Dissecting the Effects of SGD Noise in Distinct Regimes of Deep Learning

Understanding when the noise in stochastic gradient descent (SGD) affects generalization of deep neural networks remains a challenge, complicated by the fact that networks can operate in distinct training regimes. Here we study how the magnitude of this noise $T$ affects performance as the size of the training set $P$ and the scale of initialization $\alpha$ are varied. For gradient descent, $\alpha$ is a key parameter that controls if the network is `lazy' ($\alpha\gg 1$) or instead learns features ($\alpha\ll 1$). For classification of MNIST and CIFAR10 images, our central results are: (i) obtaining phase diagrams for performance in the $(\alpha,T)$ plane. They show that SGD noise can be detrimental or instead useful depending on the training regime. Moreover, although increasing $T$ or decreasing $\alpha$ both allow the net to escape the lazy regime, these changes can have opposite effects on performance. (ii) Most importantly, we find that key dynamical quantities (including the total variations of weights during training) depend on both $T$ and $P$ as power laws, and the characteristic temperature $T_c$, where the noise of SGD starts affecting performance, is a power law of $P$. These observations indicate that a key effect of SGD noise occurs late in training, by affecting the stopping process whereby all data are fitted. We argue that due to SGD noise, nets must develop a stronger `signal', i.e. larger informative weights, to fit the data, leading to a longer training time. The same effect occurs at larger training set $P$. We confirm this view in the perceptron model, where signal and noise can be precisely measured. Interestingly, exponents characterizing the effect of SGD depend on the density of data near the decision boundary, as we explain.