Disentangling feature and lazy training in deep neural networks

Two distinct limits for deep learning have been derived as the network width h -> infinity, depending on how the weights of the last layer scale with h. In the neural tangent Kernel (NTK) limit, the dynamics becomes linear in the weights and is described by a frozen kernel Theta (the NTK). By contrast, in the mean-field limit, the dynamics can be expressed in terms of the distribution of the parameters associated with a neuron, that follows a partial differential equation. In this work we consider deep networks where the weights in the last layer scale as alpha h(-1/2) at initialization. By varying alpha and h, we probe the crossover between the two limits. We observe two the previously identified regimes of 'lazy training' and 'feature training'. In the lazy-training regime, the dynamics is almost linear and the NTK barely changes after initialization. The feature-training regime includes the mean-field formulation as a limiting case and is characterized by a kernel that evolves in time, and thus learns some features. We perform numerical experiments on MNIST, Fashion-MNIST, EMNIST and CIFAR10 and consider various architectures. We find that: (i) the two regimes are separated by an alpha* that scales as 1/root h. (ii) Network architecture and data structure play an important role in determining which regime is better: in our tests, fully-connected networks perform generally better in the lazy-training regime, unlike convolutional networks. (iii) In both regimes, the fluctuations delta F induced on the learned function by initial conditions decay as delta F similar to 1/root h, leading to a performance that increases with h. The same improvement can also be obtained at an intermediate width by ensemble-averaging several networks that are trained independently. (iv) In the feature-training regime we identify a time scale t(1) similar to root h alpha, such that for t < t(1) the dynamics is linear. At t similar to t(1), the output has grown by a magnitude root h and the changes of the tangent kernel parallel to Delta Theta parallel to become significant. Ultimately, it follows parallel to Delta Theta parallel to similar to(root h alpha)(-a) for ReLU and Softplus activation functions, with a < 2 and a -> 2 as depth grows. We provide scaling arguments supporting these findings.