$\alpha$-Stable convergence of heavy-tailed infinitely-wide neural networks

• URL: http://arxiv.org/abs/2106.11064v1
• Date: Fri, 18 Jun 2021 01:36:41 GMT
• Title: $\alpha$-Stable convergence of heavy-tailed infinitely-wide neural networks
• Authors: Paul Jung, Hoil Lee, Jiho Lee, and Hongseok Yang
• Abstract summary: infinitely-wide multi-layer perceptrons (MLPs) are limits of standard feed-forward neural networks. We show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit.
• Score: 8.880921123362294
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider infinitely-wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with i.i.d. samples from either a light-tailed (finite variance) or heavy-tailed distribution in the domain of attraction of a symmetric $\alpha$-stable distribution, where $\alpha\in(0,2]$ may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric $\alpha$-stable distribution having the same $\alpha$ parameter of that layer. We then extend a recent result of Favaro, Fortini, and Peluchetti (2020), to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric $\alpha$-stable distributions.

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