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Beyond Surrogates: A Quantitative Analysis for Inter-Metric Relationships

Yuanhao Pu, Defu Lian, Enhong Chen

TL;DR

A unified theoretical framework designed to quantify the relationships between metrics is proposed, which categorize metrics into different classes to facilitate a comparative analysis across different mathematical forms and interrogates these relationships through Bayes-Optimal Set and Regret Transfer.

Abstract

The Consistency property between surrogate losses and evaluation metrics has been extensively studied to ensure that minimizing a loss leads to metric optimality. However, the direct relationship between different evaluation metrics remains significantly underexplored. This theoretical gap results in the "Metric Mismatch" frequently observed in industrial applications, where gains in offline validation metrics fail to translate into online performance. To bridge this disconnection, this paper proposes a unified theoretical framework designed to quantify the relationships between metrics. We categorize metrics into different classes to facilitate a comparative analysis across different mathematical forms and interrogates these relationships through Bayes-Optimal Set and Regret Transfer. Through this framework, we provide a new perspective on identifying the structural asymmetry in regret transfer, enabling the design of evaluation systems that are theoretically guaranteed to align offline improvements with online objectives.

Beyond Surrogates: A Quantitative Analysis for Inter-Metric Relationships

TL;DR

A unified theoretical framework designed to quantify the relationships between metrics is proposed, which categorize metrics into different classes to facilitate a comparative analysis across different mathematical forms and interrogates these relationships through Bayes-Optimal Set and Regret Transfer.

Abstract

The Consistency property between surrogate losses and evaluation metrics has been extensively studied to ensure that minimizing a loss leads to metric optimality. However, the direct relationship between different evaluation metrics remains significantly underexplored. This theoretical gap results in the "Metric Mismatch" frequently observed in industrial applications, where gains in offline validation metrics fail to translate into online performance. To bridge this disconnection, this paper proposes a unified theoretical framework designed to quantify the relationships between metrics. We categorize metrics into different classes to facilitate a comparative analysis across different mathematical forms and interrogates these relationships through Bayes-Optimal Set and Regret Transfer. Through this framework, we provide a new perspective on identifying the structural asymmetry in regret transfer, enabling the design of evaluation systems that are theoretically guaranteed to align offline improvements with online objectives.
Paper Structure (34 sections, 13 theorems, 36 equations, 1 figure, 3 tables)

This paper contains 34 sections, 13 theorems, 36 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Consistency: From Metrics to Surrogate Optimization
  4. Analysis of Evaluation Metrics
  5. Preliminaries
  6. Surrogate Risk and Metric Risk
  7. Bayes-Optimality and Regret
  8. Regret Transfer
  9. Main Results
  10. Intra-group Cohesion
  11. Inter-group Bayes-Optimal Convergence and Hierarchy
  12. Inter-group Regret Transfer Results
  13. Pointwise $\leftrightarrow$ Pairwise/Listwise
  14. Asymmetry between Pairwise and Listwise Transfer
  15. Experiments
  16. ...and 19 more sections

Key Result

Corollary 3.1

A surrogate loss $\mathcal{L}$ is Bayes-consistent with a metric $\mathcal{M}$ if, for any sequence of predictors $\{f_k\}_{k=1}^{\infty}$, the vanishing of the surrogate regret implies the vanishing of the metric regret:

Figures (1)

  • Figure 1: Regret space visualization for Simulated Pointwise, Pairwise, and Listwise losses ($n=1,000$). Left: 3D regret landscape. Right: Cross-section of Acc vs. NDCG regret.

Theorems & Definitions (28)

  • Definition 3.1: Bayes-Optimal Predictor Set
  • Definition 3.2: Bayes-Optimal Inclusion and Equivalence
  • Definition 3.3: Metric Regret
  • Corollary 3.1: Bayes-Consistency
  • Definition 3.4: Regret Transfer Function
  • Theorem 4.1: Alignment and One-way Inclusion
  • Theorem 4.2: Truncation Monotonicity
  • Theorem 4.3: Bayes-Optimal Set Relations across Groups
  • Theorem 4.4: Pointwise Transfer Failure
  • proof
  • ...and 18 more