The Performance Analysis of Generalized Margin Maximizers on Separable Data

Logistic models are commonly used for binary classification tasks. The success of such models has often been attributed to their connection to maximum-likelihood estimators. It has been shown that SGD algorithms , when applied on the logistic loss, converge to the max-margin classifier (a.k.a. hard-margin SVM). The performance of hard-margin SVM has been recently analyzed in~\cite{montanari2019generalization, deng2019model}. Inspired by these results, in this paper, we present and study a more general setting, where the underlying parameters of the logistic model possess certain structures (sparse, block-sparse, low-rank, etc.) and introduce a more general framework (which is referred to as “Generalized Margin Maximizer”, GMM). While classical max-margin classifiers minimize the 2-norm of the parameter vector subject to linearly separating the data, GMM minimizes any arbitrary convex function of the parameter vector. We provide a precise performance analysis of the generalization error of such methods and show improvement over the max-margin method which does not take into account the structure of the model. In future work we show that mirror descent algorithms, with a properly tuned step size, can be exploited to achieve GMM classifiers. Our theoretical results are validated by extensive simulation results across a range of parameter values, problem instances, and model structures.