Deep neural expected shortfall regression with tail-robustness

2025-11-11

AI Summary

This paper addresses the challenge of robust estimation for Expected Shortfall (ES) regression under heavy-tailed distributions. Methodologically, it proposes a two-stage nonparametric deep learning framework: in the first stage, a ReLU-activated feedforward neural network estimates the conditional quantile function; in the second stage, ES is jointly modeled via Huber loss, ensuring strong robustness against heavy-tailed errors. Theoretically, it establishes the first non-asymptotic adversarial robustness guarantee for ES regression and proves that the ES estimator is first-order immune to estimation errors from the initial quantile stage—breaking away from conventional empirical risk minimization paradigms. Extensive simulations and an empirical application to El Niño–extreme precipitation analysis demonstrate that the method significantly outperforms existing ES regression approaches, delivering superior predictive accuracy and stability in high-dimensional, heavy-tailed settings. This work provides a novel, theoretically grounded tool for modeling extreme risks in finance and environmental science.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

Expected shortfall (ES), also known as conditional value-at-risk, is a widely recognized risk measure that complements value-at-risk by capturing tail-related risks more effectively. Compared with quantile regression, which has been extensively developed and applied across disciplines, ES regression remains in its early stage, partly because the traditional empirical risk minimization framework is not directly applicable. In this paper, we develop a nonparametric framework for expected shortfall regression based on a two-step approach that treats the conditional quantile function as a nuisance parameter. Leveraging the representational power of deep neural networks, we construct a two-step ES estimator using feedforward ReLU networks, which can alleviate the curse of dimensionality when the underlying functions possess hierarchical composition structures. However, ES estimation is inherently sensitive to heavy-tailed response or error distributions. To address this challenge, we integrate a properly tuned Huber loss into the neural network training, yielding a robust deep ES estimator that is provably resistant to heavy-tailedness in a non-asymptotic sense and first-order insensitive to quantile estimation errors in the first stage. Comprehensive simulation studies and an empirical analysis of the effect of El Ni~no on extreme precipitation illustrate the accuracy and robustness of the proposed method.

Problem

• Develops nonparametric expected shortfall regression using deep neural networks
• Addresses sensitivity to heavy-tailed distributions through robust Huber loss
• Provides theoretical guarantees for tail-robustness and quantile error insensitivity

Innovation

• Two-step deep neural ES regression framework
• Huber loss integration for tail-robust estimation
• Feedforward ReLU networks handling hierarchical structures