Enhanced $H$-Consistency Bounds

TL;DR

This paper develops a flexible framework for deriving improved $H$-consistency bounds in surrogate loss analysis, leading to tighter finite-sample guarantees across various classification and ranking tasks.

Contribution

It relaxes previous assumptions to establish more general and favorable $H$-consistency bounds, encompassing existing results and enabling new bounds in multiple scenarios.

Findings

Derived more favorable $H$-consistency bounds for classification and ranking.

Unified framework subsuming existing bounds as special cases.

Applicable to scenarios like Tsybakov noise conditions and bipartite ranking.

Abstract

Recent research has introduced a key notion of $H$-consistency bounds for surrogate losses. These bounds offer finite-sample guarantees, quantifying the relationship between the zero-one estimation error (or other target loss) and the surrogate loss estimation error for a specific hypothesis set. However, previous bounds were derived under the condition that a lower bound of the surrogate loss conditional regret is given as a convex function of the target conditional regret, without non-constant factors depending on the predictor or input instance. Can we derive finer and more favorable $H$-consistency bounds? In this work, we relax this condition and present a general framework for establishing enhanced $H$-consistency bounds based on more general inequalities relating conditional regrets. Our theorems not only subsume existing results as special cases but also enable the derivation of more favorable bounds in various scenarios. These include standard multi-class classification, binary and multi-class classification under Tsybakov noise conditions, and bipartite ranking.