Scenario Approach with Post-Design Certification of User-Specified Properties

Algo Carè; Marco C. Campi; Simone Garatti

stat.MEFeb 17, 2026

Scenario Approach with Post-Design Certification of User-Specified Properties

Authors: Algo Carè, Marco C. Campi, Simone Garatti

Organizations: Department of Information Engineering - University of Brescia, via Branze 38, 25123 Brescia, Italia · Dipartimento di Elettronica, Informazione e Bioingegneria - Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italia

Abstract

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.

Figures & tables

Panel of Figure 1 — Fig. 1: Feedback configuration for the pole placement problem.

Panel of Figure 2 — Fig. 2: A classic pendulum

Fig. 3: Locations in the complex plane of the closed-loop system poles corresponding to the $N=2000$ scenarios in problem ( 17 ). The dotted line is the border of the desired conic sector ${\cal S}_{-0.7,0.5}$ . The black dots in $-1\pm j$ and $-3\pm 3j$ are the poles of the reference polynomial.

Fig. 4:

\overline{F_{\epsilon}}(\ell-h^{\ast}_{N})

and

\underline{F_{\epsilon}}(\ell-h^{\ast}_{N})

(solid lines) vs. actual cumulative distribution function of

\|\eta(T)\|_{\infty}

(dashed lines) for

\rho=1

(top) and

\rho=0.05

(bottom).

Generalization Bounds

Generalization bounds quantify how training performance can differ from population or test performance as sample size and model complexity change.

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Scenario theory for multi-criteria data-driven decision making

Simone Garatti, Lucrezia Manieri, Alessandro Falsone, Algo Carè+2

Dipartimento di Elettronica Informazione e Ingegneria, Politecnico di Milano, Milano, Italy · Dipartimento di Ingegneria dell’Informazione, Universit`a degli studi di Brescia, Brescia, Italy

Decision-Making under Uncertainty Multi-Criteria Decision Analysis Robust Optimization

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The scenario approach provides a powerful data-driven framework for designing solutions under uncertainty with rigorous probabilistic robustness guarantees. Existing theory, however, primarily addresses assessing robustness with respect to a single appropriateness criterion for the solution based on a dataset, whereas many practical applications - including multi-agent decision problems - require the simultaneous consideration of multiple criteria and the assessment of their robustness based on multiple datasets, one per criterion. This paper develops a general scenario theory for multi-criteria data-driven decision making. A central innovation lies in the collective treatment of the risks associated with violations of individual criteria, which yields substantially more accurate robustness certificates than those derived from a naive application of standard results. In turn, this approach enables a sharper quantification of the robustness level with which all criteria are simultaneously satisfied. The proposed framework applies broadly to multi-criteria data-driven decision problems, providing a principled, scalable, and theoretically grounded methodology for design under uncertainty.

Abstract

Generalization Bounds

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