Normalization effects on shallow neural networks and related asymptotic expansions
This work provides a theoretical understanding of normalization effects on the statistical output of shallow neural networks, which is important for researchers and practitioners optimizing network performance.
This paper investigates shallow neural networks trained with stochastic gradient descent, characterizing their performance as the number of hidden units and gradient descent steps approach infinity. It shows that both bias and variance decrease as the number of hidden units increases, and that variance decays as the network's normalization approaches the mean-field normalization. Numerical studies on MNIST and CIFAR10 confirm improved accuracy with mean-field normalization.
We consider shallow (single hidden layer) neural networks and characterize their performance when trained with stochastic gradient descent as the number of hidden units $N$ and gradient descent steps grow to infinity. In particular, we investigate the effect of different scaling schemes, which lead to different normalizations of the neural network, on the network's statistical output, closing the gap between the $1/\sqrt{N}$ and the mean-field $1/N$ normalization. We develop an asymptotic expansion for the neural network's statistical output pointwise with respect to the scaling parameter as the number of hidden units grows to infinity. Based on this expansion, we demonstrate mathematically that to leading order in $N$, there is no bias-variance trade off, in that both bias and variance (both explicitly characterized) decrease as the number of hidden units increases and time grows. In addition, we show that to leading order in $N$, the variance of the neural network's statistical output decays as the implied normalization by the scaling parameter approaches the mean field normalization. Numerical studies on the MNIST and CIFAR10 datasets show that test and train accuracy monotonically improve as the neural network's normalization gets closer to the mean field normalization.