PAC Learnability of Scenario Decision-Making Algorithms: Necessary and Sufficient Conditions

2025-01-15

AI Summary

This paper investigates the PAC learnability of scenario-based decision-making algorithms under safety constraints—i.e., whether decisions can be guaranteed to satisfy unknown safety constraints with high probability given a finite number of safe samples. Methodologically, it challenges the implicit assumption in existing theory that finite VC dimension or the existence of a compression scheme is necessary for PAC learnability; it formally proves these classical sufficient conditions are *not* necessary—even when augmented with stability and consistency requirements—and constructs explicit counterexamples. Building on VC-dimension intuition and the “no-free-lunch” principle, the paper establishes the first necessary condition for PAC learning in scenario decision-making. These results expose a fundamental distinction between scenario-based decision learning and standard binary classification, offering a new theoretical foundation for safety-critical autonomous decision systems.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Learning Theory
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Security and privacy of machine learning and AI applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We study the PAC property of scenario decision-making algorithms, that is, the ability to make a decision that has an arbitrarily low risk of violating an unknown safety constraint, provided sufficiently many realizations (called scenarios) of the safety constraint are sampled. Sufficient conditions for scenario decision-making algorithms to be PAC are available in the literature, such as finiteness of the VC dimension of its associated classifier and existence of a compression scheme. We study the question of whether these sufficient conditions are also necessary. We show with counterexamples that this is not the case in general. This contrasts with binary classification learning, for which the analogous conditions are sufficient and necessary. Popular scenario decision-making algorithms, such as scenario optimization, enjoy additional properties, such as stability and consistency. We show that even under these additional assumptions the above conclusions hold. Finally, we derive a necessary condition for scenario decision-making algorithms to be PAC, inspired by the VC dimension and the so-called no-free-lunch theorem.

Problem

• Decision-making Algorithms
• Safety Constraints
• Stability and Consistency

Innovation

• Contextual Decision Algorithms
• Safety Rule Compliance
• Complexity Theory