Optimistic Policy Learning under Pessimistic Adversaries with Regret and Violation Guarantees

Real-world decision-making systems operate in environments where state transitions depend not only on the agent's actions, but also on \textbf{exogenous factors outside its control}--competing agents, environmental disturbances, or strategic adversaries--formally, $s_{h+1} = f(s_h, a_h, \bar{a}_h)+ω_h$ where $\bar{a}_h$ is the adversary/external action, $a_h$ is the agent's action, and $ω_h$ is an additive noise. Ignoring such factors can yield policies that are optimal in isolation but \textbf{fail catastrophically in deployment}, particularly when safety constraints must be satisfied. Standard Constrained MDP formulations assume the agent is the sole driver of state evolution, an assumption that breaks down in safety-critical settings. Existing robust RL approaches address this via distributional robustness over transition kernels, but do not explicitly model the \textbf{strategic interaction} between agent and exogenous factor, and rely on strong assumptions about divergence from a known nominal model. We model the exogenous factor as an \textbf{adversarial policy} $\barπ$ that co-determines state transitions, and ask how an agent can remain both optimal and safe against such an adversary. \emph{To the best of our knowledge, this is the first work to study safety-constrained RL under explicit adversarial dynamics}. We propose \textbf{Robust Hallucinated Constrained Upper-Confidence RL} (\texttt{RHC-UCRL}), a model-based algorithm that maintains optimism over both agent and adversary policies, explicitly separating epistemic from aleatoric uncertainty. \texttt{RHC-UCRL} achieves sub-linear regret and constraint violation guarantees.