Gaussian Universality in Neural Network Dynamics with Generalized Structured Input Distributions

Abstract: Bridging the gap between the practical performance of deep learning and its theoretical foundations often involves analyzing neural networks through stochastic gradient descent (SGD). Expanding on previous research that focused on modeling structured inputs under a simple Gaussian setting, we analyze the behavior of a deep learning system trained on inputs modeled as Gaussian mixtures to better simulate more general structured inputs. Through empirical analysis and theoretical investigation, we demonstrate that under certain standardization schemes, the deep learning model converges toward Gaussian setting behavior, even when the input data follow more complex or real-world distributions. This finding exhibits a form of universality in which diverse structured distributions yield results consistent with Gaussian assumptions, which can support the theoretical understanding of deep learning models.