Phase transitions for the existence of unregularized M-estimators in single index models

This paper studies phase transitions for the existence of unregularized M-estimators under proportional asymptotics where the sample size n and feature dimension p grow proportionally with n/p → δ ∈ (1, ∞). We study the existence of M-estimators in single-index models where the response y i depends on covariates x i ∼ N (0, I p ) through an unknown index w ∈ R p and an unknown link function. An explicit expression is derived for the critical threshold δ ∞ that determines the phase transition for the existence of the M-estimator, generalizing the results of Candès & Sur (2020) for binary logistic regression to other single-index models. Furthermore, we investigate the existence of a solution to the nonlinear system of equations governing the asymptotic behavior of the M-estimator when it exists. The existence of solution to this system for δ > δ ∞ remains largely unproven outside the global null in binary logistic regression. We address this gap with a proof that the system admits a solution if and only if δ > δ ∞ , providing a comprehensive theoretical foundation for proportional asymptotic results that require as a prerequisite the existence of a solution to the system.