Distributionally Robust Optimization as a Scalable Framework to Characterize Extreme Value Distributions (2408.00131v1)

Abstract: The goal of this paper is to develop distributionally robust optimization (DRO) estimators, specifically for multidimensional Extreme Value Theory (EVT) statistics. EVT supports using semi-parametric models called max-stable distributions built from spatial Poisson point processes. While powerful, these models are only asymptotically valid for large samples. However, since extreme data is by definition scarce, the potential for model misspecification error is inherent to these applications, thus DRO estimators are natural. In order to mitigate over-conservative estimates while enhancing out-of-sample performance, we study DRO estimators informed by semi-parametric max-stable constraints in the space of point processes. We study both tractable convex formulations for some problems of interest (e.g. CVaR) and more general neural network based estimators. Both approaches are validated using synthetically generated data, recovering prescribed characteristics, and verifying the efficacy of the proposed techniques. Additionally, the proposed method is applied to a real data set of financial returns for comparison to a previous analysis. We established the proposed model as a novel formulation in the multivariate EVT domain, and innovative with respect to performance when compared to relevant alternate proposals.

• The paper introduces DRO estimators that robustify max-stable EVT distributions using optimal transport and neural network methodologies.
• It validates the approach on synthetic data, showing enhanced error performance and preservation of critical structural properties.
• In real-world financial applications, the framework outperforms traditional models by more reliably quantifying extreme event risks.

Distributionally Robust Optimization as a Scalable Framework to Characterize Extreme Value Distributions

The paper, titled "Distributionally Robust Optimization as a Scalable Framework to Characterize Extreme Value Distributions," offers a comprehensive approach towards developing distributionally robust optimization (DRO) estimators specifically tailored for multidimensional Extreme Value Theory (EVT) statistics. This work addresses significant challenges associated with modeling extreme events, where data scarcity and potential model misspecification present substantial obstacles. The authors propose DRO as a means to construct estimators that can robustify these tail distributions in a pragmatic and theoretically sound manner.

Introduction

Extreme events are pivotal across multiple domains such as finance, climate science, and medicine, due to their significant impacts despite their rarity. EVT provides a foundation for estimating the distribution of these extreme events by focusing on the limiting behavior of maxima of random samples, leading to max-stable distributions. However, these models assume large sample sizes, an assumption that is often violated in practical scenarios where extreme data points are scarce.

The primary contribution of this paper is the formulation of DRO estimators that incorporate semi-parametric max-stable constraints within the space of point processes, thereby enhancing the robustness and generalizability of these estimators. This novel approach aims to mitigate over-conservatism and improve out-of-sample performance by maintaining the max-stable properties inherent to EVT.

Theoretical Framework

Max-Stable Distributions

Max-stable distributions are central to EVT as they provide a semi-parametric class that captures the asymptotic distributional limits of maxima of i.i.d samples. The paper elaborates on the properties of these distributions, including their decomposition through radial and spectral components, which allows for structured modeling of extremal dependence.

Distributionally Robust Optimization (DRO)

DRO is framed as a zero-sum game where the statistician aims to mitigate the impact of adversarial perturbations that deviate from a nominal or baseline distribution. By leveraging the Wasserstein distance to quantify these perturbations, the authors ensure that the robustified distribution respects the extrapolative properties of EVT. The theoretical formulation is expressed in a primal-dual setup, making the approach computationally feasible.

Methodology

The paper explores two primary methods for robustifying max-stable distributions: one based on optimal transport for point processes, and another leveraging a neural network architecture. The optimal transport approach ensures that adversarial perturbations explore non-parametric models while preserving the MEV characteristics. Using a neural network provides flexibility in parameterizing model uncertainty via the Wasserstein metric.

Results

The proposed methodologies are validated against synthetic and real-world data. The synthetic datasets are specifically constructed to introduce pathological scenarios that challenge traditional EVT assumptions, confirming the enhanced performance and generalizability of the robustified models. In real-world applications, such as financial returns, the DRO framework demonstrates superior risk estimation compared to baseline models, proving its practical relevance.

1. Synthetic Data: Experiments with synthetic data underscore the robustness and improved error performance of the DRO-constrained approaches compared to unconstrained models. Results indicate that constrained models manage to retain essential structural properties while enhancing robustness against model uncertainty.
2. Real Data: The application to financial returns illustrates that the DRO framework is capable of more reliably estimating risks within a high-stake domain. This experiment highlights the practical value of adopting such robustification techniques in real-world risk-sensitive applications.

Implications and Future Directions

The theoretical and empirical outcomes from this paper suggest several implications for both practical and theoretical developments in EVT and DRO. Practically, the robustified models are poised to offer better risk measures in fields where extreme events are of critical concern. Theoretically, the bridge built between optimal transport, neural networks, and EVT paves the way for further explorations, particularly in extending the framework to more complex and high-dimensional settings.

The paper suggests future research could address optimizing policies under worst-case risk scenarios, such as portfolio optimization using CVaR, as well as extending max-stable processes to more generalized forms. Additionally, other potential directions include addressing computational challenges associated with the primal-dual optimization, especially in cases with small perturbation budgets.

Conclusion

In summary, this paper makes substantial contributions by providing a scalable, theoretically robust, and empirically validated framework for handling extreme value distributions through distributionally robust optimization. By integrating advanced methodologies like optimal transport and neural networks within an EVT context, the authors offer a powerful tool that enhances both the reliability and applicability of statistical models dealing with extremes. This work undoubtedly opens new avenues for responses to rare, high-impact events across diverse disciplines.