mlsanityData to 7 Oct 2026

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

arXiv 1804.09753 ↗First posted 25 Apr 2018stat.ME stat.MLcited by 45

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Abstract (arXiv)

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 rightarrow kappa, we show that if the problem is sufficiently high dimensional in the sense that kappa > hMLE, then the MLE does not exist with probability one. Conversely, if kappa < hMLE, the MLE asymptotically exists with probability one.

Key facts. Cited by 45 papers in the map (data through 7 Oct 2026). First posted on arXiv 25 Apr 2018.

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Citing papers per year: 2021: 10; 2022: 9; 2023: 10; 2024: 9; 2025: 5; 2026: 2.

45 papers.

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  13. 2023Regularized Linear Regression for Binary Classificationcited by 7
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  28. 2026Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Designcited by 1
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  30. 2024Statistical Inference in Classification of High-dimensional Gaussian Mixturecited by 0
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  32. 2025Multi-Environment GLAMP: Approximate Message Passing for Transfer Learning with Applications to Lasso-based Estimatorscited by 0
  33. 2026Minimax Optimal Estimator and Improved Error Rate for the MLE in Logistic Regression with Gaussian Designcited by 0