Large-margin classifiers are popular methods for classification. We derive the asymptotic expression for the generalization error of a family of large-margin classifiers in the limit of both sample size $n$ and dimension $p$ going to $\infty$ with fixed ratio $\alpha=n/p$. This family covers a broad range of commonly used classifiers including support vector machine, distance weighted discrimination, and penalized logistic regression. Our result can be used to establish the phase transition boundary for the separability of two classes. We assume that the data are generated from a single multivariate Gaussian distribution with arbitrary covariance structure. We explore two special choices for the covariance matrix: spiked population model and two layer neural networks with random first layer weights. The method we used for deriving the closed-form expression is from statistical physics known as the replica method. Our asymptotic results match simulations already when $n,p$ are of the order of a few hundreds. For two layer neural networks, we reproduce the recently developed `double descent' phenomenology for several classification models. We also discuss some statistical insights that can be drawn from these analysis.

中文翻译：

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使用基于边际的分类方法大规模分析学习中的泛化误差

大边界分类器是流行的分类方法。我们在样本大小 $n$ 和维度 $p$ 以固定比率 $\alpha=n/p$ 达到 $\infty$ 的限制下，推导出了一系列大边界分类器的泛化误差的渐近表达式。该系列涵盖了广泛的常用分类器，包括支持向量机、距离加权歧视和惩罚逻辑回归。我们的结果可用于建立两个类的可分离性的相变边界。我们假设数据是从具有任意协方差结构的单个多元高斯分布生成的。我们探索协方差矩阵的两个特殊选择：尖峰种群模型和具有随机第一层权重的两层神经网络。我们用于推导闭式表达式的方法来自称为复制方法的统计物理学。当 $n,p$ 是几百个数量级时，我们的渐近结果已经与模拟相匹配。对于两层神经网络，我们为几个分类模型重现了最近开发的“双下降”现象。我们还讨论了可以从这些分析中得出的一些统计见解。