One-Bit Quantization and Sparsification for Multiclass Linear Classification With Strong Regularization

We study the use of linear regression for multiclass classification in the over-parametrized regime where some of the training data is mislabeled. In such scenarios it is necessary to add an explicit regularization term, $\lambda f(\cdot)$, for some convex function $f(\cdot)$, to avoid overfitting the mislabeled data. In our analysis, we assume that the data is sampled from a Gaussian Mixture Model with equal class sizes, and that a proportion of the training labels is corrupted for each class. Under these assumptions, we prove that the best classification performance is achieved when $f(\cdot)=\|\cdot\|^{2}_{2}$ and $\lambda\to\infty$. We then proceed to analyze the classification errors for $f(\cdot)=\|\cdot\|_{1}$ and $f(\cdot)=\|\cdot\|_{\infty}$ in the large $\lambda$ regime and notice that it is often possible to find sparse and one-bit solutions, respectively, that perform almost as well as the one corresponding to $f(\cdot)=\|\cdot\|_{2}^{2}$.