Contextual Distributionally Robust Optimization with Causal and Continuous Structure: An Interpretable and Tractable Approach

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 the optimal policy, we propose a soft regression forest (SRF) decision rule that preserves the interpretability of classical decision trees while remaining parametric and differentiable. To solve Causal-SDRO with parametric decision rules, we present an efficient stochastic compositional gradient algorithm that converges to an $\varepsilon$-stationary point at a rate of $\mathcal{O}(\varepsilon^{-4})$, matching the convergence rate of standard stochastic gradient descent. Numerical results demonstrate the superior performance and interpretability of our method.