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Learning the Center and Radius of Wasserstein Ambiguity Sets for Data-Driven Decision Making

Junjie Guo

arXiv:2608.18123v1math.OC

TL;DR

SAA is asymptotically consistent under standard regularity conditions, but can overfit the empirical sample and perform poorly out of sample for small or moderate N. The framework uses a predictive nominal distribution and a separately estimated radius, with coverage-based guarantees, tractable formulations, stability analysis, and split-conformal marginal calibration. 80.850% to 95.275% saved Sinkhorn-proxy containment in the synthetic mean–CVaR core after conformal calibration, but experiments show no uniform decision-value advantage.

  • Problem

    SAA is asymptotically consistent under standard regularity conditions, but can overfit the empirical sample and perform poorly out of sample for small or moderate N.

    Source: §1 · p. 2

  • Method

    The framework uses a predictive nominal distribution and a separately estimated radius, with coverage-based guarantees, tractable formulations, stability analysis, and split-conformal marginal calibration.

    Source: §1 · p. 2, §11 · p. 33

  • Results

    80.850% to 95.275% saved Sinkhorn-proxy containment in the synthetic mean–CVaR core after conformal calibration, but experiments show no uniform decision-value advantage.

    Source: §11 · p. 33

  • Takeaways & Limitations

    Calibration can support statistical reliability, but improving one link—center quality, scenario approximation, radius calibration, or optimization—need not improve the final decision.

    Source: §11 · p. 33

    The oracle radius target is unavailable in ordinary single-environment applications, so supervised radius learning needs known-law simulated tasks, repeated environments, or a controllable observable proxy.

    Source: §10 · p. 32

Abstract

from arXiv · show

Wasserstein distributionally robust optimization (DRO) is commonly built around the empirical distribution, with the ambiguity radius selected from a concentration bound. Although this construction provides useful statistical guarantees, it can be conservative and does not fully exploit predictive information about the underlying distribution or the difficulty of a particular decision problem. We develop a more flexible framework in which a predictive model determines the nominal distribution and a separate model estimates a data-dependent radius. The key requirement is not that the ambiguity set be centered at the empirical distribution, but that it contain the unknown data-generating distribution with the desired probability. We establish finite-sample guarantees and asymptotic consistency for arbitrary learned centers, derive tractable reformulations for non-uniform discrete predictive distributions, separate predictive-model and scenario-discretization errors, and prove stability under simultaneous perturbations of the center and radius. We further characterize the oracle conditional-quantile radius as the smallest conditionally valid rule and introduce a split-conformal procedure for finite-sample marginal calibration. Experiments on newsvendor problems, synthetic portfolios, distribution shifts, and real financial data show that learned and calibrated ambiguity sets can improve reliability, but do not automatically yield smaller radii or better decisions. Overall, the proposed framework treats calibration as a practical mechanism for reliable decision making rather than a universal guarantee of improved optimization performance.

1 Introduction

The paper replaces empirical centering and a fixed concentration-based radius with a predictive center and a separately calibrated, data-dependent radius. Its contribution is a coverage-based framework with guarantees, tractable formulations, stability results, and experiments showing that reliable calibration does not ensure better decisions.

1 Introduction

  • Finite-sample validity and asymptotic consistency extend to learned centers, with radius dominance and Lipschitz regret and certificate bounds.The framework also distinguishes universal distributional coverage from task-specific certificate coverage and gives guarantees when radius undercoverage is nonzero.Source: §1 · pp. 2–3
  • The method separates predictive-model error from scenario-discretization error and establishes robust-value stability under joint center and radius perturbations.The two errors add in the radius.Source: §1 · p. 3
  • The oracle conditional quantile is the pointwise smallest conditionally valid radius, and split conformal supplies finite-sample marginal calibration.Approximate radius predictors can be corrected with safety margins.Source: §1 · p. 3
  • Learned and calibrated ambiguity sets can support reliable coverage, but experiments find no uniform radius-efficiency or decision-value advantage.The framework's central requirement is coverage of the unknown distribution by the random ambiguity set, not empirical centering.Source: §1 · pp. 2–3

2 Related Literature

  • Prior work motivates Wasserstein ambiguity sets and adaptive radii, while this paper focuses on jointly decomposing center learning, discretization, calibration, stability, and efficiency.The authors do not claim data-adaptive ambiguity sets are new.Source: §2 · p. 3

3 Notation and Preliminaries

  • The notation treats the nominal distributions, learned radius, and data-driven decisions as random under the product law unless stated otherwise.Additional independent predictive-model or scenario-generator randomness is included by enlarging the probability space.Source: §3 · p. 4

4 Classical Empirical Wasserstein DRO

Classical empirical Wasserstein DRO uses a concentration-based radius to obtain a finite-sample certificate. That radius is universal and safe, but can be loose because it ignores realized data geometry and problem difficulty.

4 Classical Empirical Wasserstein DRO

  • The classical empirical Wasserstein DRO construction centers its ambiguity set at the empirical law and seeks a finite-sample upper certificate.The section recalls the construction of Esfahani and Kuhn (2018).Source: §4 · p. 4

4.1 Measure-concentration radius

  • With the concentration-based radius, the empirical robust value bounds the out-of-sample cost of the robust optimizer with probability at least 1 − β.The construction assumes the stated light-tail condition and uses a measure-concentration result.Source: §4.1 · pp. 4–5

4.2 Limitations of the a priori radius

  • The universal radius may be loose because it depends on sample size, confidence, dimension, and tail constants rather than realized data geometry.Its rate can be pessimistic in high dimensions, motivating data-adaptive alternatives.Source: §4.2 · p. 5

5 Worst-Case Expectation Duality for a General Discrete Nominal Distribution

The paper extends Wasserstein worst-case expectation duality from equal-weight empirical centers to arbitrary discrete nominal laws. Under stated strong-duality and loss conditions, it also gives a finite convex reformulation.

5 Worst-Case Expectation Duality for a General Discrete Nominal Distribution

  • The worst-case expectation duality applies to a general discrete nominal distribution with non-uniform weights, including the empirical law as a special case.The result assumes strong duality for the associated moment problem, which holds under standard upper-semicontinuity and linear-growth conditions.Source: §5 · pp. 5–6

5.1 Finite convex reformulation

  • Under the concave-piece loss assumption and strong-duality conditions, the non-uniform-weight problem admits a finite convex reformulation.This is the non-uniform-weight version of the convex reduction for Wasserstein DRO.Source: §5.1 · pp. 6–7
  • For piecewise-affine loss over the full space, the reformulation sets each scenario epigraph value to the maximum affine piece and the transport multiplier to the largest dual norm.This specializes the result to Ξ = R^m.Source: §5.1 · pp. 7–8

6 Predictive Nominal Distributions

This section replaces the empirical center with a learned predictive distribution, provided its Wasserstein error can be calibrated. It establishes coverage, consistency, stability, and error bounds for predictive-center DRO.

6.1 Finite-sample guarantee

  • A calibrated predictive center yields finite-sample coverage whenever its ambiguity ball contains the true distribution with the required probability.The empirical distribution is not essential; the center may be any random nominal distribution for which a high-probability Wasserstein error bound can be calibrated.Source: §6.1 · p. 8

6.2 Predictive radius dominance over empirical radius

  • Predictive radii can dominate empirical radii exactly or approximately, including confidence-radius and high-probability comparisons.An empirical radius r with coverage at least 1 − β yields a predictive radius r + ∆N with the same coverage.Source: §6.2 · pp. 9–10

6.3 Perfect and approximately perfect predictive models

  • A perfect predictive model reduces DRO to the true optimization problem at radius zero; approximate perfection is relevant when Wasserstein error tends to zero.For continuous P and finite-discrete predictive centers, exact equality usually has probability zero.Source: §6.3 · p. 10

6.4 Predictive asymptotic consistency

  • As the radius vanishes and coverage holds eventually, predictive DRO values converge to the true value; under additional regularity, optimizer accumulation points are true optimizers.The optimizer conclusion requires the additional conditions that X is closed and the loss is lower semicontinuous in x.Source: §6.4 · p. 11

6.5 Lipschitz regret and certificate bounds

  • Lipschitz-loss assumptions yield predictive regret and certificate-gap bounds, with the regret bound depending on the center error and ambiguity radius.The paper also derives a confidence-level regret comparison under radius dominance.Source: §6.5 · pp. 12–13

6.6 Stability with respect to the center and radius

  • Under a uniform Lipschitz loss, DRO values are stable to simultaneous perturbations of the nominal center and radius.The joint stability bound also holds for the robust value at any fixed decision.Source: §6.6 · p. 14

6.7 Learning and scenario-discretization errors

  • Scenario discretization adds a separate Wasserstein error to predictive-model error, and ignoring it can lead to systematic undercoverage.Bounds for model generalization and scenario discretization can be transferred to DRO through the coverage, consistency, and regret results.Source: §6.7 · p. 14

7 Learned Predictive Radii

This section studies learned, data-dependent Wasserstein radii around predictive centers. It provides validity and consistency results, characterizes the smallest conditionally valid radius, and gives split-conformal calibration under exchangeability.

7.1 Finite-sample validity of learned radii

  • A learned radius is finite-sample valid when it covers the true center error with the required probability, and its coverage lower-bounds DRO reliability.The result links radius coverage to the probability that the resulting DRO decision satisfies the stated reliability criterion.Source: §7.1 · p. 15

7.2 Comparison with measure-concentration radius

  • A learned radius can retain validity while yielding a lower certificate than a measure-concentration radius under the stated comparison conditions.The comparison assumes the learned and measure-concentration radii use the same nominal center.Source: §7.2 · pp. 15–16

7.3 Regret and certificate-gap bounds with random radii

  • The section derives regret and certificate-gap bounds for random learned radii, including rate transfer when their coverage probability tends to one.The rate-transfer result assumes the learned radius is O_P(aN) for a deterministic sequence aN.Source: §7.3 · pp. 16–17

7.4 Asymptotic consistency with learned radii

  • Learned-radius DRO values converge to the true value when coverage holds eventually and the radius vanishes; under optimizer regularity, accumulation points are true optimizers.A Borel–Cantelli condition supplies one route to eventual coverage almost surely.Source: §7.4 · p. 17

7.5 Oracle conditional-quantile radius

  • The oracle conditional quantile is valid and pointwise smallest among conditionally valid radius rules, minimizing every increasing functional of the radius.Its minimality includes minimizing expected radius.Source: §7.5 · p. 18

7.7 Split-conformal calibration of learned radii

  • Split conformal provides finite-sample marginal radius calibration when calibration instances and the new instance are exchangeable conditional on training data.Ordinary split conformal is restricted to independent task draws; chronological rolling data alone does not justify its coverage guarantee.Source: §7.7 · pp. 19–20

7.8 Distributional radius versus certificate radius

  • Certificate-valid radii can be smaller than distributionally valid radii, but certificate-radius models are task-specific to the loss, feasible set, and optimization problem.Distributional coverage is sufficient but not necessary for the DRO certificate.Source: §7.8 · p. 20

8 Predictive DRO Reformulations for Portfolio Optimization

This section formulates portfolio optimization with a Wasserstein ambiguity set for a discrete predictive return distribution.

8 Predictive DRO Reformulations for Portfolio Optimization

  • The portfolio problem uses m assets, a long-only simplex, and a mean-CVaR loss with α denoting the upper-tail probability.The CVaR representation follows Rockafellar and Uryasev (2000).Source: §8 · p. 21
  • With an ℓ1 Wasserstein ground norm, the discrete predictive-center reformulation is a linear program after expressing the dual-norm constraints linearly.A deterministic proportional transaction-cost term can be added directly to the objective without changing the inner Wasserstein reformulation.Source: §8 · p. 21

9 Computational Experiments

The experiments separate predictive-law accuracy, ambiguity-set calibration, and downstream decisions, finding mixed results across controlled, shifted, sequential, and real-data settings. Calibration can improve reliability, but does not ensure smaller radii or better decisions.

9.1 Protocol, estimands, and claim taxonomy

  • The protocol separates training, calibration, validation, and testing, and labels grid-based and Sinkhorn discrepancy outcomes as numerical diagnostics rather than exact population coverage.Test tasks are evaluated once after model and comparison choices are fixed; oracle radii use test information only as unattainable benchmarks.Source: §9.1 · p. 22

9.2 Center learning and finite-scenario approximation

  • GMM numerical model error falls from 6.941 at N = 20 to 1.530 at N = 500, while fixed-radius containment rises to 94.225% at M/N = 10 but remains below the 95% target.Center-learning gains are not uniform across model classes, and increasing scenario count does not eliminate undercoverage.Source: §9.2 · p. 23, Figure 1 · p. 23

9.3.1 One-dimensional newsvendor

  • Conformal attains 95.506% pooled numerical containment, but raw GBR reaches 95.575%, and both rules produce the same mean regret of 1.2223.All seven candidate radii produce the same decision in each of 100 validation cells; cellwise conformal containment ranges from 84.384% to 98.298%.Source: §9.3.1 · p. 24, Table 2 · p. 24

9.3.2 Synthetic mean–CVaR portfolios

  • In synthetic portfolios, conformal raises proxy containment by 14.425 percentage points versus the raw radius, yet shows no statistically detectable paired-regret improvement versus bootstrap.The conformal-minus-bootstrap regret difference is 9.68 × 10^-7 (paired interval approximately [0, 2.46 × 10^-6], Wilcoxon p = 0.240); multivariate containment remains an entropically regularized proxy.Source: Table 2 · p. 24, §9.3.2 · pp. 25–26

9.4 Contextual and sequential distribution shift

  • Conditional conformal proxy containment is 70.325% versus 88.800% for unconditional empirical split conformal, with mean regret 0.002374 versus 0.002162.Under interpolation, conditional containment is 94.7%, but it falls to 19.4% under deliberate misspecification; the safety margin raises aggregate containment only to 81.375%.Source: §9.4 · p. 26, Table 3 · p. 26
  • Adaptive sequential methods recover after the abrupt shift, whereas the fixed exchangeable-split diagnostic has no recorded recovery; these are diagnostics on one dependent trajectory, not exchangeable guarantees.The adaptive controller covers 1,423 of 1,500 time points and recovers after 123 observations; exponential weighting covers 1,420 and recovers after 292.Source: §9.4 · p. 26, Figure 4 · p. 28

9.5 Rolling real-data portfolios

  • Gaussian-safety reaches future-block proxy rates of 93.981% for FF49 and 95.370% for ETFs, while FF49 operational reliability is 91.667% and return intervals overlap competing methods.Safety adjustment raises future-block proxy rates relative to raw and fixed radii, but these differences are descriptive rather than inferential dominance claims.Source: §9.5 · p. 29, Table 4 · p. 29

9 Computational Experiments

  • Across experiments, calibration can improve reliability but does not uniformly improve radius efficiency or decisions; scenario error, distribution shift, and real-data constraints qualify the gains.Adaptive sequential procedures recover after an abrupt shift, while severe misspecification and real-data stress remain substantive failure modes.Source: §9 · p. 21, §9.6 · p. 31

10 Limitations and Scope

The guarantees depend on how the center and radius are learned and on the validity of the discrepancy and calibration procedures. The experiments support qualified conclusions rather than exact multivariate coverage or universal decision superiority.

10 Limitations and Scope

  • Multivariate portfolio and contextual containment uses an entropically regularized discrepancy proxy, so the paper makes no exact multivariate distributional-coverage claim.Only 52.0% of paired synthetic-portfolio label calculations meet the saved convergence criterion; the exact-assignment sensitivity has 10 converged cases with 25.4% mean relative error.Source: §10 · p. 32
  • Finite scenario generation is a separate approximation: in the fixed-radius newsvendor diagnostic, numerical containment is 94.225% at M = 10N, below the 95% target.The newsvendor calculation is subject to uncertified grid error.Source: §10 · p. 32
  • Split conformal guarantees marginal validity across exchangeable tasks, not conditional validity within every distribution family, volatility level, or dimension.The contextual trajectory is nonexchangeable, so its weighted and adaptive procedures have sequential, assumption-dependent interpretations.Source: §10 · p. 32
  • In the unrestricted-support W1 newsvendor, the radius changes the robust value but not the optimizer; calibration therefore need not improve decisions.The synthetic portfolio calibration improves proxy containment without detectable regret improvement, while the conditional contextual model underperforms the unconditional empirical benchmark.Source: §10 · p. 32
  • The financial study covers long-only, fully invested portfolios with monthly rebalancing, proportional costs, two asset universes, and one historical period.It omits market impact, taxes, borrowing constraints, short sales, and implementation-capacity effects.Source: §10 · p. 32
  • Financial comparators solve different objectives, and the backtest is reproducible from frozen processed inputs although the cached raw TIP payload cannot be reconstructed byte-for-byte.Reported results are tied to hash-locked processed inputs.Source: §10 · p. 32

11 Conclusion

The paper develops a modular theory and calibration framework for learned-center Wasserstein DRO, while its experiments show that reliability gains do not establish universal decision-value gains. Its practical conclusion is that center quality, approximation, calibration assumptions, and decision sensitivity must be evaluated separately.

11 Conclusion

  • The theory establishes guarantees for general random centers, tractable reformulations for non-uniform discrete laws, error decomposition, joint center-radius stability, and oracle conditional-quantile radius optimality.A finite-sample-correct split-conformal construction supplies marginal calibration across exchangeable tasks.Source: §11 · p. 33
  • Across the experiments, learned or calibrated ambiguity sets can improve reliability but do not show universal radius efficiency or decision-value superiority.The synthetic portfolio calibration improves proxy containment without detectable regret improvement; contextual conditional modeling performs worse in aggregate, and financial return intervals do not support performance dominance.Source: §11 · p. 33
  • Calibration can support statistical reliability without guaranteeing decision value; the framework treats predictive accuracy, ambiguity-set coverage, and downstream performance as distinct.The paper identifies side information, observable or conservatively approximated radius targets, deployment-matched calibration assumptions, and decision sensitivity as conditions for learned ambiguity sets to be most compelling.Source: §11 · p. 33

A Additional Linear Reformulation for Polyhedral Support

The appendix gives an additional linear reformulation for polyhedral support and identifies a norm condition under which the resulting formulation is a linear program.

A Additional Linear Reformulation for Polyhedral Support

B Algorithmic Templates

The algorithmic templates separate fitting a predictive center from selecting its radius, and provide a split-conformal procedure for radius calibration across exchangeable tasks.

B Algorithmic Templates

  • The predictive-center Wasserstein DRO template takes training data, a confidence level, a predictive distribution model, and a radius rule as inputs.Source: B Algorithmic Templates · p. 34
  • The template fits the predictive distribution on the training data and computes or predicts a radius before returning the decision and certificate.It also returns a center-error proxy and the radius.Source: B Algorithmic Templates · p. 34
  • The conformal template fits the radius predictor on proper-training tasks and calibrates it using separate calibration tasks.It sets k = ⌈(T + 1)(1 − β)⌉, uses the corresponding score order statistic when k ≤ T, and otherwise sets the correction to +∞ before solving DRO.Source: B Algorithmic Templates · p. 34

C Additional Experimental Exhibits

This appendix gathers diagnostic, real-data path, ablation, and computational exhibits. It cautions that multivariate containment and future-block financial quantities are proxies, not distributional-coverage estimates.

C Additional Experimental Exhibits

  • All 1,714,585 recorded newsvendor checks pass at the frozen numerical tolerance.The checks cover slack in the normalized-regret bound, scenario-error triangle inequality, and center–radius stability bound.Source: Figure 7 · p. 35
  • In the paired synthetic mean–CVaR experiment, large differences in discrepancy-proxy containment do not translate into detectable differences in core decision loss.All core configurations have estimated reliability of one, so the reliability–radius frontier distinguishes conservatism but not certificate reliability.Source: Figure 9 · p. 36, Figure 8 · p. 36
  • The financial path exhibits show gross and net wealth for FF49 and cross-asset ETF universes, while drawdowns are descriptive historical outcomes, not independent stress-test replications.Net wealth applies the frozen proportional transaction cost; drawdown plots mark global-financial-crisis, COVID-crash, and inflation-stress windows.Source: Figure 10 · p. 37, Figure 11 · p. 38
  • Primary ablations vary calibration rules, scenario multipliers, fixed-radius targets, empirical-radius rules, calibration-task counts, and centers at a common radius.Every containment ordinate is explicitly a numerical proxy.Source: Figure 12 · p. 39
  • Additional ablations vary feature scaling, selection, model capacity, labels, radius rules, discrepancy and certificate proxies, training share, and real-data future-block radius rules.Complete numerical rows are retained in the replication archive.Source: Figure 13 · p. 40
  • The discrepancy diagnostics compare debiased-Sinkhorn with sliced-W1 and exact-assignment references, while computational exhibits report solve times, decision rates, feasibility residuals, and solver iterations.Exact-assignment discrepancies underscore that multivariate containment is a numerical proxy rather than exact Wasserstein coverage; failed attempted solves remain in relevant rate denominators.Source: C Additional Experimental Exhibits · p. 35, Figure 14 · p. 41, Figure 15 · p. 42

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