Organizations: School of Management and Engineering, Nanjing University, Nanjing, China · School of Mathematics, Nanjing University, Nanjing, China · Stern School of Business, New York University, New York, USA
Offline contextual pricing often relies on logged observational data, but faces challenges from distributional shifts between training and deployment environments. Distributionally robust optimization (DRO) provides a principled approach to off-policy evaluation and learning (OPE/L). However, existing methods are mostly limited to discrete actions. Recent work has explored DRO for continuous treatments using inverse propensity weighting (IPW), while such IPW-based estimators can be sensitive to the convergence rate of propensity score estimates, particularly when estimated nonparametrically, which may lead to larger estimation errors and regret. In this work, we develop a doubly robust (DR) framework for distributionally robust OPE/L in continuous pricing settings. For evaluation, we propose a localized DR estimator that addresses the computational challenges of worst-case expectations by fitting only a small number of regressions, comparable to standard non-robust DR, while achieving semiparametric efficiency under mild product rate conditions. For learning, we leverage the inherent smoothness of demand noise to handle pricing-specific discontinuities in revenue outcomes (e.g., threshold-based purchase decisions), establishing a finite-sample regret bound of O~p(T−s/(2s+1)) for smoothness orders s=1,2. This bound improves upon existing regret rates in existing DRO-based off-policy learning (OPL) for continuous treatments. Extensive experiments under various levels of distribution shift validate our proposed framework.
Figures & tables
Paper
Action Space
Distributional Shifts?
Regret Bound
Kitagawa and Tetenov (2018) ; Athey and Wager (2021) ; Mbakop and Tabord-Meehan (2021)
Binary
×
Op(T−1/2)
Dudík et al. (2011) ; Swaminathan and Joachims (2015) ; Kallus (2018) ; Zhou et al. (2023)
Discrete
×
Op(T−1/2)
Kallus and Zhou (2018) ; Colangelo and Lee (2026) ; Ai et al. (2026)
Continuous
×
Op(T−s/(2s+1))
Kallus et al. (2022) ; Si et al. (2023) ; Wang et al. (2025)
Discrete
√
Op(T−1/2)
Leung et al. (2025)
Continuous
√
Op(T−1/4)
Ours
Continuous
√
O~p(T−s/(2s+1))
Table 1: Comparison of convergence rates in the OPL literature. We report regret upper bounds with respect to sample size T , treating policy class complexity measures (e.g., entropy integrals) as constant factors. For our work and Ai et al. (2026) , s=1,2 corresponds to the smoothness order of the outcome functions, while Colangelo and Lee (2026) and Leung et al. (2025) require three-times differentiability ( s=3 ) of conditional outcome densities w.r.t. treatment. Here O~ hides logarithmic factors in T .
Algorithm 1 Localized Doubly Robust DRO OPE for Continuous Pricing (LDR 2 O 2 PE-CP)
Figure 1: Comparison of LDR 2 O 2 PE-CP against DRO-IPW in the distributionally robust OPE task. Results are shown for different bandwidth choices and test environments δ∈{0.1,0.2,0.3} with δtrain=0.2 and 2500 test samples. The x-axis is the training sample size T , and the y-axis is the mean squared error (MSE) of the estimated distributionally robust value Rδ .
Figure 2: Comparison of CDR 2 O 2 PL-CP against the non-robust DR and DRO-IPW in the distributionally robust OPL task. Results are shown for different bandwidth choices and test environments δ∈{0.1,0.2,0.3} with δtrain=0.2 and 2500 test samples. The x-axis is the training sample size T , and the y-axis is the distributionally robust policy value Rδ .
Method
R^adv
π^Lin
6.201±0.683
π^DRO−IPW
6.254±0.873
π^CDR2O2PL−CP
6.350±0.598
Table 2: Comparison of robust performance in Expedia study.
We propose a framework for contextual distributionally robust optimization (DRO) that considers the causal and continuous structure of the underlying distribution, and we develop an interpretable and tractable decision rule. We first introduce the causal Sinkhorn discrepancy (CSD), an entropy-regularized causal Wasserstein distance that encourages continuous transport plans while preserving causal consistency. We then formulate a contextual DRO model with a CSD-based ambiguity set, termed Causal Sinkhorn DRO (Causal-SDRO), and derive its strong dual reformulation, where the worst-case distribution is characterized as a mixture of Gibbs distributions. To obtain an (infinite-dimensional) optimal policy, we propose a soft regression forest (SRF) decision rule: it preserves the interpretability of classical decision trees while being fully parametric, differentiable, and Lipschitz-smooth, enabling intrinsic interpretation from both global and local perspectives. To solve the Causal-SDRO with parametric decision rules, we develop an efficient stochastic compositional gradient algorithm that converges to an ε-stationary point at a rate of O(ε−4), matching that of standard stochastic gradient descent. Finally, we validate our method through numerical experiments on synthetic and real-world datasets, demonstrating its superior performance and interpretability.
Fenglin Zhang, Jie Wang
School of Artificial Intelligence The Chinese University of Hong Kong, Shenzhen Shenzhen, 518172, China · School of Artificial Intelligence, School of Data Science The Chinese University of Hong Kong, Shenzhen Shenzhen, 518172, China
We study Bayesian Optimisation (BO) in settings where the objective function is influenced by uncontrollable environmental contexts governed by an unknown probability distribution. In practice, the contextual distribution must be estimated from empirical data, a process that inherently introduces distributional mismatch, producing sub-optimal results. While Distributionally Robust Optimisation (DRO) provides a framework to mitigate these risks, existing robust BO methods frequently suffer from high computational complexity, rely on discretisation of continuous context spaces, or impose restrictive assumptions on the structure of the ambiguity set. To overcome these limitations, we propose Ensemble Distributionally Robust Bayesian Optimisation (EDRBO). Our framework leverages the expressive power of ensemble surrogate models to approximate the black-box function while simultaneously accounting for contextual uncertainty. By utilising Wasserstein ball as ambiguity sets, EDRBO provides a robustified acquisition function that remains computationally tractable and natively handles continuous context spaces. We establish a rigorous theoretical foundation for our approach by proving sublinear cumulative regret guarantees of order O(γTT), where γT represents the maximum information gain within the ensemble. Finally, we provide extensive empirical evaluations that corroborate our theory and demonstrate the state-of-the-art performance of EDRBO.
We study distributionally robust optimization (DRO) for robust inference when the worst-case distribution is continuous, leading to significant computational challenges due to the infinite-dimensional nature of the optimization problem. Unlike traditional discrete DRO approaches, which often suffer from scalability issues, limited generalization, and costly worst-case inference, our framework exploits Brenier's theorem to characterize the least favorable distribution as the pushforward of a transport map from a continuous reference measure. This characterization motivates our study of the minimax problem in Wasserstein space. We propose an iterative algorithmic framework with multiple variants and establish global convergence guarantees under mild assumptions, deriving complexity bounds in terms of subgradient evaluations and inexact Jordan-Kinderlehrer-Otto updates. Numerical results with neural network-based transport maps demonstrate that the proposed method enables both stable training of robust classifiers and effective worst-case inference for classification tasks.
Linglingzhi Zhu, Yunqin Zhu, Yao Xie
1H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology