In this paper, we introduce a framework for contextual distributionally robust optimization (DRO) that considers the causal and continuous structure of the underlying distribution by developing interpretable and tractable decision rules that prescribe decisions using covariates. We first introduce the causal Sinkhorn discrepancy (CSD), an entropy-regularized causal Wasserstein distance that encourages continuous transport plans while preserving the 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 solve the corresponding infinite-dimensional policy optimization, we propose the Soft Regression Forest (SRF) decision rule, which approximates optimal policies within arbitrary measurable function spaces. The SRF 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 $\varepsilon$-stationary point at a rate of $O(\varepsilon^{-4})$, matching the convergence rate 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.

翻译：本文提出了一种基于上下文分布鲁棒优化（DRO）的框架，通过发展利用协变量制定决策的可解释且易处理的决策规则，考虑了潜在分布的因果与连续结构。我们首先引入因果Sinkhorn散度（CSD），这是一种熵正则化的因果Wasserstein距离，在保持因果一致性的同时鼓励连续运输方案。随后，我们构建了基于CSD的模糊集上下文DRO模型，称为因果Sinkhorn DRO（Causal-SDRO），并推导出其强对偶重构形式，其中最坏情况分布被刻画为吉布斯分布的混合。为求解相应的无穷维策略优化问题，我们提出软回归森林（SRF）决策规则，该规则可在任意可测函数空间内逼近最优策略。SRF在保持经典决策树可解释性的同时，具有完全参数化、可微且Lipschitz光滑的特性，能从全局和局部视角实现内在解释。针对带参数决策规则的Causal-SDRO求解，我们开发了一种高效的随机复合梯度算法，该算法以$O(\varepsilon^{-4})$的速率收敛到$\varepsilon$-平稳点，与标准随机梯度下降的收敛速度相匹配。最后，通过在合成数据集和真实世界数据集上的数值实验，验证了该方法优异的性能与可解释性。