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Stopping criteria for Stochastic Gradient Descent (SGD) methods playimportant roles from enabling adaptive step size schemes to providing rigor fordownstream analyses such as asymptotic inference. Unfortunately, currentstopping criteria for SGD methods are often heuristics that rely on asymptoticnormality results or convergence to stationary distributions, which may fail toexist for nonconvex functions and, thereby, limit the applicability of suchstopping criteria. To address this issue, in this work, we rigorously developtwo stopping criteria for SGD that can be applied to a broad class of nonconvexfunctions, which we term Bottou-Curtis-Nocedal functions. Moreover, as aprerequisite for developing these stopping criteria, we prove that the gradientfunction evaluated at SGD's iterates converges strongly to zero forBottou-Curtis-Nocedal functions, which addresses an open question in the SGDliterature. As a result of our work, our rigorously developed stopping criteriacan be used to develop new adaptive step size schemes or bolster otherdownstream analyses for nonconvex functions.

Stopping Criteria for, and Strong Convergence of, Stochastic Gradient Descent on Bottou-Curtis-Nocedal Functions