Mean-field Analysis of Piecewise Linear Solutions for Wide ReLU Networks

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F22%3A00358262" target="_blank" >RIV/68407700:21230/22:00358262 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.48550/arXiv.2111.02278" target="_blank" >https://doi.org/10.48550/arXiv.2111.02278</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.48550/arXiv.2111.02278" target="_blank" >10.48550/arXiv.2111.02278</a>

Result language
angličtina
Original language name
Mean-field Analysis of Piecewise Linear Solutions for Wide ReLU Networks
Original language description
Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean- field view, and consider a two-layer ReLU network trained via noisy-SGD for a univariate regularized regression problem. Our main result is that SGD with vanishingly small noise injected in the gradients is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of knot"points { i.e., points where the tangent of the ReLU network estimator changes { between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient ow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the knot"points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory.
Czech name
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Czech description
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Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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