and a
stationary phase in which iterates remain bounded in that region around a
minimum point. We construct a statistical diagnostic test for convergence to
the stationary phase using the inner product between successive gradients and
demonstrate that the proposed diagnostic works well. We theoretically and
empirically characterize how momentum can affect the test statistic of the
diagnostic, and how the test statistic captures a relatively sparse signal
within the gradients in convergence. Finally, we demonstrate an application to
automatically tune the learning rate by reducing it each time stationarity is
detected, and show the procedure is robust to mis-specified initial rates.

Machine Learning;Machine Learning