Approximation Power of Multi-Layer Random Features in Neural Networks

核心概念

Multi-layer neural networks with randomly initialized weights can approximate functions that belong to the reproducing kernel Hilbert space (RKHS) defined by the Neural Network Gaussian Process (NNGP) kernel. The number of neurons required to achieve a certain approximation error is determined by the RKHS norm of the target function.

摘要

The paper investigates the approximation power of multi-layer feedforward neural networks (NNs) with randomly initialized weights. It shows that in the infinite width limit, such NNs behave like a Gaussian Random Field whose covariance function is the Neural Network Gaussian Process (NNGP) kernel.
The key insights are:

The RKHS defined by the NNGP kernel contains only functions that can be well approximated by the NN architecture.

To achieve a certain approximation error, the required number of neurons in each layer is defined by the RKHS norm of the target function.

The approximation can be constructed from a supervised dataset by a random multi-layer representation of the input vector, together with training of the last layer's weights.

For a 2-layer NN and a domain equal to an (n-1)-dimensional sphere in Rn, the paper compares the number of neurons required by Barron's theorem and the multi-layer features construction. It shows that if the eigenvalues of the integral operator of the NNGP decay slower than k^(-n-2/3), then the multi-layer features construction guarantees a more succinct neural network approximation than Barron's theorem.

Computational experiments verify the theoretical findings and show that realistic neural networks can easily learn target functions even when the theorems do not provide any guarantees.

統計資料

There are no key metrics or important figures used to support the author's key logics.

引述

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從以下內容提煉的關鍵洞見

Multi-layer random features and the approximation power of neural networks

by Rustem Takha... 於 arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17461.pdf

深入探究

What are the implications of the findings for the design and training of practical neural network architectures

The findings have significant implications for the design and training of practical neural network architectures. The research demonstrates that functions with moderate norms in the Reproducing Kernel Hilbert Space (RKHS) defined by the Neural Network Gaussian Process (NNGP) kernel can be effectively approximated by corresponding neural networks. This suggests that when designing neural network architectures, particularly for tasks where the target functions fall within this category, utilizing architectures that align with the properties of the NNGP kernel can lead to efficient approximation.
In practical terms, this means that for functions that can be well approximated by the architecture, the required number of neurons in each layer can be determined by the RKHS norm of the target function. This insight can guide the selection of network architectures, activation functions, and training strategies to optimize performance and efficiency in neural network applications. By leveraging the theoretical guarantees provided by the multi-layer features construction, practitioners can design neural networks that are tailored to effectively approximate specific types of functions, leading to improved performance and generalization capabilities.

How do the theoretical guarantees provided by the multi-layer features construction compare to other recent advances in the approximation theory of neural networks

The theoretical guarantees provided by the multi-layer features construction offer a unique perspective on the approximation theory of neural networks, particularly in the context of the NNGP kernel. By establishing a connection between the RKHS defined by the NNGP and the approximation capabilities of neural network architectures, the research sheds light on the relationship between function complexity and network design.
Comparing these guarantees to other recent advances in approximation theory, such as Barron's theorem and the Neural Tangent Kernel (NTK) theory, reveals the distinct advantages and insights offered by the multi-layer features construction. While Barron's theorem provides bounds on the approximation capabilities of neural networks based on the Barron norm, the multi-layer features construction offers a different perspective based on the RKHS norm of the target function. This allows for a more nuanced understanding of function approximation in neural networks, particularly in the context of random features and multi-layer representations.

Can the insights from this work be extended to other types of neural network architectures beyond the fully-connected feedforward networks considered here

The insights from this work can be extended to other types of neural network architectures beyond fully-connected feedforward networks. While the research primarily focuses on the approximation power of multi-layer feedforward neural networks in the context of the NNGP kernel, the fundamental principles and theoretical frameworks can be applied to a broader range of neural network architectures.
For example, the concept of leveraging the RKHS norm of target functions to determine the required number of neurons in each layer can be adapted to different network architectures, such as convolutional neural networks (CNNs) or recurrent neural networks (RNNs). By considering the properties of the RKHS defined by the relevant kernel functions for these architectures, practitioners can tailor the design and training of neural networks to effectively approximate specific types of functions in various domains.
Overall, the foundational insights and theoretical guarantees provided by the multi-layer features construction can serve as a valuable framework for understanding and optimizing the approximation capabilities of diverse neural network architectures, paving the way for more efficient and effective neural network design and training strategies.