  The performance of stochastic gradient descent (SGD), which is the simplest
first-order optimizer for training deep neural networks, depends on not only
the learning rate but also the batch size. They both affect the number of
iterations and the stochastic first-order oracle (SFO) complexity needed for
training. In particular, the previous numerical results indicated that, for SGD
using a constant learning rate, the number of iterations needed for training
decreases when the batch size increases, and the SFO complexity needed for
training is minimized at a critical batch size and that it increases once the
batch size exceeds that size. Here, we study the relationship between batch
size and the iteration and SFO complexities needed for nonconvex optimization
in deep learning with SGD using constant or decaying learning rates and show
that SGD using the critical batch size minimizes the SFO complexity. We also
provide numerical comparisons of SGD with the existing first-order optimizers
and show the usefulness of SGD using a critical batch size. Moreover, we show
that measured critical batch sizes are close to the sizes estimated from our
theoretical results.
