Linearized two-layers neural networks in high dimension
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We consider the problem of learning an unknown function f_ on the d-dimensional sphere with respect to the square loss, given i.i.d. samples {(y_i, x_i)}_i< n where x_i is a feature vector uniformly distributed on the sphere and y_i=f_( x_i). We study two popular classes of models that can be regarded as linearizations of two-layers neural networks around a random initialization: (RF) The random feature model of Rahimi-Recht; (NT) The neural tangent kernel model of Jacot-Gabriel-Hongler. Both these approaches can also be regarded as randomized approximations of kernel ridge regression (with respect to different kernels), and hence enjoy universal approximation properties when the number of neurons N diverges, for a fixed dimension d. We prove that, if both d and N are large, the behavior of these models is instead remarkably simpler. If N = o(d^2), then RF performs no better than linear regression with respect to the raw features x_i, and NT performs no better than linear regression with respect to degree-one and two monomials in the x_i. More generally, if N= o(d^ℓ+1) then RF fits at most a degree-ℓ polynomial in the raw features, and NT fits at most a degree-(ℓ+1) polynomial.
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