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

Published online by Cambridge University Press:  03 July 2023

Paul Jung*
Affiliation:
Sam Houston State University
Hoil Lee*
Affiliation:
KAIST
Jiho Lee*
Affiliation:
Korea Science Academy of KAIST
Hongseok Yang*
Affiliation:
KAIST and Institute for Basic Science
*
*Postal address: Department of Mathematics and Statistics, 1905 University Ave, Huntsville, TX 77340, USA. Email address: phj001@shsu.edu
***Postal address: Department of Mathematics and Computer Sciences, Korea Science Academy of KAIST, 105-47, Baegyanggwanmun-ro, Busanjin-gu, Busan 47162, Republic of Korea. Email address: efidiaf@gmail.com
***Postal address: Department of Mathematics and Computer Sciences, Korea Science Academy of KAIST, 105-47, Baegyanggwanmun-ro, Busanjin-gu, Busan 47162, Republic of Korea. Email address: efidiaf@gmail.com
****Postal address: School of Computing and Kim Jaechul Graduate School of AI, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea; Discrete Mathematics Group, Institute for Basic Science, 55 Expo-ro, Yuseong-gu, Daejeon 34126, Republic of Korea. Email address: hongseok.yang@kaist.ac.kr

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 independent and identically distributed (i.i.d.) samples from either a light-tailed (finite-variance) or a 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 as that layer. Non-stable heavy-tailed weight distributions are important since they have been empirically seen to emerge in trained deep neural nets such as the ResNet and VGG series, and proven to naturally arise via stochastic gradient descent. The introduction of heavy-tailed weights broadens the class of priors in Bayesian neural networks. In this work we 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, $\alpha\in(0,2]$.

Type
Original Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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