A model of double descent for high-dimensional binary linear classification

Abstract 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 on logistic loss. For this model, we investigate the dependence of the classification error on the ratio $\kappa =p/n$. First, building on known deterministic results on the implicit bias of gradient descent, we uncover a phase-transition phenomenon for the case of Gaussian features: the classification error of the gradient descent solution is the same as that of the maximum-likelihood solution when $\kappa <\kappa _\star $, and that of the support vector machine when $\kappa>\kappa _\star $, where $\kappa _\star $ is a phase-transition threshold. Next, using the convex Gaussian min–max theorem, we sharply characterize the performance of both the maximum-likelihood and the support vector machine solutions. Combining these results, we obtain curves that explicitly characterize the classification error for varying values of $\kappa $. The numerical results validate the theoretical predictions and unveil double-descent phenomena that complement similar recent findings in linear regression settings as well as empirical observations in more complex learning scenarios.

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.