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 once 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 $p=O(N)$.
Asymptotics of Non-Convex Generalized Linear Models in High-Dimensions: A proof of the replica formula
Information Learning and Physics Laboratory, École Polytechnique Fédérale de Lausanne (EPFL)（信息学习与物理实验室，瑞士联邦理工学院（EPFL））；Statistical Physics of Computation Laboratory, École Polytechnique Fédérale de Lausanne (EPFL)（计算统计物理实验室，瑞士联邦理工学院（EPFL））；Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon (ENS)（数学纯与应用单位，里昂高等师范学院（ENS））
Analysis of overfitting in the regularized Cox model
Institute for Mathematical and Molecular Biomedicine, King’s College London（伦敦国王学院数学与分子生物医学研究所）；King’s College London（伦敦国王学院）；Saddle Point Science Ltd（鞍点科学有限公司）
Replica Field Theory for a Generalized Franz--Parisi Potential of Inhomogeneous Glassy Systems: New Closure and the Associated Self-Consistent Equation
