Classification is often the first problem described in introductory machine learning classes. Generalization guarantees of classification have historically been offered by Vapnik-Chervonenkis theory. Yet those guarantees are based on intractable algorithms, which has led to the theory of surrogate methods in classification. Guarantees offered by surrogate methods are based on calibration inequalities, which have been shown to be highly sub-optimal under some margin conditions, failing short to capture exponential convergence phenomena. Those "super" fast rates are becoming to be well understood for smooth surrogates, but the picture remains blurry for non-smooth losses such as the hinge loss, associated with the renowned support vector machines. In this paper, we present a simple mechanism to obtain fast convergence rates and we investigate its usage for SVM. In particular, we show that SVM can exhibit exponential convergence rates even without assuming the hard Tsybakov margin condition.

翻译：分类通常是机器学习入门课程中首先描述的问题。历史上，分类的泛化保证由Vapnik-Chervonenkis理论提供。然而，这些保证基于难以实现的算法，这导致了分类中替代方法理论的发展。替代方法提供的保证基于校准不等式，已证明这些不等式在某些边际条件下高度次优，无法捕捉指数收敛现象。这些"超"快速收敛速率对于光滑替代损失函数已逐渐被理解，但对于与著名支持向量机相关联的非光滑损失（如合页损失）而言，其图像仍不清晰。在本文中，我们提出了一种获得快速收敛速率的简单机制，并研究了其在支持向量机中的应用。特别地，我们表明，即使不假设严格的Tsybakov边际条件，支持向量机也能表现出指数收敛速率。