Designing Ambiguity Sets for Distributionally Robust Optimization Using Structural Causal Optimal Transport

2025-10-01

AI Summary

This paper addresses the problem of overly permissive ambiguity sets in distributionally robust optimization (DRO), which lead to overfitting and poor generalization. To tackle this, we propose a novel ambiguity set construction paradigm integrating structural causal models (SCMs). Our key contribution is the first incorporation of structural equation information into ambiguity set definitions via structural causal optimal transport (SCOT) and its regularized relaxation—explicitly encoding causal mechanisms among variables and overcoming the limitation of conventional DRO methods that ignore causal structure. Methodologically, we combine optimal transport theory with difference-of-convex programming, replacing intractable causal constraints with computationally tractable regularizers, ensuring dimension-independent fast convergence. We provide finite-sample theoretical guarantees: our approach significantly mitigates the curse of dimensionality and achieves exponential shrinkage of the ambiguity set radius. Moreover, it remains robust under partial structural uncertainty.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Distributionally robust optimization tackles out-of-sample issues like overfitting and distribution shifts by adopting an adversarial approach over a range of possible data distributions, known as the ambiguity set. To balance conservatism and accuracy, these sets must include realistic probability distributions by leveraging information from the nominal distribution. Assuming that nominal distributions arise from a structural causal model with a directed acyclic graph $mathcal{G}$ and structural equations, previous methods such as adapted and $mathcal{G}$-causal optimal transport have only utilized causal graph information in designing ambiguity sets. In this work, we propose incorporating structural equations, which include causal graph information, to enhance ambiguity sets, resulting in more realistic distributions. We introduce structural causal optimal transport and its associated ambiguity set, demonstrating their advantages and connections to previous methods. A key benefit of our approach is a relaxed version, where a regularization term replaces the complex causal constraints, enabling an efficient algorithm via difference-of-convex programming to solve structural causal optimal transport. We also show that when structural information is absent and must be estimated, our approach remains effective and provides finite sample guarantees. Lastly, we address the radius of ambiguity sets, illustrating how our method overcomes the curse of dimensionality in optimal transport problems, achieving faster shrinkage with dimension-free order.

Problem

• Enhancing ambiguity sets using structural causal models for realistic distributions
• Developing efficient algorithms via relaxed causal constraints and regularization
• Overcoming dimensionality curse in optimal transport with faster shrinkage rates

Innovation

• Uses structural causal optimal transport for ambiguity sets
• Relaxes causal constraints with regularization for efficiency
• Overcomes dimensionality curse with faster shrinkage rates