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Conformal-DRO: Distributionally Robust Optimization with Conformalized Ambiguity Set

Luhao Zhang, Shixiang Zhu

arXiv:2609.11073v1math.OCstat.ML

TL;DR

When each instance has an unobserved outcome law and contributes one observation, uncertainty about the next law persists beyond knowledge of the mixture. Conformal-DRO uses nested conformal evidence to construct a finite-sample-valid ambiguity set, with a robust value that certifies the selected decision’s expected cost.

  • Problem

    With one observation per latent law, the mixture conditional law does not determine the next instance’s law, leaving latent-law heterogeneity beyond ordinary fixed-law estimation error.

    Source: §1 · p. 1

  • Method

    Conformal-DRO inverts observable conformal validity into constraints on the future latent law, using the conformal path to set geometry and α to calibrate the radius.

    Source: §1 · p. 2

  • Results

    All Conformal-DRO variants exceed the nominal law-coverage target; increasing K from 1 to 8 reduces regret from 1.616 to 1.319, an 18.4% reduction, while maintaining law coverage near 0.95.

    Source: §1 · p. 2, §5.2 · p. 8

  • Takeaways & Limitations

    The shell reformulation makes latent-law uncertainty operationally tractable, and the resulting robust value is a finite-sample certificate for the selected decision’s expected cost.

    Source: §6 · p. 8

    The aggregation rule must be fixed independently of the final calibration sample and future observation for the stated coverage guarantee to apply directly.

    Source: §3.2 · p. 5

Abstract

from arXiv · show

Data-driven distributionally robust optimization (DRO) typically treats the conditional outcome law as fixed and uses ambiguity sets to capture estimation error. This paper studies latent distributional heterogeneity, where each instance has an unobserved law but contributes only one observation, so uncertainty persists even if the mixture law is known. We propose Conformal-DRO, which uses nested conformal regions to construct an ambiguity set for the future latent law. Under exchangeability, the set covers this law with probability at least $1-\alpha$ in finite samples, without estimating underlying latent laws or their mixing mechanism. The conformal path induces a data-driven transport geometry, while $\alpha$ determines the radius. The worst-case problem reduces to a finite linear program over conformal shells and admits sparse adversarial solutions. The resulting robust value provides a finite-sample certificate for the selected decision's expected cost.

1 Introduction

The paper addresses decisions under latent distributional heterogeneity, where one observation per instance leaves uncertainty about the next instance’s law even when the mixture law is known. Conformal-DRO uses observable conformal evidence to build a finite-sample-valid ambiguity set for that future law and a tractable robust optimization problem.

1 Introduction

  • Even if the mixture law is known, one observation per latent law leaves uncertainty about the next instance’s law that fixed-law ambiguity sets may miss.The ambiguity set should cover the future latent law, not only estimation error around a fixed conditional law.Source: §1 · pp. 1–2
  • Conformal measure inversion turns observable outcome-level miscoverage into a moment bound on the future latent law without estimating historical latent laws or their mixing mechanism.For a candidate law, probability outside a conformal region is its conditional probability of generating an uncovered outcome.Source: §1 · p. 2
  • The conformal path and its weights determine the ambiguity set’s transport geometry, while the desired latent-law miscoverage level α determines its radius.Weights can emphasize conformal levels consequential to the downstream decision.Source: §1 · p. 2
  • The adversary’s worst-case expectation reduces to probability allocation across conformal shells, with an optimal solution needing positive mass in at most two shells.The adversary allocates mass across shells and concentrates it near an outcome attaining the largest loss.Source: §1 · p. 2
  • Experiments report finite-sample validity with competitive regret, geometry that adapts to distributional shape, and diminishing robust-value improvements as resolution increases.These are the reported numerical findings; the supplied passage gives no specific values.Source: §1 · p. 2
  • Unlike approaches that treat the conditional law as fixed, the paper targets heterogeneity remaining after conditioning on observed context and constructs a set for a future random law.The approach estimates neither the mixing law nor a posterior.Source: §1 · p. 3

2 Problem Setup

The setup considers contextual decisions when each instance has an unobserved conditional outcome law drawn from an unknown mixing kernel and contributes only one observed outcome. The proposed distribution-free approach constructs a valid ambiguity set without directly estimating those laws or the mixing mechanism.

2 Problem Setup

  • The decision maker observes context x and chooses z before outcome Y is realized, with conditional expected cost as the classical objective when the outcome law is known.Source: §2 · p. 3
  • For each instance, the latent conditional law is drawn from an unknown mixing kernel, and the observed outcome is then drawn from that law.The mixing kernel is not estimated.Source: §2 · p. 3
  • Because each latent law generates only one observation, direct estimation of individual laws or calibration of a law-space ambiguity set requires additional structural assumptions.Source: §2 · p. 3
  • The method instead uses observable conformal miscoverage guarantees to construct a finite-sample-valid ambiguity set, then uses the nested conformal path as a data-driven geometry for a finite-dimensional shell reformulation.It imposes no parametric or structural assumptions on the latent-law family or mixing mechanism.Source: §2 · p. 3

3 Valid Ambiguity Set Construction

Under exchangeability, split conformal prediction supplies outcome-level validity that can be inverted into constraints on a future latent law. Aggregating constraints across conformal levels yields a finite-sample ambiguity set whose coverage requires the aggregation rule to be fixed independently of final calibration and the future observation.

3.1 Preliminaries: Conformal Prediction

  • Under exchangeability, split conformal prediction uses calibration scores to form a prediction region with an outcome-level miscoverage guarantee.The predictor is trained independently of calibration data and the next instance.Source: §3.1 · p. 4

3.2 Conformalized Ambiguity Set

  • For a candidate law, mass outside a conformal region is its leakage; averaged over future instances, leakage equals observable conformal miscoverage, which is bounded by γ.This measure-inversion step transfers outcome-level validity to a moment bound on an unobserved probability measure.Source: §3.2 · p. 4
  • Aggregating incompatibility across nested conformal levels creates a convex generalized-moment set that constrains mass allocation across regions of increasing atypicality, rather than identifying the full latent law.The aggregation rule ν determines which distinctions along the conformal path receive greater emphasis.Source: §3.2 · p. 4
  • The ambiguity set covers the future latent conditional law with finite-sample validity when exchangeability holds and ν is fixed independently of final calibration and the future observation.The guarantee is marginal over calibration data, future context, and future latent law; ν may be selected using an independent tuning dataset.Source: §3.2 · pp. 4–5

4 Distributionally Robust Framework

The framework converts robust optimization over latent laws into finite optimization over conformal shells. It also gives the robust value a finite-sample cost-certificate interpretation and allows the ambiguity geometry to be tuned subject to a resolution budget.

4 Distributionally Robust Framework

  • The framework develops a finite-dimensional shell reformulation, first evaluating worst-case expectation for a fixed aggregation rule and then jointly optimizing the rule under a complexity budget.Source: §4 · p. 5

4.1 Evaluating the Worst-Case Expectation

  • The data-generated penalty assigns statistical incompatibility to outcomes and can be interpreted as an expected-penalty budget; the aggregation rule determines the induced conformal-depth geometry.The desired coverage level fixes the radius.Source: §4.1 · p. 5
  • For fixed decision, context, and aggregation rule, the worst-case expectation is obtained by allocating probability across nonempty conformal shells subject to a linear incompatibility budget.An optimal shell-mass vector can use at most two positive components; if regional losses are attained, a worst-case law can be supported on at most two outcomes.Source: §4.1 · p. 6

4.2 Finite-Sample Robust Risk Certificate

  • The robust value provides a statistically certified upper bound on the selected decision’s expected cost with confidence 1 − α.Source: §4.2 · pp. 6–7

4.3 Joint Optimization and Reformulation

  • Joint tuning uses a finite rank grid of conformal levels; the resolution budget K limits the number of levels and the support of the tuning variables selects active levels and their weights.The grid is sufficient because the split-conformal region is constant over rank intervals.Source: §4.3 · p. 7

5 Experiments

The experiments examine learned shell geometry and decision performance under latent-law heterogeneity. Conformal-DRO refines oracle-aligned geometry as K grows, while baseline comparisons show a tradeoff between regret and protection for future realized laws.

5.1 Synthetic Examples

  • Conformal-DRO recovers population-optimal conformal levels, while increasing K refines shells and spreads weight across more quantile boundaries.The allocations are shown for the linear cost c(z, y) = y⊤z, with only y ≥ 0 and α = 0.2 displayed.Source: §5.1 · p. 7, Figure 3 · p. 8
  • Student-t3 allocations extend farther into the tail than Gaussian allocations, while finite-sample calibration preserves nested ordering and the overall profile.K controls geometric resolution; the outcome distribution determines selected shell locations and weights.Source: §5.1 · p. 8
  • At K = 100, the remaining robust-value gaps are approximately 0.054 for Gaussian and 0.176 for Student-t3, with diminishing returns as K increases.Finite-sample values show minor nonmonotonicity because tuning and calibration use independent samples; heavier Student-t3 tails make extreme quantiles more variable and operationally important.Source: §5.1 · p. 8, Figure 4 · p. 8

5.2 Comparison with Baselines

  • The baseline experiment supplies the exact mixture law to mixture-centered methods and the conformal score, isolating target mismatch from estimation error.It compares against Mixture SP and Mixture WDRO in a capacitated two-product newsvendor problem under an omitted event indicator that determines the realized law.Source: §5.2 · p. 8
  • Mixture WDRO covers only 0.345 of realized laws, while all Conformal-DRO variants exceed the 0.8 nominal law-coverage target and attain 1.000 certificate coverage.Moving from K = 1 to K = 8 reduces regret from 1.616 to 1.319 while law coverage stays near 0.95.Source: §5.2 · p. 8, Table 1 · p. 8

6 Conclusion

Conformal-DRO addresses decision-making under latent distributional heterogeneity by using outcome-level conformal validity to cover the future latent law in finite samples. Its conformal path defines a data-driven transport geometry, and its shell structure makes the optimization finite-dimensional and operationally tractable.

6 Conclusion

  • Conformal-DRO uses outcome-level conformal validity to obtain finite-sample coverage of the future latent law and a corresponding data-driven transport geometry.The prescribed miscoverage level calibrates the radius, and the shell structure reduces the infinite-dimensional DRO problem to a finite-dimensional optimization.Source: §6 · p. 8

A Proof of Theorem 1

The supplied proof passages invoke Tonelli’s theorem and Markov’s inequality to obtain (7), establish coherent upper-risk properties, and describe the conformal-depth geometry and its coverage-based radius.

A Proof of Theorem 1

B Coherent-Risk Properties

  • The conformal upper risk is coherent, including subadditivity for bounded measurable losses.The passage states the subadditivity inequality and concludes coherence.Source: B Coherent-Risk Properties · p. 11

C Geometric Interpretation of Conformalized Ambiguity Set

  • Conformal depth defines a Wasserstein-like geometry that identifies laws inducing the same distribution of conformal depth, even when the original laws differ.The source measure is an auxiliary zero-depth anchor rather than a nominal estimate of the future conditional law.Source: C Geometric Interpretation of… · p. 12
  • The radius 1/α has a coverage interpretation: the ambiguity set contains the future latent conditional law with probability at least 1 − α.A smaller α requests greater confidence and enlarges the corresponding conformal-depth Wasserstein ball.Source: C Geometric Interpretation of… · p. 12

D Proof of Theorem 2

The proof reduces worst-case expectation over the conformalized ambiguity set to optimization over shell probabilities. It also establishes a sparse worst-case law and transfers the probability guarantee to feasible and data-dependent decisions.

D Proof of Theorem 2

  • Every candidate law can be decomposed across shells, reducing the conformal upper risk to a finite optimization over shell probabilities and shell-wise worst costs.The shell partition and the constant incompatibility value within each shell support this reduction.Source: D Proof of Theorem 2 · p. 13
  • An optimal worst-case law can be chosen with positive mass in at most two shells; if regional suprema are not attained, an ε-optimal law has this support bound.Source: D Proof of Theorem 2 · pp. 14–15

E Proof of Corollary 1

  • The event used in the corollary yields the inequality simultaneously for all feasible decisions, including the data-dependent selected decision, with probability at least 1 − α.Substituting the selected decision and its robust value gives the stated probability guarantee.Source: E Proof of Corollary 1 · p. 15

F Proof of Theorem 3

The proof reduces the aggregation problem to a finite conformal-rank grid and shell representation, then establishes equivalence through an epigraph reformulation and recovery of the weights.

F Proof of Theorem 3

  • Moving each candidate level to an equivalent finite-grid level preserves the conformal region and weakly increases its incompatibility penalty, so the robust risk cannot increase.Atoms with zero incompatibility can be removed or replaced without increasing robust risk.Source: F Proof of Theorem 3 · pp. 15–16
  • Duplicate regions can be merged at the smaller level, and zero-penalty atoms can be removed or replaced without increasing robust risk.The proof thereby restricts the optimization to the stated nontrivial finite rank grid.Source: F Proof of Theorem 3 · p. 16
  • On the finite grid, the ordered conformal regions induce shells on which the incompatibility penalty is constant, enabling the finite-shell reformulation.Adjacent shells with equal penalties may be merged without changing robust value.Source: F Proof of Theorem 3 · p. 16
  • The epigraph reparameterization and reverse construction recover feasible solutions with the same objective value, establishing equivalence of the formulations.The recovered aggregation rule supplies the weights for the equivalent formulation.Source: F Proof of Theorem 3 · p. 17

G Proposed Algorithm

The algorithm tunes conformal aggregation weights on an independent sample, calibrates the resulting incompatibility score on a separate sample, and optimizes the final robust decision. Under the stated conditions, its robust value gives a finite-sample upper certificate for expected cost under the future latent law.

G Proposed Algorithm

  • An independent tuning sample selects a context-dependent aggregation rule, which is frozen before the independent final calibration sample calibrates its composite incompatibility score.The score map is evaluated using each calibration observation’s own context; the final sample supplies a scalar calibration rather than new levelwise regions.Source: G Proposed Algorithm · pp. 18–19
  • The guarantee assumes exchangeability and the stated sampling and measurability conditions; nontrivial exclusions require α > 1/(n + 1), and larger K need not improve final cost.K controls tuning-geometry resolution, with at most K + 1 nonempty shells.Source: G Proposed Algorithm · pp. 18, 20
  • Tuning uses a common finite conformal-rank grid and a cardinality-constrained routine, allowing weights and active levels to vary by context.The support budget K limits active levels; the preliminary decision selects a loss-relevant geometry and is not the final calibrated robust value.Source: G Proposed Algorithm · p. 18
  • The final ambiguity set uses calibrated penalties on the selected shells, and the shell-mass linear program yields the robust optimization under finite-loss and attainment conditions.If no nonempty shell meets the penalty threshold, the algorithm uses a conservative full-outcome-space fallback.Source: G Proposed Algorithm · pp. 19–20
  • The robust value is a finite-sample upper certificate for the selected decision’s expected cost under the future latent law, with coverage averaged over calibration, future context, and latent law.The guarantee is not conditional on each context or on a fixed realized calibration sample.Source: G Proposed Algorithm · p. 20

H Reproducibility Details for the Synthetic Experiments in Section 5.1

The synthetic experiments compare finite-sample Conformal-DRO with a population oracle for truncated Gaussian and Student-t3 outcomes. The allocations recover the oracle’s qualitative structure, while the implementation reports no Wasserstein baseline or empirical coverage evaluation.

H Reproducibility Details for the Synthetic Experiments in Section 5.1

  • The outcomes are truncated, renormalized Gaussian or Student-t3 draws, with cost c(z, y) = zy, outcome interval [−5, 5], and decision interval [0.5, 1.5].There are no covariates; each finite-sample experiment uses 500 tuning and 500 independent calibration observations, with no test sample or repeated-trial loop.Source: H Reproducibility Details for the… · p. 21
  • The saved finite-sample optimization selects z = 0.5 for every K, and the largest recorded primal–dual gap is approximately 1.07 × 10−14.The population-oracle comparator is optimized using the known truncated distribution.Source: H Reproducibility Details for the… · p. 22
  • The only executed comparator is the population oracle; no Wasserstein baseline result is reported because the experiment does not call that routine or specify its candidate-radius grid.The population oracle has direct access to the true law and is not a data-matched empirical method.Source: H Reproducibility Details for the… · p. 22
  • The implementation computes no regret, test-outcome performance, coverage, or across-trial summaries, so α = 0.2 is nominal rather than an empirical coverage rate.There is no independent test sample or repeated-trial loop.Source: H Reproducibility Details for the… · pp. 21–22
  • As K increases, Conformal-DRO refines upper-tail shells and spreads weight across more quantile boundaries; Gaussian boundaries stay concentrated while Student-t3 boundaries extend farther into the tail.Finite-sample calibration modestly shifts endpoints and weights, especially for extreme quantiles, while preserving ordering and the overall allocation profile.Source: H Reproducibility Details for the… · pp. 22–23, Figure 5 · p. 23

I Reproducibility Details for the Synthetic Experiments in Section 5.2

The experiments test decisions under latent demand-law heterogeneity in a two-product newsvendor, comparing mixture-based policies with conformal ambiguity sets. They vary the heterogeneity and protection settings, then assess realized-law costs, coverage, and the effects of slicing.

Experimental setup

  • The experiment varies δ to control latent-law heterogeneity: an unobserved event changes the realized demand law, while recorded-context observations identify only its mixture.It uses discrete and continuous demand models with the same two-product newsvendor decision problem.Source: I Reproducibility Details for the… · p. 24

Experimental design

  • Mixture methods and conformal scores receive the exact mixture Qx to remove conditional-distribution estimation error; the study compares single-level and sliced conformal sets with mixture-based benchmarks.It varies conformal levels and support budgets, evaluates expected cost and regret across future realized laws, and uses independent tuning, calibration, and test samples of 3,000, 1,000, and 2,000 instances.Source: I Reproducibility Details for the… · pp. 24–25

Results

  • When δ = 0, latent laws coincide and performance distinctions largely disappear; for example, discrete mean costs are 3.367 for both the oracle and Mixture SP.The corresponding continuous mean costs are 2.995 for both methods.Source: I Reproducibility Details for the… · p. 26
  • At δ = 2, K = 4 Conformal-DRO raises mean cost over Mixture SP by 11.5% and 14.9%, while lowering P90 cost by 19.5% and 15.2%.It also lowers CVaR0.9 by 9.3% and 8.9%, and event-regime cost by 16.8% and 19.2%, in the discrete and continuous experiments, respectively. Mixture SP and Mixture WDRO receive the exact Qx, isolating the mixture-versus-realized-law target mismatch from estimation error.Source: I Reproducibility Details for the… · p. 26
  • At δ = 2, conformal methods attain approximately 94%–96% law coverage and 100% certificate coverage, versus 34.5% and 40.6% law coverage for mixture-centered WDRO.Coverage exceeds the nominal 80% target, indicating conservatism in converting response-level validity to law-level coverage; slicing reduces the operational price of protection rather than adding coverage for its own sake.Source: I Reproducibility Details for the… · p. 26
  • With K = 4, slicing lowers mean cost by 5.5% and 5.4% and mean certificates by 18.1% and 6.8% versus a single conformal level.These results are for the discrete and continuous experiments, respectively; tail effects differ, and discrete CVaR0.9 and event cost rise moderately while law coverage remains above target.Source: I Reproducibility Details for the… · pp. 26–27
  • Increasing K from 4 to 8 changes realized operational performance only slightly, while lowering mean certificates by 7.1% and 2.5%.The respective certificate reductions are in the discrete and continuous experiments; four levels capture most operational value in these designs, while additional levels primarily refine the ambiguity geometry.Source: I Reproducibility Details for the… · p. 27
  • At α = 0.30, discrete K = 4 Conformal-DRO has mean cost 4.354, 2.3% above Mixture SP, while P90 and CVaR0.9 are 2.7% and 4.2% lower.Empirical law coverage is 0.927; increasing α moves decisions toward mixture-optimal mean performance while gradually giving up tail protection and coverage, with a non-exactly-monotone frontier.Source: I Reproducibility Details for the… · p. 27

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