A Model of Double Descent for High-dimensional Binary Linear Classification

TL;DR

This paper models high-dimensional binary classification using logistic regression with gradient descent, revealing a phase transition in error behavior and demonstrating a double descent phenomenon in classification error as the overparameterization ratio varies.

Contribution

It provides a theoretical analysis of the implicit bias of gradient descent in high-dimensional logistic regression, identifying a phase transition and characterizing the double descent phenomenon.

Findings

Classification error exhibits a phase transition at a critical overparameterization ratio.

Double descent phenomenon observed in classification error as model complexity increases.

Theoretical predictions validated by numerical experiments.

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 (GD) on logistic loss. For this model, we investigate the dependence of the classification error on the overparameterization ratio $\kappa=p/n$. First, building on known deterministic results on the implicit bias of GD, we uncover a phase-transition phenomenon for the case of Gaussian features: the classification 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 the SVM 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.