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 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. Non-rigorous 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 non-learnable regions. We believe that this random version of GLMs can serve as a challenging benchmark for multi-purpose algorithms. This paper is divided in two parts that can be read independently: The first part (main part) presents the model and main results, discusses some applications and sketches the main ideas of the proof. The second part (supplementary informations) is much more detailed and provides more examples as well as all the proofs.
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