We study the supervised learning problem under either of the following two models: (1) Feature vectors @math are @math -dimensional Gaussians and responses are @math for @math an unknown quadratic function; (2) Feature vectors @math are distributed as a mixture of two @math -dimensional centered Gaussians, and @math 's are the corresponding class labels. We use two-layers neural networks with quadratic activations, and compare three different learning regimes: the random features (RF) regime in which we only train the second-layer weights; the neural tangent (NT) regime in which we train a linearization of the neural network around its initialization; the fully trained neural network (NN) regime in which we train all the weights in the network. We prove that, even for the simple quadratic model of point (1), there is a potentially unbounded gap between the prediction risk achieved in these three training regimes, when the number of neurons is smaller than the ambient dimension. When the number of neurons is larger than the number of dimensions, the problem is significantly easier and both NT and NN learning achieve zero risk.
Chizat and Bach @cite_12 clarified that any nonlinear statistical model can be approximated by a linear one in an early () training regime. The basic argument is quite simple. Given a model @math with parameters @math , we can Taylor-expand around a random initialization @math . Setting @math , we get Here the second approximation holds since, for many random initializations, @math because of random cancellations. The resulting model @math is linear, with random features.
@cite_12
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In a series of recent theoretical works, it has been shown that strongly over-parameterized neural networks trained with gradient-based methods could converge linearly to zero training loss, with their parameters hardly varying. In this note, our goal is to exhibit the simple structure that is behind these results. In a simplified setting, we prove that "lazy training" essentially solves a kernel regression. We also show that this behavior is not so much due to over-parameterization than to a choice of scaling, often implicit, that allows to linearize the model around its initialization. These theoretical results complemented with simple numerical experiments make it seem unlikely that "lazy training" is behind the many successes of neural networks in high dimensional tasks.