The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression

Abstract

This paper rigorously establishes that the existence of the maximum likelihood estimate (MLE) in high-dimensional logistic regression models with Gaussian covariates undergoes a sharp “phase transition.” We introduce an explicit boundary curve hMLE, parameterized by two scalars measuring the overall magnitude of the unknown sequence of regression coefficients, with the following property: in the limit of large sample sizes n and number of features p proportioned in such a way that p/n → κ, we show that if the problem is sufficiently high dimensional in the sense that κ > hMLE, then the MLE does not exist with probability one. Conversely, if κ < hMLE, the MLE asymptotically exists with probability one.