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

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.

Improved Compression Bounds for Scenario Decision Making

Scenario decision making offers a flexible way of making decision in an uncertain environment while obtaining probabilistic guarantees on the risk of failure of the decision. The idea of this approach is to draw samples of the uncertainty and make a decision based on the samples, called "scenarios". The probabilistic guarantees take the form of a bound on the probability of sampling a set of scenarios that will lead to a decision whose risk of failure is above a given maximum tolerance. This bound can be expressed as a function of the number of sampled scenarios, the maximum tolerated risk, and some intrinsic property of the problem called the "compression size". Several such bounds have been proposed in the literature under various assumptions on the problem. We propose new bounds that improve upon the existing ones without requiring stronger assumptions on the problem.