  We investigate how the final parameters found by stochastic gradient descent
are influenced by over-parameterization. We generate families of models by
increasing the number of channels in a base network, and then perform a large
hyper-parameter search to study how the test error depends on learning rate,
batch size, and network width. We find that the optimal SGD hyper-parameters
are determined by a "normalized noise scale," which is a function of the batch
size, learning rate, and initialization conditions. In the absence of batch
normalization, the optimal normalized noise scale is directly proportional to
width. Wider networks, with their higher optimal noise scale, also achieve
higher test accuracy. These observations hold for MLPs, ConvNets, and ResNets,
and for two different parameterization schemes ("Standard" and "NTK"). We
observe a similar trend with batch normalization for ResNets. Surprisingly,
since the largest stable learning rate is bounded, the largest batch size
consistent with the optimal normalized noise scale decreases as the width
increases.
