Constructing Multiclass Classifiers using Binary Classifiers Under Log-Loss

TL;DR

This paper analyzes methods for building multiclass classifiers from binary classifiers under log-loss, providing regret bounds and introducing a new leverage-hierarchical approach with improved performance demonstrated through simulations.

Contribution

It offers theoretical regret bounds for one-vs-all and hierarchical methods and introduces a leverage-hierarchical method with potential for lower log-loss.

Findings

Upper bound on multiclass regret for OVA method

Exact regret expression for hierarchical classification

Leverage-hierarchical method shows improved performance in simulations

Abstract

The construction of multiclass classifiers from binary elements is studied in this paper, and performance is quantified by the regret, defined with respect to the Bayes optimal log-loss. We discuss two known methods. The first is one vs. all (OVA), for which we prove that the multiclass regret is upper bounded by the sum of binary regrets of the constituent classifiers. The second is hierarchical classification, based on a binary tree. For this method we prove that the multiclass regret is exactly a weighted sum of constituent binary regrets where the weighing is determined by the tree structure. We also introduce a leverage-hierarchical classification method, which potentially yields smaller log-loss and regret. The advantages of these classification methods are demonstrated by simulation on both synthetic and real-life datasets.