We study finite-horizon MDP planning under \emph{root-based} (resolute) risk objectives that apply a rank-dependent functional to the distribution of total returns. Such objectives are non-linear in the return distribution and generally break Bellman optimality, so direct optimization by scenario-tree enumeration is intractable. We propose \textbf{ERQDP}, an enumeration-free and sampling-free method that solves a rank--quantile surrogate via exact DP (Dynamic Programming), evaluates candidate policies exactly by DP over return Probability Mass Functions (PMFs) on a discretized return grid (with an explicit rounding bound), and refines the surrogate in an anytime loop that reports an explicit upper--lower gap (certificate) for the target objective up to discretization budgets. Across tested benchmarks, ERQDP returns certified solutions or explicit residual gaps, enables fast risk-parameter sweeps with substantial runtime gains, and supports both risk-averse and risk-seeking behaviors.
We study general-utility Markov decision processes (GUMDPs) with risk-aware objectives. In this framework, an agent aims to optimize a risk measure of the distribution of objective values, where the objective function depends on the frequency of visitation of states induced by the agent's policy. First, we motivate, propose, and formalize risk-aware GUMDPs, which enable agents and decision makers to trade off expected performance by risk aversion while benefiting from the rich set of objectives that can be cast under the framework of GUMDPs. We focus our attention on the entropic risk measure (ERM). Second, we show how we can solve risk-aware GUMDPs with ERM objectives by resorting to online planning techniques. In particular, we propose an approach based on Monte Carlo Tree Search (MCTS) to provably solve risk-aware GUMDPs up to any desired accuracy. Third, we provide a set of experimental results showcasing that our approach is successful when optimizing for a spectrum of risk-aware behaviors in the context of GUMDPs under diverse tasks (standard MDPs, maximum state entropy exploration, imitation learning, and multi-objective MDPs).
Developing efficient function-approximation methods for policy evaluation is a fundamental challenge in risk-aware reinforcement learning. Existing approaches either focus on restrictive classes of risk measures or rely on access to a simulator, limiting their applicability in fully online settings. In this work, we propose computationally efficient online learning algorithms for policy evaluation in Markov decision processes (MDPs) with dynamic utility-based shortfall risk (UBSR) measures under linear function approximation. Specifically, we introduce the UBSR-TD algorithm, establish conditions under which it converges almost surely, and develop several variants designed to accelerate convergence. Our formulation shows that existing policy evaluation algorithms for risk-neutral MDPs can be readily adapted to dynamic UBSR settings by incorporating a loss function into the temporal-difference error. Numerical experiments support our theoretical findings, and an application to a perishable inventory management problem with shelf-life uncertainty demonstrates the practical effectiveness of the proposed methods.
In online reinforcement learning, data scarcity creates epistemic uncertainty that makes robustness important early in learning, whereas sufficient exploration is needed to learn the true-environment optimal policy. We study this time-varying robustness--exploration trade-off through a quantile Bayesian risk-aware Markov decision process (BR-MDP), in which the quantile level controls how posterior uncertainty enters the Bellman backup. We characterize this control through an asymptotic normality result for the difference between the quantile BR-MDP value and the value in the true environment. The result implies that upper/lower-tail quantiles induce optimism/pessimism towards epistemic uncertainty, and the magnitude of the optimism/pessimism decreases as data accumulate. Building on this characterization, we propose an online Bayesian risk-aware algorithm with an adaptive quantile schedule that emphasizes robustness early and gradually encourages exploration of less-visited state--action pairs. We establish sublinear Bayesian regret bounds with respect to both the true optimal value and the optimal BR-MDP robust value. Numerical experiments demonstrate strong performance in both exploration-demanding and exploration-costly environments.