核分类的误差缩放律：在源条件与容量条件下
Error Scaling Laws for Kernel Classification under Source and Capacity Conditions
AI总结：
针对满足源与容量条件的数据集，在高斯设计下推导了SVM与岭分类的误分类误差缩放律，并验证其能准确描述真实数据的学习曲线。
摘要：
我们考虑核分类问题。虽然对于某些分类器，已知预测误差随样本数量衰减速率的最坏情况界，但这些界往往无法准确描述真实数据集的学習曲线。在本工作中，我们考虑满足标准源条件和容量条件的重要数据集类别，并通过数值实验表明该类别包含多个真实数据集。在高斯设计下，我们推导了误分类（预测）误差随源系数和容量系数的衰减速率。我们针对两种标准的核分类设置，即最大化间隔的支持向量机（SVM）和岭分类，进行推导，并对比这两种方法。我们发现，我们的速率能够紧密描述此类数据集的学習曲线，并且在真实数据上也能观察到。我们的结果还可视为对核分类缩放律指数的显式预测，该缩放律在某些真实数据集上是准确的。
英文摘要：
We consider the problem of kernel classification. While worst-case bounds on the decay rate of the prediction error with the number of samples are known for some classifiers, they often fail to accurately describe the learning curves of real data sets. In this work, we consider the important class of data sets satisfying the standard source and capacity conditions, comprising a number of real data sets as we show numerically. Under the Gaussian design, we derive the decay rates for the misclassification (prediction) error as a function of the source and capacity coefficients. We do so for two standard kernel classification settings, namely margin-maximizing Support Vector Machines (SVM) and ridge classification, and contrast the two methods. We find that our rates tightly describe the learning curves for this class of data sets, and are also observed on real data. Our results can also be seen as an explicit prediction of the exponents of a scaling law for kernel classification that is accurate on some real datasets.
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