Infinite-width limit of deep linear neural networks

• URL: http://arxiv.org/abs/2211.16980v1
• Date: Tue, 29 Nov 2022 18:37:14 GMT
• Title: Infinite-width limit of deep linear neural networks
• Authors: L\'ena\"ic Chizat, Maria Colombo, Xavier Fern\'andez-Real, Alessio Figalli
• Abstract summary: We study the infinite-width limit of deep linear neural networks with random parameters. We show that the linear predictors converge at an exponential rate to the minimal $ell$-norm minimizer of the risk.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper studies the infinite-width limit of deep linear neural networks initialized with random parameters. We obtain that, when the number of neurons diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a gradient descent on an infinitely wide deterministic linear neural network. Moreover, even if the weights remain random, we get their precise law along the training dynamics, and prove a quantitative convergence result of the linear predictor in terms of the number of neurons. We finally study the continuous-time limit obtained for infinitely wide linear neural networks and show that the linear predictors of the neural network converge at an exponential rate to the minimal $\ell_2$-norm minimizer of the risk.

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