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

ICML 2025 · Learning theory and algorithms › Sample complexity · approach: Minimax estimation rates · 1 citations (OpenAlex)

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Shows that the nonlinear system characterizing unregularized M-estimators in single-index models with Gaussian covariates has a unique solution if and only if the proportional-asymptotic threshold delta_infty is exceeded, generalizing the phase transition for existence of the M-estimator beyond binary logistic regression.

Problem

Existence phase transition of unregularized M-estimators under proportional asymptotics (n/p -> delta) for single-index models with Gaussian covariates, and whether the nonlinear system governing the asymptotics has a solution beyond the global null.

Approach

Derives an explicit threshold delta_infty from conic geometry and the Gaussian kinematic formula for general losses (logistic, binomial, Poisson), and links existence of the M-estimator and of the characterizing nonlinear system to an infinite-dimensional convex optimization problem in a Hilbert space, proving the system has a solution iff delta > delta_infty.

Key results

• Phase transition threshold delta_infty derived for general single-index losses and shown to predict existence of the M-estimator in Poisson and binomial simulations (n=1000, 20 repetitions).
• The nonlinear system has a unique solution if and only if delta > delta_infty, proven beyond the global null for the first time.
• Simulations for binomial regression with q in {1,2,4,8,16} match the predicted threshold 1/delta_infty.

Measured effects reported in the paper

Limitations

• Results are asymptotic and restricted to Gaussian covariates with identity covariance.
• Assumes strictly convex, C^1 losses; non-differentiable losses require a different threshold (delta_perfect).

Experimental setup

Metrics: fraction of datasets where MLE exists

Builds on: Convex Gaussian min-max theorem Gaussian kinematic formula Candes and Sur phase transition for logistic regression

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