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Enhanced $H$-Consistency Bounds

Anqi Mao, Mehryar Mohri, Yutao Zhong

TL;DR

The paper introduces a flexible framework for ${\mathscr H}$-consistency bounds by incorporating hypothesis- and instance-dependent factors $\alpha$, $\beta$, and $\gamma$, enabling finer, non-asymptotic guarantees that go beyond classical convexity-based bounds. It develops two fundamental tools for both convex and concave bound regimes and applies them to standard multiclass classification, low-noise binary and multiclass regimes, and bipartite ranking, deriving tighter bounds for constrained losses, exponential and logistic surrogates, and even revealing negative results for hinge losses. Notably, under Tsybakov and Massart noise conditions, the bounds exhibit faster, near-linear rates rather than the universal square-root rate, highlighting the distributional dependence of learning guarantees. The authors further connect these bounds to generalization by providing Rademacher-based guarantees in the constrained, low-noise, and ranking settings, and discuss practical implications such as the AdaBoost-RankBoost relationship and calibration properties. Overall, the work offers a broad, distribution-aware toolkit for deriving sharper, non-asymptotic learning guarantees across classification and ranking tasks with diverse surrogate losses.

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.

Enhanced $H$-Consistency Bounds

TL;DR

The paper introduces a flexible framework for -consistency bounds by incorporating hypothesis- and instance-dependent factors , , and , enabling finer, non-asymptotic guarantees that go beyond classical convexity-based bounds. It develops two fundamental tools for both convex and concave bound regimes and applies them to standard multiclass classification, low-noise binary and multiclass regimes, and bipartite ranking, deriving tighter bounds for constrained losses, exponential and logistic surrogates, and even revealing negative results for hinge losses. Notably, under Tsybakov and Massart noise conditions, the bounds exhibit faster, near-linear rates rather than the universal square-root rate, highlighting the distributional dependence of learning guarantees. The authors further connect these bounds to generalization by providing Rademacher-based guarantees in the constrained, low-noise, and ranking settings, and discuss practical implications such as the AdaBoost-RankBoost relationship and calibration properties. Overall, the work offers a broad, distribution-aware toolkit for deriving sharper, non-asymptotic learning guarantees across classification and ranking tasks with diverse surrogate losses.

Abstract

Recent research has introduced a key notion of -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 -consistency bounds? In this work, we relax this condition and present a general framework for establishing enhanced -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.
Paper Structure (31 sections, 35 theorems, 115 equations, 3 tables)

This paper contains 31 sections, 35 theorems, 115 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. New fundamental tools for ${\mathscr H}$-consistency bounds
  4. Standard multi-class classification
  5. Classification under low-noise conditions
  6. Binary classification
  7. Multi-class classification
  8. Bipartite ranking
  9. Fundamental tools for bipartite ranking
  10. Exponential loss
  11. Logistic loss
  12. Hinge loss
  13. Discussion
  14. Conclusion
  15. Related work
  16. ...and 16 more sections

Key Result

Theorem 1

Assume that there exist a convex function $\Psi\colon \mathbb{R}_+ \to \mathbb{R}$ and two positive functions $\alpha\colon {\mathscr H} \times {\mathscr X} \to \mathbb{R}^*_+$ and $\beta\colon {\mathscr H} \times {\mathscr X} \to \mathbb{R}^*_+$ with $\sup_{x \in {\mathscr X}} \alpha(h, x) < +\inft with $\gamma(h) = [*]{\sup_{x \in {\mathscr X}} \alpha(h, x) \beta(h, x)}$. If, additionally, ${\ma

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Theorem 3
  • Theorem 4: Enhanced $\sH$-consistency bounds for constrained losses
  • Theorem 5
  • Lemma 5
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • ...and 25 more