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

Sebastian Goldt, Bruno Loureiro, Galen Reeves, Florent Krzakala, Marc Mezard, Lenka Zdeborova
Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, PMLR 145:426-471, 2022.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v145-goldt22a, title = {The Gaussian equivalence of generative models for learning with shallow neural networks}, author = {Goldt, Sebastian and Loureiro, Bruno and Reeves, Galen and Krzakala, Florent and Mezard, Marc and Zdeborova, Lenka}, booktitle = {Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference}, pages = {426--471}, year = {2022}, editor = {Bruna, Joan and Hesthaven, Jan and Zdeborova, Lenka}, volume = {145}, series = {Proceedings of Machine Learning Research}, month = {16--19 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v145/goldt22a/goldt22a.pdf}, url = {https://proceedings.mlr.press/v145/goldt22a.html}, 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. } }
Endnote
%0 Conference Paper %T The Gaussian equivalence of generative models for learning with shallow neural networks %A Sebastian Goldt %A Bruno Loureiro %A Galen Reeves %A Florent Krzakala %A Marc Mezard %A Lenka Zdeborova %B Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference %C Proceedings of Machine Learning Research %D 2022 %E Joan Bruna %E Jan Hesthaven %E Lenka Zdeborova %F pmlr-v145-goldt22a %I PMLR %P 426--471 %U https://proceedings.mlr.press/v145/goldt22a.html %V 145 %X 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.
APA
Goldt, S., Loureiro, B., Reeves, G., Krzakala, F., Mezard, M. & Zdeborova, L.. (2022). The Gaussian equivalence of generative models for learning with shallow neural networks. Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 145:426-471 Available from https://proceedings.mlr.press/v145/goldt22a.html.

Related Material