TY - JOUR

T1 - Replica analysis of overfitting in generalized linear regression models

AU - Coolen, A. C.C.

AU - Sheikh, M.

AU - Mozeika, A.

AU - Aguirre-Lopez, F.

AU - Antenucci, F.

PY - 2020/9/11

Y1 - 2020/9/11

N2 - Nearly all statistical inference methods were developed for the regime where the number N of data samples is much larger than the data dimension p. Inference protocols such as maximum likelihood (ML) or maximum a posteriori probability (MAP) are unreliable if p = O(N), due to overfitting. This limitation has for many disciplines with increasingly high-dimensional data become a serious bottleneck. We recently showed that in Cox regression for time-to-event data the overfitting errors are not just noise but take mostly the form of a bias, and how with the replica method from statistical physics one can model and predict this bias and the noise statistics. Here we extend our approach to arbitrary generalized linear regression models (GLM), with possibly correlated covariates. We analyse overfitting in ML/MAP inference without having to specify data types or regression models, relying only on the GLM form, and derive generic order parameter equations for the case of L2 priors. Second, we derive the probabilistic relationship between true and inferred regression coefficients in GLMs, and show that, for the relevant hyperparameter scaling and correlated covariates, the L2 regularization causes a predictable direction change of the coefficient vector. Our results, illustrated by application to linear, logistic, and Cox regression, enable one to correct ML and MAP inferences in GLMs systematically for overfitting bias, and thus extend their applicability into the hitherto forbidden regime p = O(N).

AB - Nearly all statistical inference methods were developed for the regime where the number N of data samples is much larger than the data dimension p. Inference protocols such as maximum likelihood (ML) or maximum a posteriori probability (MAP) are unreliable if p = O(N), due to overfitting. This limitation has for many disciplines with increasingly high-dimensional data become a serious bottleneck. We recently showed that in Cox regression for time-to-event data the overfitting errors are not just noise but take mostly the form of a bias, and how with the replica method from statistical physics one can model and predict this bias and the noise statistics. Here we extend our approach to arbitrary generalized linear regression models (GLM), with possibly correlated covariates. We analyse overfitting in ML/MAP inference without having to specify data types or regression models, relying only on the GLM form, and derive generic order parameter equations for the case of L2 priors. Second, we derive the probabilistic relationship between true and inferred regression coefficients in GLMs, and show that, for the relevant hyperparameter scaling and correlated covariates, the L2 regularization causes a predictable direction change of the coefficient vector. Our results, illustrated by application to linear, logistic, and Cox regression, enable one to correct ML and MAP inferences in GLMs systematically for overfitting bias, and thus extend their applicability into the hitherto forbidden regime p = O(N).

KW - generalized linear models

KW - overfitting

KW - regression

KW - replica method

UR - http://www.scopus.com/inward/record.url?scp=85094181537&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aba028

DO - 10.1088/1751-8121/aba028

M3 - Article

AN - SCOPUS:85094181537

SN - 1751-8113

VL - 53

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 36

M1 - 365001

ER -