# Disentangling feature and lazy learning in deep neural networks: an empirical study

Two distinct limits for deep learning as the net width h→∞ have been proposed, depending on how the weights of the last layer scale with h. In the "lazy-learning" regime, the dynamics becomes linear in the weights and is described by a Neural Tangent Kernel Θ. By contrast, in the "feature-learning" regime, the dynamics can be expressed in terms of the density distribution of the weights. Understanding which regime describes accurately practical architectures and which one leads to better performance remains a challenge. We answer these questions and produce new characterizations of these regimes for the MNIST data set, by considering deep nets f whose last layer of weights scales as α/√(h) at initialization, where α is a parameter we vary. We performed systematic experiments on two setups (A) fully-connected Softplus momentum full batch and (B) convolutional ReLU momentum stochastic. We find that (1) α^*=1/√(h) separates the two regimes. (2) for (A) and (B) feature learning outperforms lazy learning, a difference in performance that decreases with h and becomes hardly detectable asymptotically for (A) but is very significant for (B). (3) In both regimes, the fluctuations δ f induced by initial conditions on the learned function follow δ f∼1/√(h), leading to a performance that increases with h. This improvement can be instead obtained at intermediate h values by ensemble averaging different networks. (4) In the feature regime there exists a time scale t_1∼α√(h), such that for t≪ t_1 the dynamics is linear. At t∼ t_1, the output has grown by a magnitude √(h) and the changes of the tangent kernel ΔΘ become significant. Ultimately, it follows ΔΘ∼(√(h)α)^-a for ReLU and Softplus activation, with a<2 & a→2 when depth grows.

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