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

We investigate the Probably Approximately Correct (PAC) property of scenario decision algorithms, which refers to their ability to produce decisions with an arbitrarily low risk of violating unknown safety constraints, provided a sufficient number of realizations of these constraints are sampled. While several PAC sufficient conditions for such algorithms exist in the literature -- such as the finiteness of the VC dimension of their associated classifiers, or the existence of a compression scheme -- it remains unclear whether these conditions are also necessary. In this work, we demonstrate through counterexamples that these conditions are not necessary in general. These findings stand in contrast to binary classification learning, where analogous conditions are both sufficient and necessary for a family of classifiers to be PAC. Furthermore, we extend our analysis to stable scenario decision algorithms, a broad class that includes practical methods like scenario optimization. Even under this additional assumption, we show that the aforementioned conditions remain unnecessary. Furthermore, we introduce a novel quantity, called the dVC dimension, which serves as an analogue to the VC dimension for scenario decision algorithms. We prove that the finiteness of this dimension is a PAC necessary condition for scenario decision algorithms. This allows to (i) guide algorithm users and designers to recognize algorithms that are not PAC, and (ii) contribute to a comprehensive characterization of PAC scenario decision algorithms.