Quantitative Gaussian Random Field Approximation of Wide Random Neural Networks

Core Concepts

Quantitative bounds on the Wasserstein distance between the output of wide random neural networks and the corresponding Gaussian random field.

Abstract

The paper derives upper bounds on the Wasserstein distance (W1), with respect to the sup-norm, between any continuous Rd-valued random field indexed by the n-sphere and the corresponding Gaussian random field. This is achieved by developing a novel Gaussian smoothing technique based on covariance functions constructed using powers of Laplacian operators. The authors then apply this general result to obtain the first bounds on the Gaussian random field approximation of wide random neural networks of any depth and with Lipschitz activation functions. The bounds are explicitly expressed in terms of the widths of the network and moments of the random weights. Improved bounds are also obtained when the activation function has three bounded derivatives. The key steps are: Construct a Gaussian smoothing random field S with an explicit Cameron-Martin space, using the Laplacian operator on the n-sphere. Develop a general Stein's method based approach to bound the W1 distance between a random field F and a Gaussian random field G, by first bounding the distance in a smoother metric dF and then transferring the bound to W1 using the Gaussian smoothing. Apply the general result to wide random neural networks, bounding the W1 distance between the network output F and the corresponding Gaussian random field G. This involves a recursive application of the Stein's method approximation, combined with the Gaussian smoothing. The results provide quantitative information about when the limiting Gaussian behavior of wide random neural networks "kicks in", without any restrictions on the depth, activation function, or weight distributions (beyond moment conditions).

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Gaussian random field approximation via Stein's method with applications to wide random neural networks

by Krishnakumar... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2306.16308.pdf

Deeper Inquiries

Can the dependence of the bounds on the layer widths be further improved, potentially matching the optimal central limit rate

The dependence of the bounds on the layer widths, as presented in Theorem 1.2 and Theorem 1.4, can potentially be further improved to match the optimal central limit rate. One approach to achieve this improvement is by exploring the relationship between the widths of the layers and the rate of convergence in the multivariate Central Limit Theorem (CLT). The current bounds in the theorems are based on specific assumptions and conditions, such as the Lipschitz activation function and moment bounds on the weights. By refining these assumptions and conditions, it may be possible to enhance the bounds on the layer widths. To match the optimal central limit rate, it would be essential to carefully analyze the convergence properties of wide random neural networks and their Gaussian approximations. This analysis could involve investigating the scaling behavior of the widths of the layers relative to each other and identifying the critical factors that influence the rate of convergence. By refining the assumptions and techniques used in the proofs of Theorem 1.2 and Theorem 1.4, it may be feasible to achieve bounds that exhibit the optimal central limit rate, providing a more accurate representation of the limiting behavior of wide random neural networks.

How can the techniques developed here be extended to study the limiting behavior of wide random neural networks with heavy-tailed weight distributions

The techniques developed in the study of Gaussian random field approximation for wide random neural networks can be extended to investigate the limiting behavior of networks with heavy-tailed weight distributions. Heavy-tailed distributions, such as stable distributions, introduce additional complexity due to their non-Gaussian nature and the presence of heavier tails compared to Gaussian distributions. To study the limiting behavior of wide random neural networks with heavy-tailed weight distributions, it would be necessary to adapt the existing methods to accommodate the characteristics of heavy-tailed distributions. This adaptation may involve modifying the covariance functions, adjusting the smoothing techniques, and considering the impact of heavy tails on the convergence properties of the networks. By incorporating the properties of heavy-tailed distributions into the analysis, it would be possible to provide quantitative Gaussian approximation results for networks with heavy-tailed weights, shedding light on their asymptotic behavior and convergence to Gaussian random fields.

What are the implications of the quantitative Gaussian approximation results for the training dynamics of neural networks, especially in the context of the observed phase transition from Gaussian to non-Gaussian limits under different step-size regimes

The quantitative Gaussian approximation results obtained in the study have significant implications for understanding the training dynamics of neural networks, particularly in the context of the observed phase transition from Gaussian to non-Gaussian limits under different step-size regimes. These results provide a rigorous framework for analyzing the accuracy of approximating wide random neural networks with Gaussian random fields, offering insights into the qualitative and quantitative behavior of the networks during training. By quantifying the error in approximating neural networks with Gaussian random fields, the results enable a deeper understanding of the training process and the convergence properties of the networks. The observed phase transition from Gaussian to non-Gaussian limits under varying step-size regimes highlights the complex dynamics of neural network training and the interplay between network architecture, weight distributions, and training algorithms. The quantitative bounds obtained in the study offer a systematic way to evaluate the approximation accuracy and convergence behavior of neural networks, paving the way for further research on the training dynamics and behavior of deep learning models.