Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning (1908.08729v2)

Abstract: Many decision problems in science, engineering and economics are affected by uncertain parameters whose distribution is only indirectly observable through samples. The goal of data-driven decision-making is to learn a decision from finitely many training samples that will perform well on unseen test samples. This learning task is difficult even if all training and test samples are drawn from the same distribution -- especially if the dimension of the uncertainty is large relative to the training sample size. Wasserstein distributionally robust optimization seeks data-driven decisions that perform well under the most adverse distribution within a certain Wasserstein distance from a nominal distribution constructed from the training samples. In this tutorial we will argue that this approach has many conceptual and computational benefits. Most prominently, the optimal decisions can often be computed by solving tractable convex optimization problems, and they enjoy rigorous out-of-sample and asymptotic consistency guarantees. We will also show that Wasserstein distributionally robust optimization has interesting ramifications for statistical learning and motivates new approaches for fundamental learning tasks such as classification, regression, maximum likelihood estimation or minimum mean square error estimation, among others.

• The paper introduces a convex optimization framework using Wasserstein distances to define ambiguity sets for robust decision-making under uncertainty.
• It establishes finite sample bounds and asymptotic consistency, ensuring reliable out-of-sample performance in high-dimensional settings.
• The approach mitigates overfitting by regularizing models and enables efficient reformulations, benefiting tasks such as classification, regression, and covariance estimation.

Overview of "Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning"

The paper "Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning" authored by Daniel Kuhn and colleagues explores the theoretical framework and computational techniques of Wasserstein distributionally robust optimization (DRO). The paper addresses the challenge of decision-making under uncertainty when the probability distribution of uncertain parameters is observed only through finite samples. The focus is on robust optimization that ensures high performance under the worst-case distribution within a Wasserstein distance from a nominal distribution formed from these samples.

Key Concepts and Methodology

The core idea in this work is the application of Wasserstein distance—a metric from the field of optimal transport—to define ambiguity sets in distributionally robust optimization. A distributionally robust optimization problem formulated with a Wasserstein ambiguity set seeks decisions that remain robust under the most adversarial distribution within this set. This approach accounts for deviations between the empirical distribution (derived from the training data) and the true underlying distribution of data.

Strong Points of Wasserstein DRO

1. Tractability: The authors argue that Wasserstein DRO problems can often be solved via convex optimization formulations. This is a significant advance as it allows for polynomial-time solutions, which is essential for scalability in practical applications.
2. Consistent Out-of-sample Guarantees: Wasserstein DRO offers rigorous guarantees for out-of-sample performance, with the authors providing both finite sample bounds and asymptotic consistency results.
3. Robustness Against Overfitting: By incorporating ambiguity into the optimization process, Wasserstein DRO effectively acts as a regularizer, preventing overfitting to training data—a problem prevalent in many machine learning models.
4. Insights for Statistical Learning: The approach motivates new solutions for classical learning problems such as classification, regression, and estimation by framing them as optimization problems under distributional uncertainty.

Computational Aspects and Numerical Examples

The paper explores computational tractability concerns, proving that Wasserstein DRO problems can be efficiently reformulated as finite-dimensional convex programs in many scenarios. The authors provide reformulations for empirical and elliptical distributions, discuss approximation techniques for large-scale scenarios, and elucidate dual space computations that reveal the structure of worst-case distributions.

For instance, when the loss function in a decision problem is quadratic, the authors illustrate that the worst-case risk evaluation can be solved via a semidefinite program (SDP), making the DRO problem computationally manageable even for high-dimensional data.

Applications in Machine Learning

The authors exemplify the applicability of Wasserstein DRO in machine learning tasks, such as:

• Classification: By minimizing the worst-case expected misclassification error, the approach improves generalization by accounting for data variability. It can emulate regularization effects similar to Lasso and Ridge regressions.
• Regression: Applying similar DRO frameworks, regression models can be made robust against sampling errors and possess regularization-like properties, leading to parsimonious model complexity.
• Covariance Estimation: For covariance matrix estimation, the approach offers a robust alternative to traditional maximum likelihood estimations by considering distributional robustness, useful in finance and econometrics.

Theoretical Implications and Future Directions

The work opens several avenues for future research in robust optimization and machine learning:

• Extensive exploration of adaptive metric choices in defining the Wasserstein distance, potentially leading to better domain-specific DRO models.
• The use of Wasserstein DRO for enhancing ensemble methods by integrating distributional robustness into model aggregation tasks.
• Further theoretical explorations into using DRO for nonlinear and deep learning models to improve their robustness and interpretability.

Conclusion

In summary, the paper presents a quantitative and efficient framework for tackling uncertainties in decision-making and learning problems using Wasserstein distributionally robust optimization. It combines theoretical rigor with practical tractability, making it a valuable reference for researchers and practitioners aiming to enhance model robustness amidst uncertain environmental parameters. The implications on regularization, computational tractability, and robust performance metrics make it a cornerstone contribution to the field of robust optimization and data-driven decision-making.