Kernel and Deep Regimes in Overparametrized Models

A recent line of work studies overparametrized neural networks in the ``kernel regime,'' i.e.~when the network behaves during training as a kernelized linear predictor, and thus training with gradient descent has the effect of finding the minimum RKHS norm solution. This stands in contrast to other studies which demonstrate how gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. Building on an observation by Chizat and Bach, we show how the scale of the initialization controls the transition between the ``kernel'' (aka lazy) and ``deep'' (aka active) regimes and affects generalization properties in multilayer homogeneous models. We provide a complete and detailed analysis for a simple two-layer model that already exhibits an interesting and meaningful transition between the kernel and deep regimes, and we demonstrate the transition for more complex matrix factorization models.