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：本文提出了一种上下文分布鲁棒优化框架，该框架通过开发利用协变量制定决策的可解释且易处理的决策规则，考虑了底层分布的因果与连续结构。我们首先引入了因果Sinkhorn差异，这是一种熵正则化的因果Wasserstein距离，它在保持因果一致性的同时鼓励连续的传输方案。随后，我们构建了一个基于CSD模糊集的上下文DRO模型，称为因果Sinkhorn分布鲁棒优化，并推导出其强对偶重构形式，其中最坏情况分布被表征为吉布斯分布的混合。为了解决相应的无限维策略优化问题，我们提出了软回归森林决策规则，它能在任意可测函数空间中逼近最优策略。SRF保持了经典决策树的可解释性，同时具有完全参数化、可微分且Lipschitz光滑的特性，从而支持从全局和局部视角进行内在解释。为了求解具有参数化决策规则的Causal-SDRO，我们开发了一种高效的随机组合梯度算法，该算法以$O(\varepsilon^{-4})$的速率收敛到$\varepsilon$-稳定点，与标准随机梯度下降的收敛速率相匹配。最后，我们通过在合成数据集和真实数据集上的数值实验验证了所提方法的优越性能和可解释性。