A Model of Double Descent for High-dimensional Binary Linear Classification
This provides theoretical insights into overparameterization effects in binary classification, complementing empirical findings in machine learning.
The authors studied logistic regression with gradient descent on Gaussian features, showing that classification error transitions from matching maximum-likelihood to max-margin solutions at a critical overparameterization ratio, and they theoretically characterized this error to reveal double-descent phenomena.
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 $κ=p/n$. 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 $κ<κ_\star$, and that of the max-margin (SVM) solution when $κ>κ_\star$. 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 $κ$. 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.