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

TL;DR

This paper develops a novel framework for distributionally robust optimization that incorporates causal and continuous structures, introducing interpretable decision rules and efficient algorithms with strong theoretical guarantees.

Contribution

It proposes the Causal Sinkhorn discrepancy for causal Wasserstein distance, formulates a new DRO model, and introduces the Soft Regression Forest decision rule for interpretability and tractability.

Findings

The Causal Sinkhorn DRO effectively captures causal and continuous structures.

The Soft Regression Forest provides interpretable and differentiable decision rules.

Numerical experiments show superior performance and interpretability.

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.