Equivalence between conformal prediction and a posteriori scenario optimization methods

Establish how to construct an equivalence, in terms of PAC guarantees, between conformal prediction with a priori parameter selection and scenario optimization methods that rely on a posteriori evaluation—including nonconvex scenario optimization, post-design verification in the scenario approach, and scenario optimization with constraint relaxation—for convexly parameterized data-driven reachable set estimation.

Background

The paper develops formal connections among PAC guarantees for three approaches in data-driven reachability analysis: conformal prediction, scenario optimization with sample discarding, and the holdout method. It proves that empirical conformal coverage yields PAC bounds equivalent to the holdout method and shows that split conformal prediction with training-conditional coverage can be parameterized to match scenario optimization with sample discarding, producing structurally parallel PAC guarantees and even identical reachable sets under specific parameter choices.

However, the authors note that several widely used scenario optimization variants rely on a posteriori evaluation (e.g., nonconvex scenario optimization, post-design verification, and constraint relaxation). Because conformal prediction parameters are chosen a priori, the paper states that it is unclear how to construct an equivalence between conformal prediction and these a posteriori scenario optimization methods, leaving open the task of deriving a formal equivalence of PAC guarantees in that setting.