Risk-Sensitive Reinforcement Learning with Smoothed Quantile Objectives

Reinforcement Learning (RL) has achieved tremendous success in recent years. However, the classical foundations of RL do not account for the risk sensitivity of the objective function, which is critical in various fields, including healthcare, finance, etc. A popular approach to incorporate risk sensitivity is to optimize a specific quantile of the cumulative reward distribution. However, exact quantile objectives are non-smooth and can change abruptly under small perturbations of the return distribution, making them difficult to optimize reliably when the transition model must be learned from data. Motivated by this instability, we develop UCB-BQRL, a model-based optimistic learning algorithm that maintains confidence sets for the transition kernel and plans using a lower-buffered quantile criterion. The buffered criterion smooths the exact quantile objective by averaging nearby lower quantiles, thereby improving stability under transition-estimation error. To compute the buffered-quantile policy at each episode, we introduce EVI-BQ, an exact dynamic-programming procedure. We establish a high-probability regret bound for UCB-BQRL, which up to logarithmic factors scales as $\mathcal{O}(\mathrm{e}^{τ/ρ_τ}+H^2\sqrt{SAT})$, where $ρ_τ$ is denoted as the root-level left-plateau threshold, which is a problem-dependent constant. Further, we establish an information-theoretic lower bound of $Ω(H/ρ_τ\sqrt{AT})$ for the regret of any algorithm dealing with a quantile objective function. Finally, we prove that the exact point-quantile evaluation and exact lower-buffered quantile evaluation are PP-hard under polynomial-time Turing reductions, even for a fixed policy in a two-state, one-action finite-horizon MDP.

Robust Risk Under Evolving Uncertainty: A Wasserstein Counterpart of the Entropic Value-at-Risk

An agent still learning its environment should be cautious while ignorant and bold once confident. The entropic value-at-risk captures this through a robust-optimization identity---a confidence level fixes the radius of a relative-entropy ball of alternative models---but that ball cannot reach catastrophes the nominal deems impossible, precisely what a safe agent must hedge. We instead use an optimal-transport ball and study the coherent risk measure it induces, the Wasserstein entropic value-at-risk. It has a variational dual mirroring the entropic formula (an inverse temperature becomes a transport price), occupies a definite place in the risk hierarchy, and provably accounts for the reachable catastrophes the entropic measure ignores; we verify both dualities numerically. Driving the transport radius by belief entropy then yields a closed-form robust dynamic-programming operator whose caution contracts as the belief sharpens, with a certified safety sandwich and a sharp safety switch.

Quantifying Risk Under Evolving Uncertainty: Belief-Dependent Robustness for Safe Sequential Decision Making

How cautious should an agent be while it is still learning its environment? We propose RATTL (Risk-Adversarial Total-Reward Learning), which ties caution to epistemic uncertainty: the agent holds a Bayesian posterior over unknown dynamics and plans against a Wasserstein ambiguity set whose radius is a monotone function of that posterior. The radius contracts with evidence, so behaviour interpolates continuously between worst-case robustness and risk-neutral total-reward maximization. The design follows the duality underlying the Entropic Value-at-Risk, which converts the choice of a risk level into the choice of an ambiguity radius. We show the resulting planning problem is well posed under transience and compactness conditions, and prove a Safety Sandwich: the RATTL value lies between the uninformed robust value and the full- knowledge optimum, with a gap that vanishes as the posterior concentrates. In a canonical binary-hazard instance, the induced criterion reduces to Conditional Value-at-Risk at a level set by the posterior entropy. A worked example shows the agent deferring the efficient action until a sharp identification threshold. RATTL targets runtime safety for agents, including LLM-based systems, acting under uncertainty.

Finite-Time Analysis of Discounted Exponential-Utility Reinforcement Learning

Discounted exponential utility provides a principled criterion for risk-sensitive sequential decision-making, but its nonlinear structure complicates reinforcement learning. A recent work \citep{thoppe2026reinforcement} addressed this difficulty by introducing a Bellman-compatible surrogate and two model-free fixed-point algorithms for optimizing it over stationary policies. However, their main convergence results are asymptotic. In this work, we establish finite-time rates of $\tilde{O} (1/\sqrt{n})$ for the aforementioned two algorithms under asynchronous Markovian sampling, where $n$ is the iteration index and $\tilde{O}$ hides logarithmic expressions. Importantly, we employ parameter-free choices for the stepsize parameter to derive these rate results. For the algorithmically simpler one-timescale method, the main challenge is that its update equation is not directly aligned with the contraction geometry of its underlying power-law operator. We overcome this mismatch by exploiting the boundedness, monotonicity, and homogeneity of the operator to obtain a local pseudo-contraction property for the relative-error dynamics. We then use a Moreau-envelope-based Lyapunov function and Polyak--Ruppert averaging to obtain the stated convergence rate with parameter-free stepsizes. For the two-timescale method, the main challenge is to control a tracking error on the faster timescale. These results provide the first finite-time guarantees for model-free discounted exponential-utility reinforcement learning.

A Noise-Robust Elicit-to-Optimize Framework for Distortion Riskmetrics via Inverse Reinforcement Learning

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.