Distributionally Robust Optimization (2411.02549v3)

Abstract: Distributionally robust optimization (DRO) studies decision problems under uncertainty where the probability distribution governing the uncertain problem parameters is itself uncertain. A key component of any DRO model is its ambiguity set, that is, a family of probability distributions consistent with any available structural or statistical information. DRO seeks decisions that perform best under the worst distribution in the ambiguity set. This worst case criterion is supported by findings in psychology and neuroscience, which indicate that many decision-makers have a low tolerance for distributional ambiguity. DRO is rooted in statistics, operations research and control theory, and recent research has uncovered its deep connections to regularization techniques and adversarial training in machine learning. This survey presents the key findings of the field in a unified and self-contained manner.

• The paper establishes robust theoretical foundations by systematically defining ambiguity sets and duality frameworks.
• It presents both analytical and numerical methodologies to solve worst-case risk models effectively.
• The paper highlights practical applications in finance, supply chain, and machine learning, bridging optimization with statistical regularization.

Distributionally Robust Optimization: An Insightful Overview

The survey "Distributionally Robust Optimization" presents a comprehensive exploration of Distributionally Robust Optimization (DRO), a field rooted in mathematical optimization, statistics, operations research, and control theory. This essay provides an expert overview of the paper's key findings, implications, and future directions within the DRO landscape.

Summary of Key Findings

The paper systematically addresses how DRO models handle uncertainty in probability distributions governing problem parameters. A crucial aspect of DRO is its reliance on ambiguity sets, which encompass probability distributions consistent with structural or statistical information. The DRO framework seeks decisions optimized for the worst-case distribution within these sets.

The paper is structured to cover foundational elements such as the formulation of ambiguity sets, duality theories for expectation and risk problems, and methodologies for solving nature's subproblem both analytically and numerically.

1. Ambiguity Sets:
   • Moment Ambiguity Sets: Defined through known moments, these sets can range from basic Markov and Chebyshev sets to more complex ones accounting for uncertainties in moments.
   • -Divergence Ambiguity Sets: Facilitated by entropy functions, these sets offer a parametrized measure of discrepancy from a reference distribution.
   • Optimal Transport Ambiguity Sets: Exemplified by Wasserstein distances, these sets provide a measure grounded in transportation costs between distributions.
2. Duality Theories:
   • The survey elucidates duality theories applicable to worst-case expectation and risk problems. It presents weak and strong duality results contingent on the properties of ambiguity sets and loss functions.
   • For moment ambiguity sets and -divergence sets, both weak and strong dual problems are formulated, deriving insightful connections between DRO and regularization in machine learning.
3. Analytical Solutions and Numerical Techniques:
   • Analytical solutions for nature's subproblem are explored under conditions like concavity or convexity of loss functions within specified ambiguity sets.
   • The survey details numerical solution methods, emphasizing robust optimization tactics and statistical guarantees.

Implications and Future Directions

Practical Implications

1. Optimization Under Uncertainty: DRO provides robust modeling for optimization in uncertain environments, impacting finance, supply chain management, and machine learning.
2. Data-Driven Decision Making: The aligned methodologies with statistical learning principles allow for effective integration of DRO models in machine learning, paving the way for designing algorithms that are inherently resistant to distributional shifts.

Theoretical Implications

1. Connections to Risk and Regularization: The survey highlights connections between DRO and advanced statistical methods, such as regularization and coherent risk measures, suggesting a theoretical framework that enriches both fields.
2. Unified Mathematical Framework: By correlating DRO with traditional convex optimization and duality principles, the paper places DRO as a robust extension of these classical theories, contributing to a unified framework for optimization and decision-making under uncertainty.

Future Directions

1. Expanded Ambiguity Sets: The exploration of non-standard divergence and transport methods suggests a fertile ground for future research, possibly leading to DRO models that capture even finer nuances of real-world unpredictability.
2. Computational Advancements: With DRO models' complexity, continued evolution of computational techniques, particularly those leveraging advancements in numerical optimization and machine learning, remain imperative.
3. Theoretical Extensions: Further exploration into the intersectionality of DRO with fields like game theory and adversarial learning could yield innovative approaches to integrating endogenous and exogenous uncertainties within decision-making processes.

Conclusion

The survey on Distributionally Robust Optimization provides a rich, academically rigorous discourse on DRO's foundations, advancements, and practical applications. By thoroughly examining DRO's theoretical underpinnings, this paper contributes to establishing DRO both as a vital extension of optimization disciplines and a catalyst for future research on uncertainty and data-driven decision-making. The implications drawn herein suggest much potential for ongoing innovation at the confluence of mathematical, statistical, and computational methodologies.