Calibrating Ambiguity Set via Diagnostic Transport for Distributionally Robust Optimization

Wenbin Zhou; Elizabeth Cucuzzella; Shixiang Zhu

stat.MLOct 7, 2026

Calibrating Ambiguity Set via Diagnostic Transport for Distributionally Robust Optimization

Authors: Wenbin Zhou, Elizabeth Cucuzzella, Shixiang Zhu

Organizations: Heinz College of Information Systems and Public Policy, Carnegie Mellon University · Machine Learning Department, Carnegie Mellon University · Department of Statistics and Data Science, Carnegie Mellon University

Abstract

Distributionally robust optimization (DRO) protects decisions against distributional uncertainty by optimizing over an ambiguity set, but poorly aligned set geometry can require large radii and yield overly conservative decisions. We introduce diagnostic-transport DRO (DT-DRO), which uses held-out calibration data to adapt the ambiguity-set geometry to observed predictive errors. DT-DRO uses the conditional probability integral transform cumulative distribution function to diagnose systematic probability misallocation and translates this information into an outcome-level transport that jointly adjusts the ambiguity-set center and ground cost. The resulting formulation admits a computationally tractable dual reformulation. Theoretically, we derive valid ambiguity radii and decision-risk guarantees that tighten as estimation and approximation errors vanish, and show that DT-DRO can eliminate the nonvanishing robustness floor caused by model misspecification. Synthetic experiments and a power-outage application demonstrate improved decision quality, particularly under structural and tail misspecification.

Figures & tables

Figure 1 : A synthetic illustration of diagnostic transport in DT-DRO. The heatmap visualizes the diagnostic discrepancy $\kappa$ , scaling with the shaded area between the PIT-CDF and the identity line. Gray arrows indicate the local transport directions in schematic distribution-space coordinates, illustrating how diagnostic information reshapes the ambiguity set. The transported ambiguity set (blue region) contracts toward the true distribution $\mathbb{P}_{x}^{\star}$ , and the corresponding worst-case distribution $\mathbb{Q}_{z}^{\star,\rm DT}$ yields lower regret than the worst-case distribution $\mathbb{Q}_{z}^{\star}$ obtained from the original ambiguity set.

Figure 2 : Overview of the DT-DRO procedure. The calibration data

\mathcal{D}

and base predictive model

\mathbb{P}_{b,x}

are used to estimate the conditional PIT-CDF

G_{x}

and construct the diagnostic transport

T_{x}

. The conditional P–P plot visualizes the discrepancy between

G_{x}

and perfect calibration, while the red arrows illustrate the calibration corrections induced by solving ( 4 ). The learned transport reshapes the ambiguity set into

\mathcal{U}_{x}^{\rm DT}(\varepsilon)

; together with the calibrated radius

\varepsilon

, this transported set enters the dual optimization to produce the final robust decision.

Figure 3 : Performance under four forms of predictive misspecification. Top row : mean expected regret at the target context as the calibration-sample size increases, with shaded regions indicating

95\%

confidence intervals over

100

replications. Bottom row : the true conditional density and mean fitted densities from different baselines.

Figure 4 : Rolling perfect-information (PI) regret in the sequential outage experiment for (a) all outages and (b) severe outages, defined as durations exceeding the weighted 95th percentile of the static training sample. Curves report the 12-month rolling weighted mean regret for each decision method, with shaded bands denoting

95\%

confidence intervals from 300 substation-cluster bootstrap replications.

Explore similar work

Jul 10, 2026cs.LG

Learning Predictive Ambiguity Sets for Decision-Focused Distributionally Robust Optimization

Predict-then-optimize systems usually compress uncertainty into a point forecast and then solve a downstream optimization problem as if the forecast were reliable. Distributionally robust optimization (DRO) offers protection against misspecification, but the ambiguity set is often centered at historical samples and uses a fixed radius. We propose \emph{learned predictive ambiguity sets} (LPAS): a deep contextual model outputs a finite nominal scenario distribution, a state-dependent Wasserstein radius, and optionally an anisotropic ground metric. These outputs define a contextual ambiguity set that feeds a DRO decision layer. The radius is trained by a combination of conditional quantile calibration, size regularization, and downstream decision loss, so that robustness is adaptive rather than globally fixed. We derive the finite dual form used by the decision layer, present a staged training algorithm, and evaluate the method on distributionally robust portfolio optimization with 20 S&P 500 constituents from 2018--2026. The proposed method substantially improves over equal-weight, predict-then-optimize, and historical Wasserstein DRO baselines, achieving 26.28% annualized return, Sharpe ratio 1.30, final wealth 1.61, and lower tail loss than a deep fixed-radius DRO baseline while using a smaller average radius. The results show that learned ambiguity radii can recover most of the performance of strong fixed-radius DRO while reducing unnecessary conservatism and improving regime adaptivity.

Junjie Guo

Rutgers University

Distributionally Robust Optimization Decision-Focused Learning Wasserstein Distributionally Robust Optimization Portfolio Optimization

Ask PDF

Apr 15, 2025math.OC

Wasserstein Distributionally Robust Regret Optimization

Distributionally robust optimization (DRO) is widely used for decision-making under uncertainty, but its adversarial focus on worst-case loss can lead to overly conservative policies. To mitigate this, we study ex-ante Distributionally Robust Regret Optimization (DRRO) with Wasserstein ambiguity sets, designed to balance robustness with upside potential. We develop a theory of Wasserstein DRRO (WDRRO) paralleling Wasserstein DRO. Under smoothness and regularity, WDRRO selects among ERM optima by a first-order gradient-discrepancy rule. If the ERM optimizer is unique, first-order sensitivity vanishes and a second-order expansion governs deviations. For convex quadratics ERM and DRRO coincide for any radius. We then study regimes where these assumptions fail: nondifferentiable max-affine losses, discrete references, and larger radii, where WDRRO can differ from ERM and WDRO. We show that computing WDRRO regret is NP-hard even without bilinear terms. Nevertheless, we develop exact algorithms, a tractable convex relaxation with guarantees, and experiments showing tightness and loss-dependent behavior.

Lukas-Benedikt Fiechtner, Jose Blanchet

Institute of Computational and Mathematical Engineering Stanford University · Institute of Computational and Mathematical Engineering Department of Management Science and Engineering Stanford University

Distributionally Robust Optimization Robust Optimization Wasserstein Distributionally Robust Optimization

Ask PDF

Aug 30, 2026math.OC

A Unified Perspective on Conformal Prediction and Wasserstein Distributionally Robust Optimization for Uncertainty Quantification

Uncertainty quantification from finite data is central to machine learning, optimization, and automation systems, where decisions must remain reliable under limited samples and test-time distribution shift. Conformal prediction (CP) and distributionally robust optimization (DRO) offer two complementary approaches: CP constructs data-dependent prediction sets with distribution-free finite-sample validity under exchangeability, while DRO optimizes worst-case performance over an ambiguity set around an empirical distribution. We develop a unified probabilistic perspective on CP and DRO by viewing both as ways to turn finite calibration data into a data-dependent quantile estimator that a test score falls below with high probability. From this perspective, CP and DRO correct the empirical quantile along two coordinates of the same family of estimators: CP inflates the quantile level, whereas DRO shifts the quantile value through an ambiguity radius. Both methods provide the same calibration-conditional guarantee for the true distribution, requiring the target coverage to hold with high probability over the calibration sample. Their constructions differ, however: CP uses a closed-form, distribution-free level correction, while DRO uses a value-space correction whose certified radius depends on properties of the unknown distribution and additionally guarantees coverage uniformly over the ambiguity set. This distinction emerges in the tails of the score distribution. Because CP relies on sparse upper-tail order statistics of the calibration samples, its level inflation barely moves the estimator when those samples are dense near the target quantile but overshoots when they are sparse, whereas a well-chosen DRO radius corrects in value space and may avoid this overshoot.

Kehan Long, Yiqi Zhao, Pol Mestres +3

Contextual Robotics Institute, University of California San Diego, La Jolla, CA, USA · Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, USA · Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA, USA +1

Distribution Shift Robustness Uncertainty Quantification Conformal Prediction Wasserstein Distributionally Robust Optimization

Ask PDF