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arXiv · 2211.16980

Infinite-width limit of deep linear neural networks

Lénaïc Chizat·Maria Colombo·Xavier Fernández-Real·Alessio Figalli
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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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Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real, Alessio Figalli. 2022-11-29. Infinite-width limit of deep linear neural networks. https://arxiv.org/abs/2211.16980

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