Minimization of Stochastic First-order Oracle Complexity of Adaptive Methods for Nonconvex Optimization

Numerical evaluations have definitively shown that, for deep learning optimizers such as stochastic gradient descent, momentum, and adaptive methods, the number of steps needed to train a deep neural network halves for each doubling of the batch size and that there is a region of diminishing returns beyond the critical batch size. In this paper, we determine the actual critical batch size by using the global minimizer of the stochastic first-order oracle (SFO) complexity of the optimizer. To prove the existence of the actual critical batch size, we set the lower and upper bounds of the SFO complexity and prove that there exist critical batch sizes in the sense of minimizing the lower and upper bounds. This proof implies that, if the SFO complexity fits the lower and upper bounds, then the existence of these critical batch sizes demonstrates the existence of the actual critical batch size. We also discuss the conditions needed for the SFO complexity to fit the lower and upper bounds and provide numerical results that support our theoretical results.