Convex Calibration Dimension for Multiclass Loss Matrices

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.

Accepted to JMLR, pending editing

References in corpus (4)

• Composite Binary Losses
• Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
• Surrogate Regret Bounds for Bipartite Ranking via Strongly Proper Losses
• Consistent Algorithms for Multiclass Classification with a Reject Option

Cited by in corpus (10)

• On the Consistency of Ordinal Regression Methods
• A Generalization Error Bound for Multi-class Domain Generalization
• Multiclass Classification Calibration Functions
• Fairness with Overlapping Groups
• Calibrated Surrogate Losses for Adversarially Robust Classification
• Unifying Lower Bounds on Prediction Dimension of Consistent Convex Surrogates
• Consistent Robust Adversarial Prediction for General Multiclass Classification
• Quantifying Learning Guarantees for Convex but Inconsistent Surrogates
• Training Over-parameterized Models with Non-decomposable Objectives
• Weston-Watkins Hinge Loss and Ordered Partitions