Interpolation property of shallow neural networks
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We study the geometry of global minima of the loss landscape of overparametrized neural networks. In most optimization problems, the loss function is convex, in which case we only have a global minima, or nonconvex, with a discrete number of global minima. In this paper, we prove that in the overparametrized regime, a shallow neural network can interpolate any data set, i.e. the loss function has a global minimum value equal to zero as long as the activation function is not a polynomial of small degree. Additionally, if such a global minimum exists, then the locus of global minima has infinitely many points. Furthermore, we give a characterization of the Hessian of the loss function evaluated at the global minima, and in the last section, we provide a practical probabilistic method of finding the interpolation point.
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