Actor-Critic Algorithm for Dynamic Expectile and CVaR

Tail-end risk measures such as static conditional value-at-risk (CVaR) are used in safety-critical applications to prevent rare, yet catastrophic events. Unlike risk-neutral objectives, the static CVaR of the return depends on entire trajectories without admitting a recursive Bellman decomposition in the underlying Markov decision process. A classical resolution relies on state augmentation with a continuous variable. However, unless restricted to a specialized class of admissible value functions, this formulation induces sparse rewards and degenerate fixed points. In this work, we propose a novel formulation of the static CVaR objective based on augmentation. Our alternative approach leads to a Bellman operator with: (1) dense per-step rewards; (2) contracting properties on the full space of bounded value functions. Building on this theoretical foundation, we develop risk-averse value iteration and model-free Q-learning algorithms that rely on discretized augmented states. We further provide convergence guarantees and approximation error bounds due to discretization. Empirical results demonstrate that our algorithms successfully learn CVaR-sensitive policies and achieve effective performance-safety trade-offs.

We study risk-sensitive reinforcement learning in finite discounted MDPs, where a generative model of the MDP is assumed to be available. We consider a family or risk measures called the optimized certainty equivalent (OCE), which includes important risk measures such as entropic risk, CVaR, and mean-variance. Our focus is on the sample complexities of learning the optimal state-action value function (value learning) and an optimal policy (policy learning) under recursive OCE. We provide an exact characterization of utility functions $u$ for which the corresponding OCE defines an objective that is PAC-learnable. We analyze a simple model-based approach and derive PAC sample complexity bounds. We establish that whenever $u$ does not have full domain $\text{dom}(u)\neq \mathbb{R}$, the corresponding problem is not PAC-learnable. Finally, we establish corresponding lower bounds for both value and policy learning, demonstrating tightness in the size $SA$ of state-action space, and for a more restricted class of utilities, we derive lower bounds that makes the dependence on the effective horizon $\frac{1}{1-γ}$ explicit. Specifically, for $\text{CVaR}_τ$ we show that the correct dependence on $τ$ is $\frac{1}{τ^2}$, thus improving by a factor of $\frac{1}τ$ over state-of-the-art although our bound has a suboptimal dependence on $\frac{1}{1-γ}$.

We propose a noise-robust elicit-to-optimize framework that integrates inverse reinforcement learning (IRL) and reinforcement learning (RL) for eliciting agents' risk preferences and optimizing policies under a broad class of risk objectives characterized by distortion riskmetrics. On the elicitation side, we propose an adaptive Bayesian IRL method that infers agents' latent risk objectives from their noisy observed decisions, explicitly allowing agents to take stochastic and suboptimal actions. We establish the existence of a finite set of distinguishing questions that identifies the preferred distortion riskmetric within the candidate class and prove that the convergence rate of the algorithm is of order $O(\exp(-cm+O(\sqrt{m\log m})))$ under general settings, where $c>0$ is a constant and $m$ denotes the number of algorithm iterations. On the optimization side, we develop a model-free RL algorithm for optimizing policies under conditional distortion riskmetrics. By representing the objective as an integral of the conditional cost quantile function with respect to the distortion function, the method unifies distortion-riskmetric objectives. We optimize diverse risk objectives by extending the Proximal Policy Optimization (PPO) algorithm with policy, value, and quantile neural networks, where the quantile network estimates the full conditional cost quantile function and enables numerical evaluation of general risk objectives. A comprehensive empirical study demonstrates the framework's elicitation accuracy and effectiveness in complex financial environments.