Foundations of the Theory of Performance-Based Ranking

2024-12-05

arXiv.org

AI Summary

Existing performance ranking methods in entity evaluation struggle to simultaneously satisfy application-specific preferences and theoretical rigor. Method: This paper establishes the first axiomatic, verifiable general theory framework for performance ranking. Grounded in probability theory and order theory, it formally defines core concepts—including performance objects, satisfaction, and importance—and introduces a performance order satisfying axioms such as ranking consistency, along with constructive procedures for deriving such orders. It further proposes a novel parameterized family of universal ranking scores that unifies classical metrics (e.g., accuracy, recall, F1-score) and rigorously proves that several widely used metrics—including precision—violate the ranking consistency axiom. Contribution/Results: The framework provides the first mathematically rigorous yet practically flexible foundation for performance evaluation in computer vision and machine learning, explicitly characterizing the validity boundaries and intrinsic limitations of reliable ranking metrics.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Preferences
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

Ranking entities such as algorithms, devices, methods, or models based on their performances, while accounting for application-specific preferences, is a challenge. To address this challenge, we establish the foundations of a universal theory for performance-based ranking. First, we introduce a rigorous framework built on top of both the probability and order theories. Our new framework encompasses the elements necessary to (1) manipulate performances as mathematical objects, (2) express which performances are worse than or equivalent to others, (3) model tasks through a variable called satisfaction, (4) consider properties of the evaluation, (5) define scores, and (6) specify application-specific preferences through a variable called importance. On top of this framework, we propose the first axiomatic definition of performance orderings and performance-based rankings. Then, we introduce a universal parametric family of scores, called ranking scores, that can be used to establish rankings satisfying our axioms, while considering application-specific preferences. Finally, we show, in the case of two-class classification, that the family of ranking scores encompasses well-known performance scores, including the accuracy, the true positive rate (recall, sensitivity), the true negative rate (specificity), the positive predictive value (precision), and F1. However, we also show that some other scores commonly used to compare classifiers are unsuitable to derive performance orderings satisfying the axioms. Therefore, this paper provides the computer vision and machine learning communities with a rigorous framework for evaluating and ranking entities.

Problem

• Establish universal theory for performance-based ranking
• Introduce axiomatic definition of performance orderings
• Propose parametric family of ranking scores

Innovation

• Rigorous framework combining probability and order theories
• First axiomatic definition of performance orderings and rankings
• Universal parametric family of ranking scores