Proportional asymptotics of piecewise exponential proportional hazards models

TL;DR

This paper rigorously analyzes the high-dimensional behavior of piecewise exponential proportional hazards models with ridge regularization, establishing convergence results and insights into regularization effects.

Contribution

It provides the first rigorous proof of the asymptotic behavior of high-dimensional piecewise exponential models using the Convex Gaussian Min-Max theorem.

Findings

Convergence of the ridge penalized log-likelihood to a saddle point.

Impact of ridge regularization on parameter estimates and prediction error.

First rigorous validation of heuristic replica method results for Cox models.

Abstract

We study the flexible piecewise exponential model in a high dimensional setting where the number of covariates $p$ grows proportionally to the number of observations $n$ and under the hypothesis of random uncorrelated Gaussian designs. We prove rigorously that the optimal ridge penalized log-likelihood of the model converges in probability to the saddle point of a surrogate objective function. The technique of proof is the Convex Gaussian Min-Max theorem of Thrampoulidis, Oymak and Hassibi. An important consequence of this result, is that we can study the impact of the ridge regularization on the estimates of the parameter of the model and the prediction error as a function of the ratio $p/n > 0$. Furthermore, these results represent a first step toward rigorously proving the (conjectured) correctness of several results obtained with the heuristic replica method for the Cox semi-parametric model.