Improved Classification Rates under Refined Margin Conditions

TL;DR

This paper introduces a partitioning technique to improve the statistical analysis of classification algorithms by separately analyzing regions near and far from the decision boundary, leading to better error bounds under certain margin conditions.

Contribution

It proposes a simple partitioning method to refine error analysis, achieving improved rates for basic classifiers under specific margin assumptions.

Findings

Refined error bounds for histogram rule and other classifiers.

Improved rates compared to SVMs and plug-in classifiers under margin conditions.

Margin conditions that relate noise to the decision boundary enhance optimal rates.

Abstract

In this paper we present a simple partitioning based technique to refine the statistical analysis of classification algorithms. The core of our idea is to divide the input space into two parts such that the first part contains a suitable vicinity around the decision boundary, while the second part is sufficiently far away from the decision boundary. Using a set of margin conditions we are then able to control the classification error on both parts separately. By balancing out these two error terms we obtain a refined error analysis in a final step. We apply this general idea to the histogram rule and show that even for this simple method we obtain, under certain assumptions, better rates than the ones known for support vector machines, for certain plug-in classifiers, and for a recently analyzed tree based adaptive-partitioning ansatz. Moreover, we show that a margin condition which sets the critical noise in relation to the decision boundary makes it possible to improve the optimal rates proven for distributions without this margin condition.