The scenario approach is an established data-driven design framework that comes equipped with a powerful theory linking design complexity to generalization properties. In this approach, data are simultaneously used both for design and for certifying the design's reliability, without resorting to a separate test dataset. This paper takes a step further by guaranteeing additional properties, useful in post-design usage but not considered during the design phase. To this end, we introduce a two-level framework of appropriateness: baseline appropriateness, which guides the design process, and post-design appropriateness, which serves as a criterion for a posteriori evaluation. We provide distribution-free upper bounds on the risk of failing to meet the post-design appropriateness; these bounds are computable without using any additional test data. Under additional assumptions, lower bounds are also derived. As part of an effort to demonstrate the usefulness of the proposed methodology, the paper presents two practical examples in H2 and pole-placement problems. Moreover, a method is provided to infer comprehensive distributional knowledge of relevant performance indexes from the available dataset.
T HE scenario approach is a well-established methodol- ogy in systems and control for data-driven design with probabilistic guarantees. The distinctive feature of the scenario approach is that data-driven designs are certified without the need for separate test datasets.
Since its introduction in the mid-2000s, [2], [3], the scenario approach has evolved into a versatile framework supporting robust optimization, [4]- [6], optimization with constraint relaxation, [7], [8], risk-averse formulations using Conditional A preliminary version of this work appeared as a conference paper in [1]. The present paper has been substantially revised and extended. In particular, besides extensive rewriting and several additions introduced throughout the manuscript, all the material in Sections III and V, as well as the numerical example in Section IV-B, is entirely new. The authors gratefully acknowledge Dr. Kristina Frizyuk for her assistance with the simulations. Paper supported by the PRIN 2022 project 2022RRNAEX “The Scenario Approach for Control and Non-Convex Design” (CUP: D53D23001440006), funded by the NextGeneration EU program (Mission 4, Component 2, Investment 1.1), by the PRIN PNRR project P2022NB77E “A data-driven cooperative framework for the management of distributed energy and water resources” (CUP: D53D23016100001), funded by the NextGeneration EU program (Mission 4, Component 2, Investment 1.1), and by the FAIR (Future Artificial Intelligence Research) project, funded by the NextGenerationEU program within the PNRR-PE-AI scheme (M4C2, Investment 1.3).
A. Carè and M.C. Campi are with the Department of Information Engineering -University of Brescia, via Branze 38, 25123 Brescia, Italia (e-mail: [algo.care,marco.campi]@unibs.it).
S. Garatti is with the Dipartimento di Elettronica, Informazione e Bioingegneria -Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italia (e-mail: simone.garatti@polimi.it).
Value at Risk (CVaR), [9], [10], [8], as well as broader decision-making strategies, [7], [8], [11]; the reader is referred to the book [12] and the review article [13] for a general background.
A large body of research, [14]- [36], has contributed to the theoretical development of the scenario approach, addressing the following central question: given a scenario design based on a finite sample of observations, what is the probability that it will generalize and remain appropriate for previously unseen situations? For the scope of the present paper, two contributions are particularly relevant. The first is paper [7], where the framework of consistent decision-making was introduced, formalizing the abstract notion of appropriateness and providing tight upper and lower bounds on the probability of inappropriateness under non-degeneracy assumptions. These bounds were further studied in an asymptotic setting in [11]. The second contribution is [8], which extended the framework by removing the non-degeneracy assumption and proved that the upper bounds remain valid in this more general setup.
This paper further advances the scenario theory by considering settings in which a design is carried out with respect to a baseline appropriateness criterion, while additional properties of interest are assessed after the design has been determined. We term these additional properties post-design appropriateness conditions. Within this extended framework, we establish upper bounds on the probability of violating postdesign appropriateness, and, under additional assumptions, derive complementary lower bounds, thereby enclosing the post-design risk within certified intervals. Importantly, all bounds are computable without resorting to any additional data besides those used during the design. These guarantees provide informative and practically relevant assessments of additional or stricter performance requirements. Typical situations where these new results can be applied include:
(i) the quantification of the probability of achieving enhanced performances beyond a guaranteed baseline. For example, the assessment of the probability that a controller meets a stricter performance requirement than those enforced at design time; (ii) certain design goals lead to mathematical problems that are difficult to handle. For instance, their formalization may involve non-convex or computationally complex optimization procedures. In such cases, one often resorts to surrogate or heuristic design criteria to obtain a solution, and the theory presented here can then be used to verify whether the original goals of interest are met by the solution obtained using the simplified scheme.
In the second part of this paper, we present two examples illustrating points (i) and (ii). Furthermore, when a cost quantifies the quality of a design in relation to an uncertain environment, we show that a suitable post-design criterion enables the assessment of the full cost distribution, thus providing a comprehensive evaluation of the scena
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