Convergence-rate analysis for classifiers is often conducted under either Tsybakov margin or Massart margin. The former is a relatively weak condition that typically yields polynomial rates, while the latter is substantially stronger but can guarantee exponential rates. In this paper, we introduce a new condition, called Boltzmann margin, that bridges the gap between these two regimes. It is weaker than Massart margin, generally stronger than Tsybakov margin, and can imply many of their properties under suitable conditions. We apply Boltzmann margin to the analysis of kNN classifiers and establish the first near-exponential convergence rates for kNN classification. We also present extensions of the main results and provide numerical evidence supporting the main theoretical implications.

翻译：分类器的收敛速率分析通常在Tsybakov间隔或Massart间隔条件下进行。前者是相对较弱的条件，通常产生多项式速率，而后者则显著更强，但能保证指数速率。本文提出了一种称为玻尔兹曼间隔的新条件，填补了这两种机制之间的空白。该条件弱于Massart间隔，通常强于Tsybakov间隔，并在适当条件下可蕴含两者的许多性质。我们将玻尔兹曼间隔应用于kNN分类器的分析，首次建立了kNN分类的近指数收敛速率。此外，我们还给出了主要结果的扩展，并提供了支持主要理论结论的数值证据。