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

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.• We provide a set of equations characterizing the asymptotic statistical properties of the estimator defined by problemThis article has been accepted for publication in IEEE Transactions on Information Theory.

Learning curves of generic features maps for realistic datasets with a teacher-student model*

“…For ridge regression, there are also precise predictions thanks to random matrix theory [12,[36][37][38][39][40][41]. A related set of results was obtained in [42] for orthogonal random matrix models. The main technical novelty of our proof is the handling of a generic loss and regularisation, not only ridge, representing convex empirical risk minimization, for both classification and regression, with the generic correlation structure of the model (1.1).…”

Perfect reconstruction of sparse signals with piecewise continuous nonconvex penalties and nonconvexity control

“…entries should be discussed for practical usage. The rotationally invariant matrix is one of the candidates to examine the effectiveness of the nonconvexity control for the general matrix [26][27][28].…”

Theoretical characterization of uncertainty in high-dimensional linear classification

“…Exact asymptotics for Bayesian estimation in generalized linear models was rigorously established in [11]. On the empirical risk minimization side, exact asymptotics based on different techniques, such as Convex Gaussian min-max theorem [5,16,19,36,38,51,52,62], Random Matrix Theory [43], GAMP [23,39] and first order expansions [14] have been used to study high-dimensional logistic regression and max-margin estimation.…”