Stopping Criteria for, and Strong Convergence of, Stochastic Gradient Descent on Bottou-Curtis-Nocedal Functions
This work provides a theoretical foundation for adaptive step size schemes and downstream analyses in nonconvex optimization, though it is incremental as it builds on existing function classes.
The paper tackles the lack of rigorous stopping criteria for Stochastic Gradient Descent (SGD) on nonconvex functions by developing two criteria applicable to Bottou-Curtis-Nocedal functions, and proves strong convergence of the gradient to zero, addressing an open question in the literature.
Stopping criteria for Stochastic Gradient Descent (SGD) methods play important roles from enabling adaptive step size schemes to providing rigor for downstream analyses such as asymptotic inference. Unfortunately, current stopping criteria for SGD methods are often heuristics that rely on asymptotic normality results or convergence to stationary distributions, which may fail to exist for nonconvex functions and, thereby, limit the applicability of such stopping criteria. To address this issue, in this work, we rigorously develop two stopping criteria for SGD that can be applied to a broad class of nonconvex functions, which we term Bottou-Curtis-Nocedal functions. Moreover, as a prerequisite for developing these stopping criteria, we prove that the gradient function evaluated at SGD's iterates converges strongly to zero for Bottou-Curtis-Nocedal functions, which addresses an open question in the SGD literature. As a result of our work, our rigorously developed stopping criteria can be used to develop new adaptive step size schemes or bolster other downstream analyses for nonconvex functions.