Neural tangent kernel analysis of shallow $\alpha$-Stable ReLU neural networks

• URL: http://arxiv.org/abs/2206.08065v1
• Date: Thu, 16 Jun 2022 10:28:03 GMT
• Title: Neural tangent kernel analysis of shallow $\alpha$-Stable ReLU neural networks
• Authors: Stefano Favaro, Sandra Fortini, Stefano Peluchetti
• Abstract summary: We consider problems for $alpha$-Stable NNs, which generalize Gaussian NNs. For shallow $alpha$-Stable NNs with a ReLU function, we show that if the NN's width goes to infinity then a rescaled NN converges weakly to an $alpha$-Stable process. Our main contribution is the NTK analysis of shallow $alpha$-Stable ReLU-NNs, which leads to an equivalence between training a rescaled NN and performing a kernel regression with an $(alpha/
• Score: 8.000374471991247
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: There is a recent literature on large-width properties of Gaussian neural networks (NNs), i.e. NNs whose weights are distributed according to Gaussian distributions. Two popular problems are: i) the study of the large-width behaviour of NNs, which provided a characterization of the infinitely wide limit of a rescaled NN in terms of a Gaussian process; ii) the study of the large-width training dynamics of NNs, which set forth an equivalence between training the rescaled NN and performing a kernel regression with a deterministic kernel referred to as the neural tangent kernel (NTK). In this paper, we consider these problems for $\alpha$-Stable NNs, which generalize Gaussian NNs by assuming that the NN's weights are distributed as $\alpha$-Stable distributions with $\alpha\in(0,2]$, i.e. distributions with heavy tails. For shallow $\alpha$-Stable NNs with a ReLU activation function, we show that if the NN's width goes to infinity then a rescaled NN converges weakly to an $\alpha$-Stable process, i.e. a stochastic process with $\alpha$-Stable finite-dimensional distributions. As a novelty with respect to the Gaussian setting, in the $\alpha$-Stable setting the choice of the activation function affects the scaling of the NN, that is: to achieve the infinitely wide $\alpha$-Stable process, the ReLU function requires an additional logarithmic scaling with respect to sub-linear functions. Then, our main contribution is the NTK analysis of shallow $\alpha$-Stable ReLU-NNs, which leads to an equivalence between training a rescaled NN and performing a kernel regression with an $(\alpha/2)$-Stable random kernel. The randomness of such a kernel is a further novelty with respect to the Gaussian setting, that is: in the $\alpha$-Stable setting the randomness of the NN at initialization does not vanish in the NTK analysis, thus inducing a distribution for the kernel of the underlying kernel regression.

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