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

• URL: http://arxiv.org/abs/2602.18053v1
• Date: Fri, 20 Feb 2026 08:10:11 GMT
• Title: On the Generalization and Robustness in Conditional Value-at-Risk
• Authors: Dinesh Karthik Mulumudi, Piyushi Manupriya, Gholamali Aminian, Anant Raj,
• Abstract summary: We develop a learning-theoretic analysis of Conditional Value-at-Risk (CVaR)-based empirical risk minimization under heavy-tailed and contaminated data.<n>We establish sharp, high-probability generalization and excess risk bounds under minimal moment assumptions.<n>We show that CVaR decisions themselves can be intrinsically unstable under heavy tails.
• Score: 12.253712889424584
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $β$-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.

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