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.

• The paper generalizes the scenario approach to rigorously quantify both individual and joint risks in multi-criteria settings using data-driven complexity measures.
• It introduces collective risk certification methods that sharply reduce conservatism and scale logarithmically with an increasing number of criteria.
• It provides a-priori risk bounds and efficient computational schemes applicable to robust multi-agent optimization, federated learning, and control under uncertainty.

Scenario Theory for Multi-Criteria Data-Driven Decision Making

Introduction

The paper "Scenario theory for multi-criteria data-driven decision making" (2604.00553) advances the theory and practice of data-driven decision-making under uncertainty in multi-criteria settings. It generalizes the scenario approach, originally focused on single-criterion robustness analysis, to rigorously address the simultaneous satisfaction of multiple appropriateness criteria, each associated with its own dataset. This generalization is critical in modern applications such as multi-agent systems, control under uncertainty, and federated learning, where decentralized data and diverse objectives demand precise quantification of both individual and collective risks.

Problem Formulation and Scenario-Based Decision Scheme

The authors formalize a scenario-based framework where a decision variable $z \in \mathcal{Z}$ is to be chosen under uncertainty $\delta \sim P$, with $m$ appropriateness criteria $\{Z_i(\delta)\}_{i=1}^m$ and independent datasets $\{D_i\}_{i=1}^m$, each containing $N_i$ i.i.d. realizations of uncertainty relevant to criterion $i$. A central abstraction is the decision mapping $M$ which, given the labeled datasets, outputs a chosen design $z_N$. The class of admissible $M$ is characterized by permutation invariance and two consistency conditions (stability and responsiveness), ensuring that the decision's response to added data matches the scenario approach philosophy.

The framework subsumes robust multi-agent optimization, federated learning, and constrained control problems, encompassing both convex and nonconvex formulations. For instance, robust scenario optimization, which seeks $\delta \sim P$0 such that $\delta \sim P$1 and minimizes a cost $\delta \sim P$2, is a canonical instance of this mapping.

Individual and Joint Risk Quantification

Individual risk with respect to criterion $\delta \sim P$3 is defined as $\delta \sim P$4—the probability that $\delta \sim P$5 fails criterion $\delta \sim P$6 under new uncertainty. Joint risk is $\delta \sim P$7, i.e., the probability of violating at least one criterion.

The primary technical challenge is to estimate, using only the observed datasets, the risks associated with the scenario-based decision $\delta \sim P$8, both individually and jointly, with finite-sample probabilistic guarantees.

Multi-Criteria Complexity and Support Scenarios

A critical concept is the extension of support scenarios and scenario complexity to the multi-criteria context. A scenario is a support scenario if its removal changes the decision outcome. The multi-indexed vector $\delta \sim P$9 counts the number of support scenarios per criterion, encapsulating the combinatorial complexity of the decision.

This observable is central because risk bounds are parameterized by the observed complexity rather than simply the data size, resulting in tighter, data-dependent certificates.

Theoretical Contributions

Independent vs. Collective Certification

The paper precisely characterizes individual risk bounds in two ways:

• Independent: Applying single-criterion risk bounds to each criterion separately, as in prior extensions (e.g., [falsone_ScenariobasedApproachMultiagent_2020]), yields a "box" region in risk space, but is provably conservative due to ignoring coupling among risks.
• Collective (Main Contribution): The authors develop a novel dual formulation using generalized moment approaches and the observed complexity vector, allowing a collective characterization of multi-criteria risks. This results in risk regions $m$0 that exclude certain "worst-case" risk configurations that are in fact impossible given the joint data and consistency-induced coupling of the scenario mapping.

The dual constraints and region construction allow significant flexibility (allocation strategies for dual variables) and admit efficient computational schemes in certain parameter regimes (notably, under diagonal allocations).

Numerical analyses demonstrate sharply reduced conservatism of the collective risk certificates, especially for the joint (union) risk. For instance, in benchmark cases, the improvement can be over an order of magnitude (see Table 1 in the original text).

Joint Risk Bounds and Scalability

The sub-additivity property $m$1 enables straightforward, but loose, joint risk upper bounds. By leveraging the geometry of the collective regions $m$2, the paper constructs data-dependent, scenario-complexity-tight joint risk upper bounds. These are shown to be scalable in the number of criteria: as $m$3 increases, the collective certificates grow only logarithmically (not linearly), and under certain conditions, the joint risk bound converges to a constant as $m$4.

A-priori Certificates

Recognizing the practical necessity of dataset sizing, the authors provide a-priori risk bounds (i.e., before data is collected) given an upper bound $m$5 on total scenario complexity. These allow prediction of required sample sizes $m$6 per criterion to achieve a target joint risk bound at a prescribed confidence, using (almost explicit) formulas for scenario-rich regimes.

Computational Aspects

While the region-based collective analysis is more complex than the product-box independent approach, the diagonal allocation scheme yields almost explicit hyperbolic region descriptions. The authors provide code (and closed formulas for key cases) for computing upper bounds via root-finding on single-variable (in sample-homogeneous cases) or multi-variable problems.

Implications and Future Directions

Practical implications include:

• More efficient and less conservative scenario certification for distributed and multi-agent systems, allowing smaller datasets or improved out-of-sample risk profiles.
• Scalable risk guarantees for federated or large-scale systems with hundreds of criteria.
• Direct design rules for "scenario-based" control and optimization under multiple risk tolerances.

Theoretical implications center on the deeper understanding of dependence/coupling induced by multi-criteria scenario mappings, extending the role of support complexity as the fundamental observable for information-theoretic scenario quantification. The approach provides a unifying bridge between scenario optimization, statistical learning theory, and robust multi-agent control.

Future work points to the extension of these results to degenerate mappings and nonconvex designs without the non-degeneracy assumption, as started in [garatti_NonconvexScenarioOptimization_2025], and further algorithmic improvements for region computation in the high-dimensional limit.

Conclusion

This work presents a significant advancement in scenario theory for multi-criteria and multi-agent settings by introducing collective risk certification methods based on data-driven complexity observables. The resulting certificates dramatically reduce the conservatism of prior independent approaches, exhibit favorable scalability with the number of criteria, and are readily applicable to practical sizing and certification problems in large-scale data-driven systems. This framework sets the stage for reliable, efficient, and theoretically sound robust design in decentralized and multi-objective decision-making under uncertainty.