🤖 AI Summary
Existing H-consistency bounds rely heavily on strong convexity assumptions, limiting their applicability and yielding loose guarantees. Method: We propose a generalized conditional regret inequality framework that—without requiring the surrogate loss lower bound to be convex—derives tighter H-consistency bounds under broader, non-convex settings with predictor- and instance-dependent conditions. By precisely modeling finite-sample relationships between surrogate and target losses (e.g., 0–1 loss) and integrating functional inequalities with statistical learning theory, we obtain unified, improved bounds. Contribution/Results: Our framework encompasses standard multiclass classification, binary/multiclass classification under Tsybakov noise, and bipartite ranking. It substantially enhances both the tightness and generality of theoretical guarantees, overcoming key limitations of prior work while extending H-consistency analysis to previously intractable non-convex and heterogeneous regimes.
📝 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.