Classification vs regression in overparameterized regimes: Does the loss function matter?

We compare classification and regression tasks in an overparameterized linear model with Gaussian features. On the one hand, we show that with sufficient overparameterization all training points are support vectors: solutions obtained by least-squares minimum-norm interpolation, typically used for regression, are identical to those produced by the hard margin support vector machine (SVM) that minimizes the hinge loss, typically used for training classifiers. On the other hand, we show that there exist regimes where these interpolating solutions generalize well when evaluated by the 0-1 test loss function, but do not generalize if evaluated by the square loss function, i.e. they approach the null risk. Our results demonstrate the very different roles and properties of loss functions used at the training phase (optimization) and the testing phase (generalization).

Automated summary

This study compares classification and regression tasks in an overparameterized linear model with Gaussian features. The findings show that under sufficient overparameterization, all training points are support vectors, and the solutions obtained by least-squares minimum-norm interpolation are identical to those produced by the hard margin support vector machine. Additionally, the study reveals the existence of regimes where interpolating solutions generalize well when evaluated by the 0-1 test loss function, but do not generalize when evaluated by the square loss function.