Convex Calibration Dimension for Multiclass Loss Matrices

TL;DR

This paper introduces the convex calibration dimension to analyze the smallest prediction space size needed for convex surrogate losses to be calibrated with respect to general multiclass loss matrices, extending calibration theory.

Contribution

It extends classification calibration to general multiclass settings and introduces the convex calibration dimension, providing bounds and analysis for designing calibrated convex surrogates.

Findings

Derived necessary and sufficient conditions for calibration.

Established bounds on the convex calibration dimension.

Applied framework to subset ranking losses, showing existence and non-existence of convex calibrated surrogates.

Abstract

We study consistency properties of surrogate loss functions for general multiclass learning problems, defined by a general multiclass loss matrix. We extend the notion of classification calibration, which has been studied for binary and multiclass 0-1 classification problems (and for certain other specific learning problems), to the general multiclass setting, and derive necessary and sufficient conditions for a surrogate loss to be calibrated with respect to a loss matrix in this setting. We then introduce the notion of convex calibration dimension of a multiclass loss matrix, which measures the smallest `size' of a prediction space in which it is possible to design a convex surrogate that is calibrated with respect to the loss matrix. We derive both upper and lower bounds on this quantity, and use these results to analyze various loss matrices. In particular, we apply our framework to study various subset ranking losses, and use the convex calibration dimension as a tool to show both the existence and non-existence of various types of convex calibrated surrogates for these losses. Our results strengthen recent results of Duchi et al. (2010) and Calauzenes et al. (2012) on the non-existence of certain types of convex calibrated surrogates in subset ranking. We anticipate the convex calibration dimension may prove to be a useful tool in the study and design of surrogate losses for general multiclass learning problems.