Bridging Conformal Prediction and Scenario Optimization: Discarded Constraints and Modular Risk Allocation

TL;DR

This paper connects conformal prediction and scenario optimization from a control systems perspective, introducing a modular risk allocation method and demonstrating its effectiveness in multi-output prediction and control tasks.

Contribution

It extends the conformal/scenario bridge to feasible sample-and-discard algorithms and introduces a modular composition rule for joint risk guarantees.

Findings

Discarded samples act as admissible exceptions in the violation law.

The modular rule enables risk distribution across multiple outputs or constraints.

Numerical results show trade-offs between safety and performance in constraint tightening.

Abstract

Scenario optimization and conformal prediction share a common goal, that is, turning finite samples into safety margins. Yet, different terminology often obscures the connection between their respective guarantees. This paper revisits that connection directly from a systems-and-control viewpoint. Building on the recent conformal/scenario bridge of \citet{OSullivanRomaoMargellos2026}, we extend the forward direction to feasible sample-and-discard scenario algorithms. Specifically, if the final decision is determined by a stable subset of the retained sampled constraints, the classical mean violation law admits a direct exchangeability-based derivation. In this view, discarded samples naturally appear as admissible exceptions. We also introduce a simple modular composition rule that combines several blockwise calibration certificates into a single joint guarantee. This rule proves particularly useful in multi-output prediction and finite-horizon control, where engineers must distribute risk across coordinates, constraints, or prediction steps. Finally, we provide numerical illustrations using a calibrated multi-step tube around an identified predictor. These examples compare alternative stage-wise risk allocations and highlight the resulting performance and safety trade-offs in a standard constraint-tightening problem.