On the Generalization and Robustness in Conditional Value-at-Risk

2026-02-20

AI Summary

This work addresses the lack of theoretical guarantees for Conditional Value-at-Risk (CVaR) learning under heavy-tailed and contaminated data, where endogenous quantile estimation often leads to threshold sensitivity and unstable decisions. Through a learning-theoretic analysis of CVaR empirical risk minimization in such settings, this study uncovers—for the first time—the error structure driven by the CVaR threshold and proposes a truncated median-of-means CVaR estimator. By integrating Bahadur–Kiefer-type expansions, minimax optimal rate analysis, techniques for β-mixing sequences, and robust statistical methods, the paper establishes high-probability generalization and excess risk bounds under weak moment conditions. The proposed estimator achieves minimax optimal convergence rates under adversarial contamination and precisely characterizes the boundary conditions that determine whether CVaR learning is generalizable or inherently unstable.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Stochastic Optimization

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: 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

Conditional Value-at-Risk (CVaR) is a widely used risk-sensitive objective for learning under rare but high-impact losses, yet its statistical behavior under heavy-tailed data remains poorly understood. Unlike expectation-based risk, CVaR depends on an endogenous, data-dependent quantile, which couples tail averaging with threshold estimation and fundamentally alters both generalization and robustness properties. In this work, we develop a learning-theoretic analysis of CVaR-based empirical risk minimization under heavy-tailed and contaminated data. We establish sharp, high-probability generalization and excess risk bounds under minimal moment assumptions, covering fixed hypotheses, finite and infinite classes, and extending to $\beta$-mixing dependent data; we further show that these rates are minimax optimal. To capture the intrinsic quantile sensitivity of CVaR, we derive a uniform Bahadur-Kiefer type expansion that isolates a threshold-driven error term absent in mean-risk ERM and essential in heavy-tailed regimes. We complement these results with robustness guarantees by proposing a truncated median-of-means CVaR estimator that achieves optimal rates under adversarial contamination. Finally, we show that CVaR decisions themselves can be intrinsically unstable under heavy tails, establishing a fundamental limitation on decision robustness even when the population optimum is well separated. Together, our results provide a principled characterization of when CVaR learning generalizes and is robust, and when instability is unavoidable due to tail scarcity.

Problem

• Conditional Value-at-Risk
• heavy-tailed data
• generalization
• robustness
• risk-sensitive learning

Innovation

• Conditional Value-at-Risk
• heavy-tailed data
• generalization bounds
• robust estimation
• Bahadur-Kiefer expansion