Enhanced $H$-Consistency Bounds
This work provides incremental improvements in theoretical guarantees for machine learning algorithms, benefiting researchers in statistical learning theory by offering tighter bounds for surrogate loss analysis.
The paper tackles the problem of deriving more favorable finite-sample guarantees for surrogate losses in classification and ranking by relaxing a key condition in existing H-consistency bounds, resulting in enhanced bounds that subsume previous results and apply to scenarios like multi-class classification and Tsybakov noise conditions.
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.