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 gradient descent algorithm, when applied on the logistic loss, converges to the max-margin classifier (a.k.a. hard-margin SVM). The performance of the max-margin classifier has been recently analyzed. 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 analysis of the performance of GMM via the solution of a system of nonlinear equations. We also provide a detailed study for three special cases: ($1$) $\ell_2$-GMM that is the max-margin classifier, ($2$) $\ell_1$-GMM which encourages sparsity, and ($3$) $\ell_{\infty}$-GMM which is often used when the parameter vector has binary entries. Our theoretical results are validated by extensive simulation results across a range of parameter values, problem instances, and model structures.
翻译：后勤模型通常用于二进制分类任务。这些模型的成功往往归因于它们与最大似值估测器的连接。已经显示,在对后勤损失应用时,梯度下降算法会与最大误差分类器(a.k.a. hard-margin SVM)相融合。最近对最大误差分类器的性能进行了分析。根据这些结果,我们在本文件中提出并研究一个更宽泛的设置,其中后勤模型的基本参数拥有某些结构(粗、块偏、低等),并引入了一个更一般性的框架(在对后勤损失应用时,梯度梯度下降算算算算算算法与最大误差分类器相融合(a.k.a.a.b.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a
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