A Sharp Phase Transition for MLE Existence in Multinomial Logistic Regression

This paper studies when multinomial logistic regression admits a finite maximum-likelihood estimator. Under a Gaussian feature model with softmax class probabilities and a fixed number of classes, we establish a sharp phase transition for MLE existence. As the sample size and feature dimension grow proportionally, the probability that the MLE is finite converges to zero or one depending on whether the sample-to-dimension ratio lies below or above a critical threshold. We characterize this threshold through a matrix Gaussian-distance optimization whose dimension depends only on the number of classes, rather than on the feature dimension. The result allows for general feature covariance and signal strength, and applies to models fitted both with and without an intercept. Although fitting an intercept adds only finitely many parameters, it can strictly increase the asymptotic existence threshold, even when the true intercept is zero. Our characterization recovers the binary logistic threshold and yields explicit thresholds when labels are uniform and independent of the features. Simulations support the predicted thresholds across signal strengths and true intercepts, and show that estimating an intercept can lead to a higher threshold.