Critical feature learning in deep neural networks

Kirsten Fischer, Javed Lindner, David Dahmen, Zohar Ringel, Michael Krämer, Moritz Helias
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Abstract

A key property of neural networks driving their success is their ability to learn features from data. Understanding feature learning from a theoretical viewpoint is an emerging field with many open questions. In this work we capture finite-width effects with a systematic theory of network kernels in deep non-linear neural networks. We show that the Bayesian prior of the network can be written in closed form as a superposition of Gaussian processes, whose kernels are distributed with a variance that depends inversely on the network width N . A large deviation approach, which is exact in the proportional limit for the number of data points $P = \alpha N \rightarrow \infty$, yields a pair of forward-backward equations for the maximum a posteriori kernels in all layers at once. We study their solutions perturbatively to demonstrate how the backward propagation across layers aligns kernels with the target. An alternative field-theoretic formulation shows that kernel adaptation of the Bayesian posterior at finite-width results from fluctuations in the prior: larger fluctuations correspond to a more flexible network prior and thus enable stronger adaptation to data. We thus find a bridge between the classical edge-of-chaos NNGP theory and feature learning, exposing an intricate interplay between criticality, response functions, and feature scale.
深度神经网络中的关键特征学习
神经网络取得成功的一个关键特性是其从数据中学习特征的能力。从理论角度理解特征学习是一个新兴领域,存在许多未决问题。在这项工作中,我们利用深入非线性神经网络的网络核的系统理论来捕捉有限宽度效应。我们证明,网络的贝叶斯先验可以以封闭形式写成高斯过程的叠加,其内核分布的方差与网络宽度 N 成反比。大偏差方法在数据点数量 $P = \alpha N \rightarrow \infty$ 的比例极限下是精确的,它可以一次性得到所有层中最大后验核的前向后向方程。我们对它们的解进行了扰动研究,以证明跨层的后向传播如何使核与目标保持一致。替代的场论表述表明,贝叶斯后验在有限宽度下的内核适应性来自于先验的波动:较大的波动对应于更灵活的网络先验,从而能够更强地适应数据。因此,我们在经典的边缘混沌 NNGP 理论和特征学习之间找到了一座桥梁,揭示了临界性、响应函数和特征尺度之间错综复杂的相互作用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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