Risk-Sensitive RL

Safe navigation for mobile robots demands policies that remain reliable under the high-consequence perception uncertainty of cluttered environments. Yet most existing safe reinforcement learning (RL) methods assess safety through average cumulative cost. Such metrics can mask dangerous tail-risk behaviors. To address this, we propose a framework that trains risk-sensitive policies through Conditional Value-at-Risk (CVaR) constrained optimization on an off-policy TD3 backbone and evaluates their safety margins post-training through neural network reachability verification. During training, the policy is optimized under CVaR constraints on cumulative costs, promoting sensitivity to high-cost tail outcomes rather than average behavior alone. After training, we compute action reachable sets under bounded observation uncertainty using Taylor Model analysis, yielding a safety rate metric that quantifies the proportion of evaluated states at which the policy's reachable action set remains within prescribed safety margins. A key finding is that policies trained with CVaR constraints maintain larger safety margins from obstacles across evaluated states. This makes them significantly more amenable to formal reachability verification. Experiments across ten navigation scenarios and six baselines show that our method achieves a 98.3% success rate, the highest safety verification rate among all compared methods, while revealing that average cost rankings and reachability-based safety rankings can diverge. This indicates that reachability verification captures risks which are missed by empirical cost metrics alone. We further validate our approach on a physical Clearpath Jackal robot, demonstrating successful sim-to-real transfer.

We study best-policy identification for finite-horizon risk-sensitive reinforcement learning under the entropic risk measure. Recent work established a constant gap in the exponential horizon dependence between lower and upper bounds on the number of samples required to identify an approximately optimal policy. Precisely, known lower bounds scale in $Ω(e^{|β| H})$ where $H$ is the horizon of the MDP, while the state-of-the-art upper bound achieves at best $O(e^{2|β| H})$ (arXiv:2506.00286v2) using a generative model. We show that this extra exponential factor can be traced to overly loose concentration control for exponential utilities. To close this open gap, we revisit the analysis of this problem through a forward-model based algorithm building on KL-based exploration bonuses that we adapt to the entropic criterion. The improvement we get is due to two main novel technical innovations. We leverage the smoothness properties of the exponential utility to derive sharper concentration bounds, and we propose a new stopping rule that exploits further this tightness to obtain a sample complexity that matches the lower bound.

Reinforcement learning (RL) for exponential-utility optimization in discounted Markov decision processes (MDPs) lacks principled value-based algorithms. We address this gap in the fixed risk-aversion setting. Building on the Bellman-type equation for exponential utility studied in \cite{porteus1975optimality}, we derive two Q-value-style extensions and show that the associated operators are contractions in the $L_\infty$ and sup-log/Thompson metrics, respectively. We characterize their fixed points and prove that the induced greedy stationary policy is optimal for the exponential-utility objective among stationary policies. These structural results lead to two model-free algorithms: a two-timescale Q-learning--style algorithm, for which we establish almost-sure convergence and provide finite-time convergence rates via timescale separation, and a one-timescale algorithm governed by a sublinear power-law operator. Since the latter does not admit a global contraction in standard metrics, we prove its convergence using delicate arguments based on local Lipschitzness, monotonicity, homogeneity, and Dini derivatives, and provide a scalar finite-time analysis that highlights the challenges in obtaining convergence rates in the vector case. Our work provides a foundation for value-based RL under exponential-utility objectives.

Optimizing dynamic risk with stochastic policies is challenging in both policy updates and value learning. The former typically requires transition perturbation, while the latter may rely on model-based approaches. To address these challenges, we propose a surrogate policy gradient without transition perturbation under softmax policy parameterization. We further develop model-free value learning methods for dynamic expectile and conditional value-at-risk by leveraging elicitability. Finally, inspired by Expected SARSA and Expected Policy Gradient, a model-free off-policy actor-critic algorithm is constructed. Empirical results in domains with verifiable risk-averse behavior show that our algorithm can learn risk-averse policy and consistently outperforms other existing methods.

Risk-aware navigation should be selective: a policy should expose evasive degrees of freedom only when the local scene admits a lower-risk feasible maneuver, and suppress them when no safer alternative exists. We show that adding one context-energy term to a port-Hamiltonian navigation policy produces a learned force channel with exactly this falsifiable signature. When the local risk field contains a feasible lower-risk direction, the induced context force activates toward it; when the apparent escape is blocked or not yet available, a route-aware gate suppresses lateral force rather than hallucinating an unsafe maneuver. A CVaR tail-risk objective focuses gradient updates on rare but consequential risk transitions. We validate the selectivity signature across four settings. In the primary delayed-required-escape benchmark, route-aware CVaR reduces premature force activation from 0.950 to 0.180 versus DWA while raising success from 0.480 to 0.810 with zero replans. On real off-road terrain (RELLIS-3D), route-aware enrichment achieves correct activation rate 0.837 and false activation rate 0.114, compared to 0.378/0.752 for scalar risk gradients. On static semantic maps (DFC2018), enrichment reduces catastrophic failure from 0.60 to 0.10 and oscillation by 90.7% while preserving path efficiency. In highway traffic, collisions drop from 100% to 0% when a lane escape is feasible; when no escape exists, the policy suppresses the lateral maneuver. The selectivity property follows from the gradient structure of the context energy rather than from training-time tuning.

Current critic-less RLHF methods aggregate multi-objective rewards via an arithmetic mean, leaving them vulnerable to constraint neglect: high-magnitude success in one objective can numerically offset critical failures in others (e.g., safety or formatting), masking low-performing "bottleneck" rewards vital for reliable multi-objective alignment. We propose Reward-Variance Policy Optimization (RVPO), a risk-sensitive framework that penalizes inter-reward variance during advantage aggregation, shifting the objective from "maximize sum" to "maximize consistency." We show via Taylor expansion that a LogSumExp (SoftMin) operator effectively acts as a smooth variance penalty. We evaluate RVPO on rubric-based medical and scientific reasoning with up to 17 concurrent LLM-judged reward signals (Qwen2.5-3B/7B/14B) and on tool-calling with rule-based constraints (Qwen2.5-1.5B/3B). By preventing the model from neglecting difficult constraints to exploit easier objectives, RVPO improves overall scores on HealthBench (0.261 vs. 0.215 for GDPO at 14B, $p < 0.001$) and maintains competitive accuracy on GPQA-Diamond without the late-stage degradation observed in other multi-reward methods, demonstrating that variance regularization mitigates constraint neglect across model scales without sacrificing general capabilities.

Quantile-based distributional reinforcement learning methods learn return distributions through sampled quantile regression, but their bootstrapped target quantiles may induce distorted or degenerate distribution estimates. We propose Robust Quantile-based Implicit Quantile Networks (RQIQN), a lightweight Wasserstein distributionally robust enhancement boosted from a quantile estimation perspective. We first reinterpret a snapshot of IQN loss as a collection of local empirical quantile estimation problems over sampled current fractions. We then robustify each local slot with a Wasserstein distributionally robust quantile estimation formulation, yielding a closed-form, fraction-dependent correction to the Bellman target. This correction directly addresses distributional degeneration: its median antisymmetry preserves the risk-neutral quantile average, while its monotonicity enlarges upper-lower quantile gaps and counteracts collapsed distributional spread. RQIQN thus regularizes quantile geometry without changing the underlying value objective or requiring additional sample set reconstruction. Finally, we empirically show that the proposed RQIQN outperforms other existing quantile-based distributional reinforcement learning algorithms in risk-sensitive navigation and Atari games.

The increasing penetration of renewable energy has introduced substantial volatility into wholesale electricity markets, complicating the optimal bidding strategies for power producers. Traditional Reinforcement Learning (RL) approaches often struggle to balance profit maximization with risk management, frequently overfitting to specific market conditions or failing to account for the stochastic spread between Day-Ahead (DA) and Real-Time (RT) settlements. To address these challenges, this paper makes two primary contributions. First, we introduce and open-source a high-fidelity gymnasium environment for two-settlement electricity market bidding. Grounded in extensive empirical data from the PJM Interconnection, the environment explicitly models the interplay between DA commitments and RT deviations, providing a standardized testbed for general and risk-sensitive agents. Second, we propose MARS-DA (Multi-Agent Regime-Switching for Day-Ahead markets), a novel hierarchical framework that orchestrates distinct sub-policies for risk management and profit seeking. MARS-DA utilizes a top-level Meta-Controller to dynamically blend the actions of two specialized base agents: a "Safe Agent" that optimizes for reliable DA allocation and a "Speculator Agent" that targets volatile RT arbitrage opportunities. Extensive experiments demonstrate that MARS-DA achieves superior risk-adjusted returns compared to state-of-the-art baselines while maintaining robust regime alignment during periods of extreme market volatility.

For a risk-averse finite-horizon Markov Decision Problem, we introduce a special class of Markov coherent risk measures, called mini-batch measures. We also define the class of multipattern risk-averse problems that generalizes the class of linear systems. We use both concepts in a feature-based $Q$-learning method with multipattern $Q$-factor approximation and we prove a high-probability regret bound of $\mathcal{O}\big(H^2 N^H \sqrt{ K}\big)$, where $H$ is the horizon, $N$ is the mini-batch size, and $K$ is the number of episodes. We also propose an economical version of the $Q$-learning method that streamlines the policy evaluation (backward) step. The theoretical results are illustrated on a stochastic assignment problem and a short-horizon multi-armed bandit problem.

We propose Frictive Policy Optimization (FPO), a framework for learning language model policies that regulate not only what to say, but when and how to intervene in order to manage epistemic and normative risk. Unlike standard alignment methods that optimize surface-level preference or task utility, FPO treats clarification, verification, challenge, redirection, and refusal as explicit control actions whose purpose is to shape the evolution of belief, commitment, and uncertainty over time. We formalize alignment as a risk-sensitive epistemic control problem in which intervention decisions are selected based on their expected effect on downstream epistemic quality rather than on immediate reward alone. We introduce a compact taxonomy of frictive interventions, a structured friction functional that operationalizes multiple alignment failure modes, and a unified family of FPO methods spanning reward shaping, preference pairing, group-relative ranking, and risk-conditioned trust regions. We further propose an evaluation framework that measures epistemic competence directly through clarification behavior, calibration, contradiction repair, refusal proportionality, and information efficiency. Together, these results provide a formal and algorithmic foundation for learning agents that are aligned not only in outcome, but in epistemic conduct.

Cooperative equilibria are fragile. When agents learn alongside each other rather than in a fixed environment, the process of learning destabilizes the cooperation they are trying to sustain: every gradient step an agent takes shifts the distribution of actions its partner will play, turning a cooperative partner into a source of stochastic noise precisely where the cooperation decision is most sensitive. We study how this co-learning noise propagates through the structure of coordination games, and find that the cooperative equilibrium, even when strongly Pareto-dominant, is exponentially unstable under standard risk-neutral learning, collapsing irreversibly once partner noise crosses the game's critical cooperation threshold. The natural response to apply distributional robustness to hedge against partner uncertainty makes things strictly worse: risk-averse return objectives penalize the high-variance cooperative action relative to defection, widening the instability region rather than shrinking it, a paradox that reveals a fundamental mismatch between the domains where robustness is applied and instability originates. We resolve this by showing that robustness should target the policy gradient update variance induced by partner uncertainty, not the return distribution. This distinction yields an algorithm whose gradient updates are modulated by an online measure of partner unpredictability, provably expanding the cooperation basin in symmetric coordination games. To unify stability, sample complexity, and welfare consequences of this approach, we introduce the Price of Paranoia as the structural dual of the Price of Anarchy. Together with a novel Cooperation Window, it precisely characterizes how much welfare learning algorithms can recover under partner noise, pinning down the optimal degree of robustness as a closed-form balance between equilibrium stability and sample efficiency.

Offline reinforcement learning learns policies from fixed datasets without further environment interaction. A key challenge in this setting is epistemic uncertainty, arising from limited or biased data coverage, particularly when the behavior policy systematically avoids certain actions. This can lead to inaccurate value estimates and unreliable generalization. Ensemble-based methods like SAC-N mitigate this by conservatively estimating Q-values using the ensemble minimum, but they require large ensembles and often conflate epistemic with aleatoric uncertainty. To address these limitations, we propose a unified and generalizable framework that replaces discrete ensembles with compact uncertainty sets over Q-values. %We further introduce an Epinet based model that directly shapes the uncertainty sets to optimize the cumulative reward under the robust Bellman objective without relying on ensembles. We also introduce a benchmark for evaluating offline RL algorithms under risk-sensitive behavior policies, and demonstrate that our method achieves improved robustness and generalization over ensemble-based baselines across both tabular and continuous state domains.

Federated reinforcement learning typically aggregates value functions or policies by parameter averaging, which emphasizes expected return and can obscure statistical multimodality and tail behavior that matter in safety-critical settings. We formalize federated distributional reinforcement learning (FedDistRL), where clients parametrize quantile value function critics and federate these networks only. We also propose TR-FedDistRL, which builds a per client, risk-aware Wasserstein barycenter over a temporal buffer. This local barycenter provides a reference region to constrain the parameter averaged critic, ensuring necessary distributional information is not averaged out during the federation process. The distributional trust region is implemented as a shrink-squash step around this reference. Under fixed-policy evaluation, the feasibility map is nonexpansive and the update is contractive in a probe-set Wasserstein metric under evaluation. Experiments on a bandit, multi-agent gridworld, and continuous highway environment show reduced mean-smearing, improved safety proxies (catastrophe/accident rate), and lower critic/policy drift versus mean-oriented and non-federated baselines.

International shipping produces approximately 3% of global greenhouse gas emissions, yet voyage routing remains dominated by heuristic methods. We present PIER (Physics-Informed, Energy-efficient, Risk-aware routing), an offline reinforcement learning framework that learns fuel-efficient, safety-aware routing policies from physics-calibrated environments grounded in historical vessel tracking data and ocean reanalysis products, requiring no online simulator. Validated on one full year (2023) of AIS data across seven Gulf of Mexico routes (840 episodes per method), PIER reduces mean CO2 emissions by 10% relative to great-circle routing. However, PIER's primary contribution is eliminating catastrophic fuel waste: great-circle routing incurs extreme fuel consumption (>1.5x median) in 4.8% of voyages; PIER reduces this to 0.5%, a 9-fold reduction. Per-voyage fuel variance is 3.5x lower (p<0.001), with bootstrap 95% CI for mean savings [2.9%, 15.7%]. Partial validation against observed AIS vessel behavior confirms consistency with the fastest real transits while exhibiting 23.1x lower variance. Crucially, PIER is forecast-independent: unlike A* path optimization whose wave protection degrades 4.5x under realistic forecast uncertainty, PIER maintains constant performance using only local observations. The framework combines physics-informed state construction, demonstration-augmented offline data, and a decoupled post-hoc safety shield, an architecture that transfers to wildfire evacuation, aircraft trajectory optimization, and autonomous navigation in unmapped terrain.

Safe Reinforcement Learning from Human Feedback (RLHF) typically enforces safety through expected cost constraints, but the expectation captures only a single statistic of the cost distribution and fails to account for distributional uncertainty, particularly under heavy tails or rare catastrophic events. This limitation is problematic when robustness and risk sensitivity are critical. Stochastic dominance offers a principled alternative by comparing entire cost distributions rather than just their averages, enabling direct control over tail risks and potential out-of-distribution failures that expectation-based constraints may overlook. In this work, we propose Risk-sensitive Alignment via Dominance (RAD), a novel alignment framework that replaces scalar expected cost constraints with First-Order Stochastic Dominance (FSD) constraints. We operationalize this constraint by comparing the target policy's cost distribution to that of a reference policy within an Optimal Transport (OT) framework, using entropic regularization and Sinkhorn iterations to obtain a differentiable and computationally efficient objective for stable end-to-end optimization. Furthermore, we introduce quantile-weighted FSD constraints and show that weighted FSD universally controls a broad class of Spectral Risk Measures (SRMs), so that improvements under weighted dominance imply guaranteed improvements in the corresponding spectral risk. This provides a principled mechanism for tuning a model's risk profile via the quantile weighting function. Empirical results demonstrate that RAD improves harmlessness over baselines while remaining competitive in helpfulness, and exhibits greater robustness on out-of-distribution harmlessness evaluations.

Provably efficient and robust equilibrium computation in general-sum Markov games remains a core challenge in multi-agent reinforcement learning. Nash equilibrium is computationally intractable in general and brittle due to equilibrium multiplicity and sensitivity to approximation error. We study Risk-Sensitive Quantal Response Equilibrium (RQRE), which yields a unique, smooth solution under bounded rationality and risk sensitivity. We propose RQRE-OVI, an optimistic value iteration algorithm for computing RQRE with linear function approximation in large or continuous state spaces. Through finite-sample regret analysis, we establish convergence and explicitly characterize how sample complexity scales with rationality and risk-sensitivity parameters. The regret bounds reveal a quantitative tradeoff: increasing rationality tightens regret, while risk sensitivity induces regularization that enhances stability and robustness. This exposes a Pareto frontier between expected performance and robustness, with Nash recovered in the limit of perfect rationality and risk neutrality. We further show that the RQRE policy map is Lipschitz continuous in estimated payoffs, unlike Nash, and RQRE admits a distributionally robust optimization interpretation. Empirically, we demonstrate that RQRE-OVI achieves competitive performance under self-play while producing substantially more robust behavior under cross-play compared to Nash-based approaches. These results suggest RQRE-OVI offers a principled, scalable, and tunable path for equilibrium learning with improved robustness and generalization.

Equalizer parameter optimization is critical for signal integrity in high-speed memory systems operating at multi-gigabit data rates. However, existing methods suffer from computationally expensive eye diagram evaluation, optimization of expected rather than worst-case performance, and absence of uncertainty quantification for deployment decisions. In this paper, we propose a distributional risk-sensitive reinforcement learning framework integrating Information Bottleneck latent representations with Conditional Value-at-Risk optimization. We introduce rate-distortion optimal signal compression achieving 51 times speedup over eye diagrams while quantifying epistemic uncertainty through Monte Carlo dropout. Distributional reinforcement learning with quantile regression enables explicit worst-case optimization, while PAC-Bayesian regularization certifies generalization bounds. Experimental validation on 2.4 million waveforms from eight memory units demonstrated mean improvements of 37.1% and 41.5% for 4-tap and 8-tap equalizer configurations with worst-case guarantees of 33.8% and 38.2%, representing 80.7% and 89.1% improvements over Q-learning baselines. The framework achieved 62.5% high-reliability classification eliminating manual validation for most configurations. These results suggest the proposed framework provides a practical solution for production-scale equalizer optimization with certified worst-case guarantees.

For robots to successfully transition from lab settings to everyday environments, they must begin to reason about the risks associated with their actions and make informed, risk-aware decisions. This is particularly true for robots performing mobile manipulation tasks, which involve both interacting with and navigating within dynamic, unstructured spaces. However, existing whole-body controllers for mobile manipulators typically lack explicit mechanisms for risk-sensitive decision-making under uncertainty. To our knowledge, we are the first to (i) learn risk-aware visuomotor policies for mobile manipulation conditioned on egocentric depth observations with runtime-adjustable risk sensitivity, and (ii) show risk-aware behaviours can be transferred through Imitation Learning (IL) to a visuomotor policy conditioned on egocentric depth observations. Our method achieves this by first training a privileged teacher policy using Distributional Reinforcement Learning (DRL), with a risk-neutral distributional critic. Distortion risk-metrics are then applied to the critic's predicted return distribution to calculate risk-adjusted advantage estimates used in policy updates to achieve a range of risk-aware behaviours. We then distil teacher policies with IL to obtain risk-aware student policies conditioned on egocentric depth observations. We perform extensive evaluations demonstrating that our trained visuomotor policies exhibit risk-aware behaviour (specifically achieving better worst-case performance) while performing reactive whole-body motions in unmapped environments, leveraging live depth observations for perception.

Risk-averse decision-making under uncertainty in partially observable domains is a central challenge in artificial intelligence and is essential for developing reliable autonomous agents. The formal framework for such problems is the partially observable Markov decision process (POMDP), where risk sensitivity is introduced through a risk measure applied to the value function, with Conditional Value-at-Risk (CVaR) being a particularly significant criterion. However, solving POMDPs is computationally intractable in general, and approximate methods rely on computationally expensive simulations of future agent trajectories. This work introduces a theoretical framework for accelerating CVaR value function evaluation in POMDPs with formal performance guarantees. We derive new bounds on the CVaR of a random variable X using an auxiliary random variable Y, under assumptions relating their cumulative distribution and density functions; these bounds yield interpretable concentration inequalities and converge as the distributional discrepancy vanishes. Building on this, we establish upper and lower bounds on the CVaR value function computable from a simplified belief-MDP, accommodating general simplifications of the transition dynamics. We develop estimators for these bounds within a particle-belief MDP framework with probabilistic guarantees, and employ them for acceleration via action elimination: actions whose bounds indicate suboptimality under the simplified model are safely discarded while ensuring consistency with the original POMDP. Empirical evaluation across multiple POMDP domains confirms that the bounds reliably separate safe from dangerous policies while achieving substantial computational speedups under the simplified model.

We consider the policy evaluation and control in a finite horizon reinforcement learning (RL) setting under an expectile-based objective. First, we derive the mean-squared error (MSE) and concentration bounds for the classic estimator of expectiles based on independent and identically distributed (i.i.d.) samples. To the best of our knowledge, expectiles have not been analyzed in the non-asymptotic regime and the bounds we derive may be of independent interest. Next, we analyze a Monte Carlo type estimator of expectile of the Markov chain underlying a given policy. We derive upper bounds that hold in expectation as well as with high probability for this estimator. Further, we show the order-optimality of our estimator by deriving a lower bound on expectile-based policy evaluation. For the problem of control, we adopt a policy gradient approach and derive a policy gradient theorem for expectiles. Using this result, we propose a gradient estimator with a $O\left(1/m\right)$ mean-squared error bounds, where $m$ is the number of trajectories. Further, under standard assumptions for policy gradient-type algorithms, we establish smoothness of the expectile-sensitive objective, in turn leading to stationary convergence rate bounds for the overall risk-sensitive policy gradient algorithm that we propose. Finally, we conduct numerical experiments to show the utility of expectiles on popular RL benchmarks.

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 introduce Canary, a risk-averse method designed to optimize Value-at-Risk (VaR) constrained reinforcement learning (RL) problems. We employ Cantelli's inequality to obtain a tractable, conservative and smooth bound on the VaR constraint based on the first two moments of the cost return. This yields a constraint estimator that remains stable with tight violation thresholds in dense cost regimes. Extending the trust-region framework of the Constrained Policy Optimization (CPO) method, we further provide worst-case bounds for both policy improvement and constraint violation during the training process. Empirically during training, Canary is the only method that reliably satisfies the VaR constraint in every environment tested.

We consider the problem of policy optimization with noisy rewards, where the policy is constrained to be in the neighborhood of a reference policy.We study the risk of the policy optimization process, defined to be the chance of obtaining a policy worse off than the reference policy. In an idealized multivariate Gaussian model for the noisy reward function, we show that the regularizer achieving optimal risk in a suitably local neighborhood of the reference policy is the quadratic norm weighted by the reward covariance matrix. This result is used to develop a new policy optimization algorithm based on regularization via reward variance estimates. Experiments across synthetic bandit models, large language models and reward models show that this algorithm yields more robust alignment outcomes than existing approaches.

Imagining potential outcomes of actions before execution helps agents make more informed decisions, a prospective thinking ability fundamental to human cognition. However, mainstream model-free Reinforcement Learning (RL) methods lack the ability to proactively envision future scenarios, plan, and guide strategies. These methods typically rely on trial and error to adjust policy functions, aiming to maximize cumulative rewards or long-term value, even if such high-reward decisions place the environment in extremely dangerous states. To address this, we propose the Prospective (ProSpec) RL method, which makes higher-value, lower-risk optimal decisions by imagining future n-stream trajectories. Specifically, ProSpec employs a dynamic model to predict future states (termed "imagined states") based on the current state and a series of sampled actions. Furthermore, we integrate the concept of Model Predictive Control and introduce a cycle consistency constraint that allows the agent to evaluate and select the optimal actions from these trajectories. Moreover, ProSpec employs cycle consistency to mitigate two fundamental issues in RL: augmenting state reversibility to avoid irreversible events (low risk) and augmenting actions to generate numerous virtual trajectories, thereby improving data efficiency. We validated the effectiveness of our method on the DMControl benchmarks, where our approach achieved significant performance improvements. Code will be open-sourced upon acceptance.

Risk-sensitive reinforcement learning (RL) is crucial for maintaining reliable performance in high-stakes applications. While traditional RL methods aim to learn a point estimate of the random cumulative cost, distributional RL seeks to estimate the entire distribution of it, leading to a unified framework for handling different risk measures. However, developing policy gradient methods for risk-sensitive distributional RL is inherently more complex as it often involves finding the gradient of a probability measure. This paper introduces a new distributional policy gradient framework for risk-sensitive RL, where we derive an analytical gradient of the probability measure of the cumulative cost. For practical implementation, we further design a categorical distributional policy gradient algorithm (CDPG) that approximates arbitrary distributions using a categorical family supported on fixed points. Using Conditional Value-at-Risk (CVaR) as the objective, we prove that the proposed CDPG converges to stationary points and establish its iteration complexities under inexact policy evaluation. Through experiments in a stochastic Cliffwalk environment, we demonstrate the effectiveness of the proposed algorithm and highlight the benefits of incorporating risk sensitivity into distributional RL.