Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models

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Jean Barbier, Florent Krzakala, Nicolas Macris, Léo Miolane, Lenka Zdeborová ;
Proceedings of the 31st Conference On Learning Theory, PMLR 75:728-731, 2018.

Abstract

Generalized linear models (GLMs) arise 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 benchmarks models in neural networks. % We evaluate the mutual information (or “free entropy”) from which we deduce the Bayes-optimal inference and generalization errors. Our analysis applies to the high-dimensional limit where both the number of samples and dimensions are large and their ratio is fixed. % Non-rigorous predictions for the optimal inference and generalization 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 non-learnable regions.

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