A Model Of Double Descent For High-Dimensional Logistic Regression

We consider a model for logistic regression where only a subset of features of size $p$ is used for training a linear classifier over $n$ training samples. The classifier is obtained by running gradient-descent (GD) on logistic-loss. For this model, we investigate the dependence of the test error on the over-parameterization ratio $\kappa=p/n$. First, building on known deterministic results on convergence properties of the GD, we uncover a phase-transition phenomenon for the case of Gaussian features: the test error of GD is the same as that of the maximum-likelihood (ML) solution when $\kappa<\kappa_\star$, and that of the max-margin (SVM) solution when $\kappa>\kappa_\star$. Next, using the convex Gaussian min-max theorem (CGMT), we sharply characterize the performance of both the ML and SVM solutions. Combining these results, we obtain curves that explicitly characterize the Test error of GD for varying values of $\kappa$. The numerical results validate the theoretical predictions and unveil ``double-descent" phenomena that complement similar recent observations in linear regression settings.