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 (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 CVaR introduced by Rockafellar and Uryasev, we develop a novel procedure for online 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 CVaR is asymptotically controlled at the target level, and that the resulting control is asymptotically tight up to a finite-sample 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.