The Performance Analysis of Generalized Margin Maximizers on Separable Data

Fariborz Salehi · Ehsan Abbasi · Babak Hassibi

Keywords: [ Convex Optimization ] [ Learning Theory ] [ Statistical Learning Theory ]

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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.

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