Adversarially Robust Control of Conditional Value-at-Risk via Rockafellar-Uryasev Conformal Inference

We present an online, distribution-free framework for controlling the Conditional Value-at-Risk ($\operatorname{CVaR}$), extending conformal tail risk control to non-stationary and adversarial environments. Unlike classical risk control methods, which rely on stationarity or linearity of expectation, our approach provides provable safety guarantees for a nonlinear tail risk functional under arbitrary data-generating processes that may drift or shift strategically over time. By leveraging deep connections between conformal tail risk control, online learning, and the variational representation of $\operatorname{CVaR}$ introduced by Rockafellar and Uryasev, we develop a novel procedure for online $\operatorname{CVaR}$ control with adversarial regret guarantees. The proposed method operates without assumptions on the underlying data-generating process, making it broadly applicable in modern high-stakes deployment settings. We prove that the realized empirical $\operatorname{CVaR}$ is asymptotically controlled at the target level, and that the resulting control is asymptotically tight up to a finite-sample ${O}(1/\sqrt{T})$ conservatism gap. We demonstrate the effectiveness of our approach on portfolio risk management and toxicity mitigation for Large Language Models (LLMs), where rare but catastrophic failures dominate system risk.

We propose a new method for keeping rare but serious failures under control in systems that make decisions over time. This is important in settings such as financial risk management or large language model deployment, where the most harmful outcomes may happen infrequently but can have large consequences. Many existing methods rely on the assumption that the data are generated in a stable way. In practice, this is often unrealistic, markets can shift, users can change their behavior, and deployed AI systems may face new or adversarial inputs. Our method is designed to work even when the environment changes over time and without requiring a model of how the data are generated. The main goal is to control the average severity of the worst outcomes, rather than only controlling average performance. Our algorithm adapts as new data arrive and automatically adjusts the level of caution needed to keep tail risk near a user-specified target. We prove that, over time, the method keeps realized tail risk under control without becoming unnecessarily conservative. We demonstrate its effectiveness in two high-stakes applications: managing tail risk in financial portfolios and toxic outputs from large language models. These examples show how our approach can help make deployed systems safer in changing and unpredictable environments.