Near-Exponential Convergence Rates for kNN Classification based on Boltzmann Margin

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 Figure 1: Example data densities on [0,1] that satisfy different margins respectively. Bayes decision boundary is 0.5.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 evidencenecessarily strong for many problems. Can there be a more supporting the main theoretical implications.