[Paper Review] Bulk-Calibrated Credal Ambiguity Sets: Fast, Tractable Decision Making under Out-of-Sample Contamination

Mengqi Chen, Thomas B. Berrett+2 more|arXiv (Cornell University)|Jan 29, 2026

Risk and Portfolio Optimization0 citations

TL;DR

The paper introduces bulk-calibrated credal ambiguity sets (LV) that translate imprecise probability into a tractable DRO objective, enabling fast robust decisions under out-of-sample contamination with data-driven bulk calibration.

ABSTRACT

Distributionally robust optimisation (DRO) minimises the worst-case expected loss over an ambiguity set that can capture distributional shifts in out-of-sample environments. While Huber (linear-vacuous) contamination is a classical minimal-assumption model for an $\varepsilon$-fraction of arbitrary perturbations, including it in an ambiguity set can make the worst-case risk infinite and the DRO objective vacuous unless one imposes strong boundedness or support assumptions. We address these challenges by introducing bulk-calibrated credal ambiguity sets: we learn a high-mass bulk set from data while considering contamination inside the bulk and bounding the remaining tail contribution separately. This leads to a closed-form, finite $\mathrm{mean}+\sup$ robust objective and tractable linear or second-order cone programs for common losses and bulk geometries. Through this framework, we highlight and exploit the equivalence between the imprecise probability (IP) notion of upper expectation and the worst-case risk, demonstrating how IP credal sets translate into DRO objectives with interpretable tolerance levels. Experiments on heavy-tailed inventory control, geographically shifted house-price regression, and demographically shifted text classification show competitive robustness-accuracy trade-offs and efficient optimisation times, using Bayesian, frequentist, or empirical reference distributions.

Motivation & Objective

• Motivate robust decision-making under distributional uncertainty and out-of-sample contamination.
• Introduce bulk-restricted credal ambiguity sets (forward LV) that yield a closed-form worst-case risk.
• Provide data-driven bulk calibration with finite-sample guarantees and a high-probability risk certificate.
• Show that IP credal sets correspond to DRO objectives with interpretable tolerance levels.

Proposed method

• Define a bulk-restricted LV credal ambiguity set around a data-driven centre distribution.
• Derive the closed-form worst-case risk: (1−ε) E_{P_c,Ξ0}[f_x(ξ)] + ε sup_{ξ∈Ξ0} f_x(ξ).
• Provide tractable LP/SOCP reformulations for common losses and bulk geometries.
• Calibrate bulk set Ξ0 using score-based selection with DKW-based risk certificates.
• Prove risk bounds that separate in-bulk robustness from tail control under Huber ε-contamination.
• Demonstrate equivalence between IP upper expectation and a DRO worst-case risk.

Figure 2 : Worst-case distributions $Q^{\star}$ for $\sup_{Q}\mathbb{E}_{\xi\sim Q}[f]$ under forward LV, reverse LV, and TV balls around a centre $\mathbb{P}_{c,\Xi_{0}}$ (loss $f$ plateaus at a small region to avoid Dirac deltas).

Experimental results

Research questions

• RQ1 How can we construct a robust optimization objective that remains well-posed in unbounded spaces under Huber ε-contamination?
• RQ2 Can we learn a bulk set with mass guarantees from data to enable tractable DRO formulations?
• RQ3 What is the relationship between imprecise probabilities (credal sets) and distributionally robust optimization in continuous spaces?
• RQ4 Do LV-based credal sets provide practical robustness with competitive performance across real-world tasks?
• RQ5 How does bulk calibration affect computational efficiency and robustness trade-offs across datasets and losses?

Key findings

• The bulk-restricted LV credal ambiguity set yields a closed-form worst-case risk: (1−ε) E_{P_c,Ξ0}[f_x(ξ)] + ε sup_{ξ∈Ξ0} f_x(ξ).
• The approach leads to tractable LP or SOCP reformulations for common losses and bulk geometries.
• Calibrating Ξ0 with data using a DKW-based score selection provides a high-probability bulk-mass certificate (1−γ with confidence 1−δ).
• Experiments on heavy-tailed inventory control, California housing regression under deployment shift, and CivilComments text classification show competitive robustness-accuracy trade-offs and faster optimisation times than baselines.
• LV-based methods often achieve superior OOS performance and shorter solve times, particularly under contamination, compared with KL-based DRO and OR-WDRO baselines.
• The framework accommodates Bayesian, frequentist, or empirical reference distributions and offers flexible centre choices.

Figure 3 : Student- $t$ newsvendor (cost: lower-left is better). Top row: OOS mean–variance frontiers for a range of $\varepsilon_{\operatorname{LV}}\in(0,1]$ ; $\varepsilon_{\operatorname{KL}}\in(0,25]$ . OR-WDRO uses $\varepsilon_{\operatorname{LV}}$ in $(0,0.5)$ . Each point represents one $\vare