Optimal errors and phase transitions in high-dimensional generalized linear models

Generalized linear models (GLMs) are used in high-dimensional machine learning, statistics, communications, and signal processing. In this paper we analyze GLMs when the data matrix is random, as relevant in problems such as compressed sensing, error-correcting codes, or benchmark models in neural n...

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Veröffentlicht in:Proceedings of the National Academy of Sciences - PNAS Jg. 116; H. 12; S. 5451
Hauptverfasser: Barbier, Jean, Krzakala, Florent, Macris, Nicolas, Miolane, Léo, Zdeborová, Lenka
Format: Journal Article
Sprache:Englisch
Veröffentlicht: United States 19.03.2019
Schlagworte:
generalized linear model
high-dimensional inference
approximate message-passing algorithm
perceptron
Bayesian inference
ISSN:1091-6490, 1091-6490
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Zusammenfassung:Generalized linear models (GLMs) are used in high-dimensional machine learning, statistics, communications, and signal processing. In this paper we analyze GLMs when the data matrix is random, as relevant in problems such as compressed sensing, error-correcting codes, or benchmark models in neural networks. We evaluate the mutual information (or "free entropy") from which we deduce the Bayes-optimal estimation and generalization errors. Our analysis applies to the high-dimensional limit where both the number of samples and the dimension are large and their ratio is fixed. Nonrigorous predictions for the optimal errors existed for special cases of GLMs, e.g., for the perceptron, in the field of statistical physics based on the so-called replica method. Our present paper rigorously establishes those decades-old conjectures and brings forward their algorithmic interpretation in terms of performance of the generalized approximate message-passing algorithm. Furthermore, we tightly characterize, for many learning problems, regions of parameters for which this algorithm achieves the optimal performance and locate the associated sharp phase transitions separating learnable and nonlearnable regions. We believe that this random version of GLMs can serve as a challenging benchmark for multipurpose algorithms.
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ISSN:1091-6490
1091-6490
DOI:10.1073/pnas.1802705116