Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (Or: How to Prove Kabashima’s Replica Formula)

Cedric Gerbelot; Alia Abbara; Florent Krzakala

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Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (Or: How to Prove Kabashima’s Replica Formula)

机译：Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (Or: How to Prove Kabashima’s Replica Formula)

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There has been a recent surge of interest in the study of asymptotic reconstruction performance in various cases of generalized linear estimation problems in the teacher-student setting, especially for the case of i.i.d standard normal matrices. Here, we go beyond these matrices, and prove an analytical formula for the reconstruction performance of convex generalized linear models with rotationally-invariant data matrices with arbitrary bounded spectrum, rigorously confirming, under suitable assumptions, a conjecture originally derived using the replica method from statistical physics. The proof is achieved by leveraging on message passing algorithms and the statistical properties of their iterates, allowing to characterize the asymptotic empirical distribution of the estimator. For sufficiently strongly convex problems, we show that the two-layer vector approximate message passing algorithm (2-MLVAMP) converges, where the convergence analysis is done by checking the stability of an equivalent dynamical system, which gives the result for such problems. We then show that, under a concentration assumption, an analytical continuation may be carried out to extend the result to convex (non-strongly) problems. We illustrate our claim with numerical examples on mainstream learning methods such as sparse logistic regression and linear support vector classifiers, showing excellent agreement between moderate size simulation and the asymptotic prediction.

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《IEEE Transactions on Information Theory》 |2023年第3期|1824-1852|共29页
作者
Cedric Gerbelot; Alia Abbara; Florent Krzakala;
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作者单位

Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France|Courant Institute of Mathematical Sciences, NYU, New York City, NY, USA;

Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France|Laboratory of Computational Biology and Theoretical Biophysics, EPFL, Lausanne, Switzerland;

IdePhics Laboratory, EPFL, Lausanne, Switzerland;

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正文语种英语
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Heuristic algorithms; Approximation algorithms; Estimation; Convergence; Signal processing algorithms; Mathematical models; Machine learning;

Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (Or: How to Prove Kabashima’s Replica Formula)

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