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

Giuseppe C. Calafiore

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.

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

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.
Paper Structure (12 sections, 5 theorems, 58 equations, 2 figures, 2 tables)

This paper contains 12 sections, 5 theorems, 58 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and problem statement
  3. Conformal prediction, calibration, and conditional risk
  4. Scenario optimization with discarded constraints
  5. A bridge lemma: reconstruction with admissible exceptions
  6. Forward bridge for discarded scenarios
  7. A modular rule for multi-output and multi-stage risk allocation
  8. Sharper bounds under coordinate independence
  9. Numerical illustrations
  10. Risk allocation across the prediction horizon
  11. From calibrated tubes to constraint tightening
  12. Conclusions

Key Result

Lemma 3

Let $Z_1,\dots,Z_{m+1}$ be exchangeable random elements of a measurable space $\mathcal{Z}$. For each $n\ge 1$, let be measurable, permutation-equivariant random index sets, interpreted respectively as a reconstruction set and an exception set. Let $\mathcal{R}$ be a measurable map that associates with each finite tuple of points in $\mathcal{Z}$ a measurable subset of $\mathcal{Z}$. Assume that,

Figures (2)

  • Figure 1: Effect of modular allocation across the horizon. Left: mean tube half-widths $q_k$. Right: mean empirical stage-wise risks $V_k(S)$. The three allocations use the same total risk budget and the same total rank sum, but redistribute conservatism across prediction steps.
  • Figure 2: Representative tightened tube for the increasing-risk allocation in the simple planning problem. The band is computed from one calibration set, the solid line is the nominal identified trajectory under the selected constant input $u^\star$, the gray lines are true trajectories of the nonlinear plant, and the dashed line is the output limit.

Theorems & Definitions (19)

  • Remark 1
  • Lemma 3: Exchangeable reconstruction with admissible exceptions
  • proof
  • Remark 4
  • Theorem 5: Forward bridge with discarded constraints
  • proof
  • Remark 6: Interpretation for control design
  • Example 7: One-dimensional order-statistic predictor
  • Proposition 8: Modular calibration rule
  • proof
  • ...and 9 more