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Scenario theory for multi-criteria data-driven decision making

Simone Garatti, Lucrezia Manieri, Alessandro Falsone, Algo Carè, Marco C. Campi, Maria Prandini

Abstract

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.

Scenario theory for multi-criteria data-driven decision making

Abstract

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.

Paper Structure

This paper contains 22 sections, 119 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation and key definitions
  3. Mathematical setup
  4. The risk certification problem
  5. Individual risks assessment
  6. Independent assessment of individual risks
  7. Collective assessment of individual risks
  8. Computational aspects
  9. Joint risk assessment
  10. Evaluation of the joint risk based on the independent assessment of individual risks
  11. Evaluation of the joint risk based on the collective assessment of individual risks
  12. A-priori joint risk assessment
  13. Scalability of the a-priori bounds as $m$ grows
  14. Conclusions
  15. Proof of Theorem \ref{['thm:region_bound']}
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Region $\mathcal{R}(\boldsymbol{k})$ (in green) obtained from Corollary \ref{['corol:region_bound']} for $m = 2$ appropriateness criteria, $\boldsymbol{N} = [800 \; 1200]^\top\space$, and different choices of $\{\lambda_{\boldsymbol{h}} \}_{\boldsymbol{h} = \boldsymbol{0}}^{\boldsymbol{H}}$ (columns) and $\boldsymbol{H} = 2\boldsymbol{N}$ (top row) or $\boldsymbol{H} = \boldsymbol{N}$ (bottom row). Each plot also shows the box region $\tilde{\mathcal{R}}(\boldsymbol{k})$ (in gray), obtained using the bound in \ref{['eq:individual-independent-bound']}. The maximum value of $|\boldsymbol{v}|$ when $\boldsymbol{v}$ belongs to each region is also reported in the legende within square brackets beside the corresponding entry.
  • Figure 2: Region $\mathcal{R}(\boldsymbol{k})$ obtained from Theorem \ref{['thm:region_bound']} for $m = 2$ appropriateness criteria, $\boldsymbol{N} = [800 \; 1200]^\top\space$, and different choices of $\{\lambda_{\boldsymbol{h}} \}_{\boldsymbol{h} = \boldsymbol{0}}^{\boldsymbol{H}}$ (columns), $\boldsymbol{H} = 2\boldsymbol{N}$ (top row) or $\boldsymbol{H} = \boldsymbol{N}$ (bottom row), and $\boldsymbol{k} = [199 \; 1]^\top\space$ (purple) or $\boldsymbol{k} = [100 \; 100]^\top\space$ (blue) or $\boldsymbol{k} = [1 \; 199]^\top\space$ (yellow). The maximum value of $|\boldsymbol{v}|$ when $\boldsymbol{v}$ belongs to each region is also reported in square brackets beside the corresponding legend entry.
  • Figure 3: Comparison between the different a-priori joint risk bounds discussed in Section \ref{['sec:scalability_apriori_bounds']}, as a function of $m$.