Abstract
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.
Abstract (translated)
在这篇论文中，我们提出了一种框架用于上下文分布鲁棒优化（DRO），该框架通过开发可解释且易于处理的决策规则来考虑潜在分布的因果和连续结构。这些决策规则使用协变量来制定决定。 首先，我们引入了因果Sinkhorn偏差（CSD），这是一种带熵正则化的因果Wasserstein距离，它鼓励连续的运输计划同时保持因果一致性。然后，我们提出了一种基于CSD不确定性集的上下文DRO模型，并将其命名为因果Sinkhorn DRO（Causal-SDRO）。我们推导了其强对偶重构形式，在其中最坏情况下的分布被描述为吉布斯分布的混合体。 为了求解相应的无限维策略优化问题，我们提出了软回归森林（SRF）决策规则。该规则可以在任意可测量的功能空间中逼近最优政策。与经典的决策树相比，SRF保持了其解释性的同时，也具有完全参数化、可微和Lipschitz光滑的特点，从而可以实现全局和局部视角的内在解释。 为了解决带有参数决策规则的Causal-SDRO问题，我们开发了一种高效的随机合成梯度算法。该算法以$O(\varepsilon^{-4})$的速度收敛到$\varepsilon$次平稳点，其速度与标准随机梯度下降一致。 最后，通过在合成和实际数据集上的数值实验验证了我们的方法的有效性和解释性，展示了它的优越性能。
URL
https://arxiv.org/abs/2601.11016
