Contextual Distributionally Robust Optimization with Causal and Continuous Structure: An Interpretable and Tractable Approach
Fenglin Zhang, Jie Wang
TL;DR
The paper addresses contextual DRO under distributional shift by preserving the causal structure and continuity of the underlying distribution through a causal Sinkhorn discrepancy, enabling robust yet non-insight-discarding decisions. It introduces the Soft Regression Forest (SRF), a differentiable, interpretable ensemble of soft decision trees that approximates optimal policies within arbitrary function spaces and supports end-to-end optimization. The inner DRO problem is tackled via a strong dual formulation, yielding a worst-case distribution that is a mixture of Gibbs distributions; a gradient-based stochastic compositional algorithm with convergence rate $\mathcal{O}(\varepsilon^{-4})$ ensures tractable learning of the parametric policy. Empirical results on Newsvendor, inventory substitution, and portfolio selection demonstrate improved out-of-sample performance and provide intrinsic interpretability through global and local measures, underscoring practical impact in decision-making under uncertainty.
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.