Beyond Modern Asymptotics for Log-Likelihood Ratios in Logistic Regression

What the math gives to ML

The paper supplies sharp, design-uniform finite-sample control of the excess log-likelihood achieved by fitting a binary logistic model, including adversarial fixed designs and parameters near separation. The transferable asset is the explicit complexity penalty d log(e n/d) plus log(1/delta), together with uniformity over the unknown target and no Hessian regularity assumption. A practical neural-network use is to freeze a learned representation, treat its binary classification head as logistic regression, and construct a likelihood-based confidence region for that head. This enables statistically justified abstention and calibration under ill-conditioned features, where asymptotic Fisher-information ellipsoids can fail.