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

TL;DR

The paper addresses out-of-sample contamination in unbounded spaces by introducing bulk-calibrated credal ambiguity sets that confine adversarial mass to a data-driven bulk \\\Xi_0 with a probabilistic mass certificate. This yields a closed-form mean+sup robust objective, enabling tractable LP/SOCP formulations for common losses and bulk geometries, and clarifies the IP-DRO relationship via forward LV, CVaR, and TV connections. A two-stage, score-based bulk calibration procedure (score fitting and DKW-based selection) provides high-probability guarantees on bulk mass, while a tunable tolerance parameter \\\varepsilon controls robustness and is chosen via validation. The framework demonstrates strong performance across heavy-tailed inventory, geographically shifted housing, and demographically shifted text classification, offering robust yet efficient decision-making under distributional shifts with flexible centre choices (Bayesian, frequentist, or empirical). The work advances robust optimisation by making contamination-based DRO well-posed, interpretable, and computationally tractable in continuous settings, with practical implications for reliability under real-world distributional shifts.

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.

Figures (13)

• Figure 1: Equivalence between IP and DRO, with an example when $\mathcal{A}$ and $\mathcal{M}$ are Huber contamination sets.
• 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).
• Figure 3: Student-$t$ newsvendor (cost: lower-left is better). Top row: OOS mean--variance frontiers for a range of $\varepsilon_{\mathop{\mathrm{LV}}\nolimits}\in(0,1]$; $\varepsilon_{\mathop{\mathrm{KL}}\nolimits}\in (0,25]$. OR-WDRO uses $\varepsilon_{\mathop{\mathrm{LV}}\nolimits}$ in $(0,0.5)$. Each point represents one $\varepsilon$ value. Some $\varepsilon_{\mathop{\mathrm{LV}}\nolimits}$ and $\varepsilon_{\mathop{\mathrm{KL}}\nolimits}$ are marked. Bottom row: $\text{MSD}=\tfrac{1}{2}(\text{OOS mean}+\text{OOS SD})$ versus $\varepsilon$; shaded regions: $95\%$ confidence bands. From left to right: $\{0,0.1,0.2\}$ contamination.
• Figure 4: Bulk sets for three two-dimensional data-generating processes (left to right): Gaussian, contaminated Gaussian, and Student-$t$. For each process we show (top to bottom): (i) the certified ellipsoidal bulk set $\widehat{\Xi}_0$ at mass $1-\gamma$ and guarantee $1-\delta$; and (ii) the certified axis-aligned box bulk set. All plots correspond to $\gamma=0.05$, $\delta=0.05$, and $n=6000$ with $\left\lvert\mathcal{D}_{\mathrm{fit}}\right\rvert = \left\lvert\mathcal{D}_{\mathrm{select}}\right\rvert = 3000$. The black ellipsoid marks the $1-\gamma$ highest density region (HDR) of each distribution.
• Figure 5: Empirical CDF and DKW lower envelope for the ellipsoid bulk sets displayed in Fig. \ref{['fig:bulk-dkw']} under three data-generating processes: Gaussian (left), contaminated Gaussian (middle), and Student-$t$ (right).

Theorems & Definitions (37)

• Theorem 2.1: Worst-case risk of support-restricted LV set, proved in Appendix \ref{['proof:worst-case-risk']}.
• Corollary 2.2: Worst-case distribution, proved in Appendix \ref{['proof:orst-case-distribution-lv']}
• Proposition 2.3: Divergence-based classification of the contamination set, proved in Appendix \ref{['proof:credal-ball']}
• Proposition 2.4: kuhn2025distributionally, Proposition 6.13
• proof : Sketch of alternative proof of Prop. \ref{['lem:TV-risk']}
• Remark 3.1
• Lemma 3.2: High-probability bulk-mass certificate, proved in Appendix \ref{['proof:bulk-coverage-dkw']}
• Remark 3.3
• Theorem 3.4: Risk certificate, proved in Appendix \ref{['proof:certificate']}
• Remark 5.1: Why an ellipsoid-interval product bulk set instead of an ellipsoid?