Phase Transitions, Optimal Errors and Optimality of Message-Passing in Generalized Linear Models
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We consider generalized linear models (GLMs) where an unknown n-dimensional signal vector is observed through the application of a random matrix and a non-linear (possibly probabilistic) componentwise output function. We consider the models in the high-dimensional limit, where the observation consists of m points, and m/n→α where α stays finite in the limit m,n→∞. This situation is ubiquitous in applications ranging from supervised machine learning to signal processing. A substantial amount of theoretical work analyzed the model-case when the observation matrix has i.i.d. elements and the components of the ground-truth signal are taken independently from some known distribution. While statistical physics provided number of explicit conjectures for special cases of this model, results existing for non-linear output functions were so far non-rigorous. At the same time GLMs with non-linear output functions are used as a basic building block of powerful multilayer feedforward neural networks. Therefore rigorously establishing the formulas conjectured for the mutual information is a key open problem that we solve in this paper. We also provide an explicit asymptotic formula for the optimal generalization error, and confirm the prediction of phase transitions in GLMs. Analyzing the resulting formulas for several non-linear output functions, including the rectified linear unit or modulus functions, we obtain quantitative descriptions of information-theoretic limitations of high-dimensional inference. Our proof technique relies on a new version of the interpolation method with an adaptive interpolation path and is of independent interest. Furthermore we show that a polynomial-time algorithm referred to as generalized approximate message-passing reaches the optimal generalization error for a large set of parameters.
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