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

Giuseppe C. Calafiore

eess.SYMar 19, 2026

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

Authors: Giuseppe C. Calafiore

Organizations: Department of Electronics and Telecommunications Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy

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.

Figures & tables

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.

allocation	$(r_{1},\ldots,r_{4})$	$(\epsilon_{1},\ldots,\epsilon_{4})$	certificate $1-\sum_{k}\delta_{k}$	mean $V_{\mathrm{traj}}$	$Q_{0.90}(V_{\mathrm{traj}})$	$Q_{0.99}(V_{\mathrm{traj}})$
increasing risk	$(0,1,2,3)$	$(0.04,0.05,0.06,0.07)$	0.9264	0.0618	0.0932	0.1288
uniform risk	$(1,1,2,2)$	$(0.055,0.055,0.055,0.055)$	0.9095	0.0613	0.0930	0.1270
decreasing risk	$(3,2,1,0)$	$(0.07,0.06,0.05,0.04)$	0.9264	0.0626	0.0940	0.1268

Table 1: Alternative horizon-wise allocations for the calibrated four-step tube. All three choices use

\sum_{k}\epsilon_{k}=0.22

and

\sum_{k}r_{k}=6

allocation	mean $u^{\star}$	$Q_{0.10}(u^{\star})$	$Q_{0.90}(u^{\star})$	mean viol. prob.	$Q_{0.90}(\text{viol. prob.})$	mean terminal output
increasing risk	0.3491	0.2992	0.3925	0.0174	0.0353	0.4190
uniform risk	0.3331	0.2801	0.3778	0.0123	0.0260	0.4004
decreasing risk	0.2373	0.1219	0.3244	0.0025	0.0068	0.2923

Table 2: Simple planning problem with calibrated constraint tightening. The controller chooses the largest constant input

u^{\star}\in[0,1]

such that

\hat{y}_{k}(y_{0},u^{\star})\leq y_{\max}-q_{k}(S)

for

k=1,\dots,4

, with

y_{0}=0.1

and

y_{\max}=0.7

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.

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Conformal Predictive Programming for Chance Constrained Optimization

Yiqi Zhao, Xinyi Yu, Matteo Sesia, Jyotirmoy V. Deshmukh+1

Thomas Lord Department of Computer Science, University of Southern California · Department of Data Sciences and Operations, University of Southern California · Automatic Control Laboratory, ETH Zürich

Stochastic Optimization Chance-Constrained Optimization Conformal Prediction Constrained Optimization

Open paper AskPDF arXiv

We propose conformal predictive programming (CPP), a framework to solve chance constrained optimization problems, i.e., optimization problems with constraints that are functions of random variables. CPP utilizes samples from these random variables along with the quantile lemma - central to conformal prediction - to transform the chance constrained optimization problem into a deterministic problem with a quantile reformulation. CPP's main strength is an independent calibration step that provides a posteriori guarantees for the solution of this problem that are of conditional and marginal nature otherwise. These guarantees even apply in settings when assumptions required for obtaining standard a priori guarantees (e.g., in scenario optimization or sample average approximation) are unavailable, difficult to compute, or conservative. Another strength of CPP is that it can easily support different variants of conformal prediction which have been (or will be) proposed within the conformal prediction community. To illustrate this, we present robust CPP to deal with distribution shifts in the random variables and Mondrian CPP to deal with class conditional chance constraints. In a series of case studies, we show the validity of the aforementioned approaches, and illustrate the advantage of CPP as compared to scenario approach.

Abstract

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