The Gaussian equivalence of generative models for learning with shallow neural networks

TL;DR

This paper demonstrates that the performance metrics of neural networks trained on data from pre-trained generative models can be accurately modeled using Gaussian equivalents, enabling theoretical analysis of realistic data scenarios.

Contribution

It introduces a Gaussian equivalence framework for analyzing neural networks trained on generative model data, extending theoretical understanding beyond simple input distributions.

Findings

Gaussian equivalence holds under certain conditions for generative models

Derived equations describe generalization performance of neural networks and kernel methods

Experimental results confirm the applicability of the theory to deep generative models

Abstract

Understanding the impact of data structure on the computational tractability of learning is a key challenge for the theory of neural networks. Many theoretical works do not explicitly model training data, or assume that inputs are drawn component-wise independently from some simple probability distribution. Here, we go beyond this simple paradigm by studying the performance of neural networks trained on data drawn from pre-trained generative models. This is possible due to a Gaussian equivalence stating that the key metrics of interest, such as the training and test errors, can be fully captured by an appropriately chosen Gaussian model. We provide three strands of rigorous, analytical and numerical evidence corroborating this equivalence. First, we establish rigorous conditions for the Gaussian equivalence to hold in the case of single-layer generative models, as well as deterministic rates for convergence in distribution. Second, we leverage this equivalence to derive a closed set of equations describing the generalisation performance of two widely studied machine learning problems: two-layer neural networks trained using one-pass stochastic gradient descent, and full-batch pre-learned features or kernel methods. Finally, we perform experiments demonstrating how our theory applies to deep, pre-trained generative models. These results open a viable path to the theoretical study of machine learning models with realistic data.