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

TL;DR

This paper rigorously derives an analytical formula for the asymptotic reconstruction performance of convex generalized linear models with rotationally-invariant data matrices, confirming a physics-inspired conjecture and extending previous results beyond Gaussian matrices.

Contribution

It provides a rigorous proof of the replica formula for convex generalized linear models with rotationally-invariant matrices, using message passing algorithms and stability analysis.

Findings

Analytical formula matches numerical simulations for logistic regression and SVM.

Convergence of 2-MLVAMP established for strongly convex problems.

Extension to non-strongly convex problems via analytical continuation.

Abstract

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.