One-Bit Quantization and Sparsification for Multiclass Linear Classification via Regularized Regression

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, λ f(w), for some convex function f(·), 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 c of the training labels is corrupted for each class. Under these assumptions, we prove that the best classification performance is achieved when f(·) = \|·\|²₂ and λ → ∞. We then proceed to analyze the classification errors for f(·) = \|·\|₁ and f(·) = \|·\|_∞ in the large λ 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(·) = \|·\|₂².