Distributional RL

RL: Reinforcement Learning

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1 paper in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

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Latest papers 29

Oct 6, 2026cs.LG

Variance-Averse $n$ -Step Offline Reinforcement Learning for Sparse Long-Horizon Environments

Generative actors are transforming offline reinforcement learning (RL) by enabling expressive policy classes that model complex action distributions. However, this expressiveness also exposes a key challenge in heterogeneous datasets: generative policies can reproduce unreliable action modes whose return distributions exhibit high variance, occasionally yielding high returns by chance but lacking consistency. Consequently, maximizing the expected

Q

-value alone is insufficient for identifying reliable actions. We propose VAN-Flow (Variance-Averse

n

-step Flow), a framework that promotes reliable actions in generative offline RL. VAN-Flow combines (i) a categorical distributional critic, (ii) a variance-averse expectation operator that smoothly reweights atom probabilities to favor actions with both high returns and low dispersion, and (iii) a flow-matching generative actor guided via rejection sampling. Unlike CVaR or mean-variance objectives, the operator redistributes probability mass over the categorical return distribution without hard truncation or auxiliary penalty terms. Across more than 40 tasks from D4RL and OGBench, VAN-Flow consistently outperforms strong baselines, with the largest gains in long-horizon and high-variance regimes where reliable action selection becomes critical.

Guhyeon Kang, Minhae Kwon

Department of Electrical and Computer Engineering Sungkyunkwan University, Republic of Korea

Distributional RL Risk-Sensitive RL Offline RL RL for Generative Models

Sep 21, 2026cs.LG

Reinforcement Learning under State and Outcome Uncertainty: A Foundational Distributional Perspective

In many real-world planning tasks, agents must tackle uncertainty about the environment's state and variability in the outcomes of any chosen policy. We address both forms of uncertainty as a first step toward safer algorithms in partially observable settings. Specifically, we extend Distributional Reinforcement Learning (DistRL)-which models the entire return distribution for fully observable domains-to Partially Observable Markov Decision Processes (POMDPs), allowing an agent to learn the distribution of returns for each conditional plan. Concretely, we introduce new distributional Bellman operators for partial observability and prove their convergence under the supremum p-Wasserstein metric. We also propose a finite representation of these return distributions via psi-vectors, generalizing the classical alpha-vectors in POMDP solvers. Building on this, we develop Distributional Point-Based Value Iteration (DPBVI), which integrates psi-vectors into a standard point-based backup procedure-bridging DistRL and POMDP planning. By tracking return distributions, DPBVI lays the foundation for future risk-sensitive control in domains where rare, high-impact events must be carefully managed. We provide source code to foster further research in robust decision-making under partial observability.

Larry Preuett, Qiuyi Zhang, Muhammad Aurangzeb Ahmad

University of Washington, Tacoma, USA · Google DeepMind, California, USA · University of Washington, Bothell, USA

Distributional RL Markov Decision Processes Partially Observable RL Partially Observable Markov Decision Processes

Aug 27, 2026stat.ML

A Finite-Sample Analysis of Quantile Temporal-Difference Learning

Quantile temporal-difference learning (QTD) is an effective method for learning return distributions through quantile approximation, yet its finite-time behavior remains poorly understood. Its update is nonlinear and nonsmooth, and the stability needed for a sharp convergence rate holds only near the target. We establish a global high-probability last-iterate guarantee for synchronous tabular QTD under general positive, nonincreasing step-size sequences and arbitrary initialization in the natural parameter range. For polynomially decaying step sizes with exponent

a\in(0,1)

, the last iterate converges to the target at rate

T^{-a/2}

in the infinity norm, up to logarithmic and lower-order terms. A suitably tuned harmonic schedule recovers the

T^{-1/2}

statistical rate up to logarithmic factors. For the

m

-quantile representation, its

\infty

-Wasserstein error scales as

\sqrt{m/T}

up to logarithmic factors, matching the leading polynomial dependence on the quantile resolution and sample size of the corresponding model-based estimator. The proof uses a two-stage global-to-local argument. From arbitrary initialization, Bellman contraction and CDF monotonicity first bring the iterate close to the target, after which, a novel variance--drift matching argument sharpens the control of accumulated noise and local contraction reduces the remaining errors, yielding the sharp rate. Simulations verify the predicted polynomial decay and assess the finite-time entrance bound.

Zijie Cheng, Xiang Li, Yang Peng +1

School of Mathematical Sciences, Peking University · Perelman School of Medicine, University of Pennsylvania · Yau Mathematical Sciences Center, Tsinghua University

Distributional RL Stochastic Approximation Temporal-Difference Learning

Aug 13, 2026stat.ML

Online Inference for Quantile Temporal Difference Learning in Distributional Reinforcement Learning

In this paper, we study how to perform statistical inference for quantile temporal difference learning (QTD) in distributional reinforcement learning. Assuming access to a generative model, we first establish functional central limit theorems for both synchronous and asynchronous QTD, which show that the averaged iterates of QTD converge weakly to a rescaled Brownian motion. We next provide online inference methods. Based on random scaling, the inference procedure constructs an asymptotically pivotal statistic for inference by using the information along the whole QTD path. Meanwhile, the proposed statistic can be computed online without storing the entire trajectory of QTD iterates. This substantially reduces the memory requirement and enables efficient statistical inference in distributional reinforcement learning.

Zijie Cheng, Yang Peng, Zhihua Zhang

School of Mathematical Sciences, Peking University · Yau Mathematical Sciences Center, Tsinghua University

Distributional RL Reinforcement Learning Temporal-Difference Learning

Jul 20, 2026cs.LG

Distributional Soft Bellman Operator under the Cramér Geometry

Distributional soft policy iteration (DSPI) provides an important framework for combining distributional reinforcement learning (DRL) with maximum-entropy control, in which the policy evaluation step is governed by a distributional soft Bellman operator acting on entropy-regularised returns. Theoretical analysis of such an evaluation step requires a probability metric under which Bellman updates can be controlled, typically by showing that the operator contracts the distance between any two candidate return-distribution estimates. In this paper, we focus on the Cramér geometry, a cumulative distribution function (CDF)-based metric with an

L^2

structure, and study whether the fixed-policy distributional soft Bellman operator has this contraction property and hence a unique fixed point under this metric. Working directly on an admissible CDF field domain, we formulate the CDF-level distributional soft Bellman operator, prove that it is a

\sqrtγ

-contraction, and obtain the corresponding unique fixed point together with convergent iterative policy evaluation. The CDF formulation also shows that this finite-Cramér-domain property follows from a uniform first-moment condition on the combined one-step reward entropy shift, rather than from separate uniform boundedness assumptions on the reward and entropy terms. We then transport the same evaluation problem to the spectral domain by conjugation, obtaining an equivalent Hilbert-space representation of the same decision process. Taken together, these results identify the Cramér-geometric Bellman fixed point associated with the policy-evaluation step of DSPI, providing a reference point for studying approximate critics, evaluation error, and critic-loss design in DSPI-style algorithms.

Keru Wang, Yixin Deng, Yao Lyu +2

School of Electrical and Electronic Engineering, University College Dublin, Dublin, Ireland · School of Vehicle and Mobility, Tsinghua University, Beijing, China · College of Artificial Intelligence, Tsinghua University, Beijing, China

Distributional RL Maximum Entropy RL

Jul 14, 2026cs.IT

Active Beyond-Diagonal RIS Empowered Heterogeneous Edge Computing: A Distributional Reinforcement Learning Approach

Active beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) enables hybrid transmitting and reflecting mode to achieve effective signal amplification and full-space coverage, thus providing a promising solution for blockage-aware uplink offloading in heterogeneous mobile edge computing (MEC) systems. However, practical hybrid mode active BD-RIS are realized by reciprocal devices, which inherently generate cross-sector energy leakage that will reshape the system-level energy-latency tradeoff. This paper studies energy-aware offloading and resource allocation for reciprocal active BD-RIS-assisted heterogeneous MEC, where offloading decisions, CPU/GPU computation allocation, transmit powers, receive processing, and active BD-RIS are tightly coupled. The resulting problem is a high-dimensional mixed integer nonconvex problem and is difficult to solve efficiently by conventional per-instance optimization. To address this challenge, we develop an end-to-end joint optimization framework based on a refined version of the distributional soft actor--critic algorithm, named as DSAC-T. By modeling return distributions rather than only expected values, DSAC-T improves policy stability under reward heterogeneity and feasibility-boundary sensitivity. Compared with other baseline algorithms, DSAC-T achieves the best energy-latency reward, the highest feasibility ratio of 81.67%, and a fast online decision time of 0.0267 s per scenario.

Tianyu Pang, Hongyu Li

Internet of Things Thrust, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China

Distributional RL Reconfigurable Intelligent Surfaces Edge Computing Wireless Communications Soft Actor-Critic Wireless Resource Allocation

Jul 13, 2026cs.AI

Auditing the Risk Claims of Distributional Reinforcement Learning

Distributional reinforcement learning agents learn full return distributions that are increasingly read at face value: for interpretability, risk-sensitive control, and safety monitoring. We ask a question theory anticipates but that has not been measured directly: are the risk claims of a trained distributional agent true? Our audit combines a decision-relevant screening metric (the excess Wasserstein gap between the top two actions, which equals the mass by which first-order stochastic dominance is violated), ground truth from snapshot-restart Monte Carlo, and a statistical harness (permutation nulls, bootstrap refutation, FDR control) without which the audit itself manufactures false conclusions. Across QR-DQN, C51, and IQN on MinAtar (33 runs), 40-95% of the strongest claimed risk trade-offs are refuted at 95% confidence, the placement of the strongest claims is statistically indistinguishable from truth-blind, and essentially no claim is confirmable: for these agents, the learned "risk" reflects a training artifact rather than environment stochasticity. The artifact is structural (fully formed early in training, uncorrelated with final score, idiosyncratic to each seed) and appears unchanged at full-Atari scale, with every top Breakout claim of a pretrained near-state-of-the-art QR-DQN refuted. Positive controls of known magnitude confirm 96-100% of real claims (correlation 0.89-0.92): the reading measures the agents, not the audit. Acting on the heads' CVaR advice at their most-flagged states ranges from beneficial to significantly worse than chance. Neither training for risk nor ensembling removes the artifact, and recalibration passes the audit only by nullifying the claims: the head is uninformative, not merely miscalibrated. We release the toolkit and document two silent pitfalls that produced convincing but wrong audits of our own.

Hari Prasad

Distributional RL Algorithmic Auditing Risk-Sensitive RL

Jul 9, 2026stat.ML

Statistical Efficiency and Inference of Quantile Distributional Reinforcement Learning

In this paper, we study quantile-based distributional reinforcement learning from the perspective of statistical efficiency. We focus on distributional policy evaluation, whose goal is to characterize the return distribution, namely the distribution of discounted cumulative rewards under a given policy. To obtain a finite-dimensional representation of the return distribution, we consider the quantile fixed point

η_m

induced by the quantile-projected distributional Bellman equation. Assuming access to a generative model, we construct an estimator

η_m^{(n)}

based on an empirical Markov decision process. For a fixed number of quantiles

m

, we establish a non-asymptotic error bound for

η_m^{(n)}

and

η_m

under the supremum

W_\infty

metric, showing that the estimation error scales as

\widetilde{O}(\sqrt{m/n})

with respect to

m

and

n

. This implies that the quantile-based distributional policy evaluation problem can be solved with sample efficiency, achieving the optimal parametric

\sqrt{n}

convergence rate. We derive the asymptotic distribution of the quantile parameters

\sqrt{n}(θ_m^{(n)}-θ_m)

and characterize the semiparametric efficiency bound, which is attained by our estimator. Beyond the fixed-dimensional setting, we investigate the asymptotic regime in which the number of quantiles diverges. We characterize the limit covariance structure and show that it matches the semiparametric efficiency bound of the nonparametric model for distributional policy evaluation, showing that quantile-based estimators remain asymptotically efficient in the infinite-dimensional limit. Finally, we establish a Berry--Esseen theorem for smooth functionals

\sqrt{n}(η_m^{(n)}(s)-η_m(s))f

, thereby providing a foundation for statistically valid inference on functionals of the quantile-projected return distribution.

Zijie Cheng, Yang Peng, Zhihua Zhang

School of Mathematical Sciences, Peking University · Yau Mathematical Sciences Center, Tsinghua University

Distributional RL Reinforcement Learning Sample-Efficient RL

Jul 2, 2026cs.LG

Learning the Supports for Categorical Critic in Reinforcement Learning

Value functions are an essential component in actor-critic based deep reinforcement learning (RL). Conventionally, these functions are trained as a regression task by minimising the mean squared error (MSE) relative to bootstrapped target values. Meanwhile, in distributional RL, a distribution of returns is modelled based on the distributional Bellman operator. This work investigates the Gaussian Histogram Loss (HL-Gauss), a recent approach that reframes value estimation as classification by encoding each scalar Bellman target as a Gaussian-smoothed categorical target. Despite its potential, applying histogram-based losses to RL presents inherent challenges, most notably the requirement to pre-define a fixed support interval, which is often complicated by the non-stationary and stochastic nature of target values typically found in RL tasks. In this work, we propose an approach that dynamically learns the lower and upper bounds of the support instead of assigning them beforehand. We derive an objective that jointly learns these bounds whilst learning the categorical representation of the scalar values, and we show that this objective forms an upper bound on the mean-squared Bellman error. Our theoretical analysis further shows that this bound is tighter than that of non-learned supports of HL-Gauss. Empirically, the proposed objective enables stable adaptation of the support interval and matches HL-Gauss-based actor-critic algorithms on most continuous-control tasks whilst improving on a subset, without requiring a pre-specified support interval.

Jen-Yen Chang, Takayuki Osa, Tatsuya Harada

Graduate School of Information Science and Technology, The University of Tokyo · RIKEN AIP

Distributional RL

Jun 30, 2026cs.RO

FastDSAC: Enhancing Policy Plasticity via Constrained Exploration for Scalable Humanoid Locomotion

Scalable reinforcement learning has popularized high-throughput sampling architectures, which significantly compresses the training time for off-policy methods in robotic locomotion. However, the rapid increase of data volume and update frequency undermines the stability of value-based methods and diminishes the plasticity of policy networks. To address these challenges, this work presents FastDSAC, a fast and high-performance variant of the Distributional Actor-Critic algorithm designed for parallel sampling scenarios. Specifically, we introduce a truncated Gaussian distribution to approximate the learned policy, which effectively excludes out-of-distribution actions that strain target value estimation while keeping necessary stochasticity for exploration. The proposed action constraint functions as an implicit regularization, which counteracts the plasticity loss typically caused by aggressive gradient updates. This preservation of network adaptability enhances sample efficiency, particularly in scenarios with a high update-to-data ratio, and accelerates the early training process. In contrast to prior fast reinforcement learning approaches that rely on discrete value distributions, our method utilizes a continuous Gaussian representation equipped with adaptive variance regulation, which improves value estimation accuracy by sampling confident and informative transitions. Extensive experiments on MuJoCo Playground and HumanoidBench demonstrate that FastDSAC not only stabilizes the overall training process but also achieves superior asymptotic performance and faster convergence compared to state-of-the-art baselines.

Guanchen Lu, Yajuan Dun, Yi Zhou +4

Chongqing University, Chongqing 400044, China · Ministry of Industry and Information Technology, Beijing 100804, China · Tsinghua University, Beijing 100084, China +1

Distributional RL Reinforcement Learning RL for Legged Locomotion Sample-Efficient RL

Jun 29, 2026cs.LG

Dual-Flow Reinforcement Learning with State-Aware Exploration

In complex continuous-control reinforcement learning tasks, multimodal optimal actions often coincide with uncertain, multimodal return distributions, making reliable value estimation and multimodal exploration challenging. Existing value estimation methods using unimodal Gaussians restrict expressiveness and yield biased estimates. Recent generative policies can represent multimodal actions but often collapse to a few modes and under-explore high-value areas of the action space. Motivated by these challenges, we propose Dual-Flow RL, a unified actor-critic framework that jointly models a continuous return distribution and a multimodal policy distribution using conditional flow matching (CFM). This design supports reliable value estimation and sustained multimodal exploration. To further enhance exploration, we introduce an Entropy-Covariance Exploration Regulator (ECER) that enables state-aware exploration regulation leveraging policy entropy and action-uncertainty covariance. Experiments on DeepMind Control Suite and Humanoid-Bench show that Dual-Flow RL achieves state-of-the-art performance on most tasks, significantly outperforming prior diffusion-based and flow-based methods.

Qijun Li, Zheng Fu, Qi Song +4

School of Vehicle and Mobility and the State Key Laboratory of Intelligent Green Vehicle and Mobility, Tsinghua University, Beijing 100084, China

Distributional RL Reinforcement Learning RL Exploration

Jun 26, 2026cs.LG

RS-Diffuser: Risk-Sensitive Diffusion Planning with Distributional Value Guidance

Offline reinforcement learning enables policy learning from fixed datasets without additional environment interaction, making it appealing for safety-critical applications where online exploration is costly or unsafe. Diffusion-based decision-making methods have recently achieved strong performance in offline RL by modeling rich, multimodal trajectory distributions. However, existing diffusion planners are typically risk-neutral and therefore may overlook rare but catastrophic outcomes that are crucial in real-world deployment. In this work, we propose RS-Diffuser, a risk-sensitive offline diffusion planning framework that combines diffusion-based trajectory generation with distributional value critics. RS-Diffuser learns a diffusion planner over future state trajectories, a separate inverse dynamics model for action decoding, and a Monte Carlo distributional critic that estimates the full return distribution of candidate plans through quantile regression. At sampling time, we incorporate a risk-sensitive guidance signal into the denoising process, using gradients computed from tail-aware objectives such as Conditional Value at Risk to steer generation toward desired risk profiles. As a result, a single trained model can flexibly produce risk-averse, risk-neutral, or risk-seeking behaviors by changing only the inference-time risk parameter. Extensive experiments on risk-sensitive D4RL and risky robot navigation benchmarks demonstrate that RS-Diffuser achieves state-of-the-art performance, improving both overall return and worst-case robustness while reducing safety violations.

Shiqiang Gong

Northwestern Polytechnical University

Distributional RL Diffusion-Based Planning Risk-Sensitive RL Offline RL

Jun 12, 2026cs.LG

DRIVE: Distributional and Retrieval-Augmented Bidding with Value Evaluation

Auto-bidding is a core component of real-time advertising systems, where decisions must optimize long-term performance under budget and cost constraints, while online exploration is prohibitively risky. Offline reinforcement learning and, more recently, Transformer-based sequence modeling have shown promise for learning bidding policies from logged data, but their unimodal and purely parametric formulations often collapse multiple effective bidding strategies into suboptimal averaged actions and perform unreliably under sparse or long-tail traffic. To mitigate these limitations, we propose DRIVE (Distributional and Retrieval-Augmented Bidding with Value Evaluation), a unified Transformer-based framework that decouples candidate action generation from decision making for offline auto-bidding. DRIVE combines distributional action modeling, retrieval-augmented candidate generation from high-quality historical decisions, and value-based evaluation to select the most promising bid at inference time. Extensive experiments on AuctionNet and additional offline reinforcement learning benchmarks demonstrate that DRIVE consistently improves bidding performance and generalizes well across multiple Transformer-based methods.

Miduo Cui, Haochen Wang, Shangqin Mao +6

Shandong University, Jinan, China · Meituan, Beijing, China

Distributional RL Automated Bidding Offline RL Real-Time Bidding

Jun 6, 2026quant-ph

QnRL: Quantum-Native Reinforcement Learning

Quantum reinforcement learning (QRL) is a promising approach to learn effective decision strategies across several applications with stochastic environments. Instead of directly modeling the random variables that govern these environments, existing QRL architectures indirectly approximate environment behavior by estimating expected outcomes, which limits their expressive power and adaptive potential. Overcoming such challenges requires a novel QRL approach that exploits the distributional nature of quantum computers to directly model environment random variables as quantum state distributions. Hence, in this paper, a novel framework dubbed quantum-native reinforcement learning (QnRL) is proposed. QnRL is a distributional RL framework that learns conditional distributions naturally in Hilbert space via superimposed and entangled quantum states. Thus, QnRL can directly model the behavior of stochastic learning environments via the natural properties of quantum systems. QnRL accomplishes this via a novel, proposed quantum amplitude kickback (QuAK) algorithm that enables comparing the

n

-th power of the

m

-th moment of multiple superimposed distributions. It is theoretically proven that a conditional action policy distribution is distilled from the moments of a quantum generative model entirely within Hilbert space via QuAK, and optimized via QnRL. This complex distribution composition is also shown to provide extra dimensions for expressing environment correlations that are unknown to purely classical and classically-sampled quantum distributional models. Experimental results across diverse environments show that QnRL achieves up to

82.9\%

higher evaluation scores, with up to

94.3\%

fewer parameters on average, more accurately estimates the expected return for unseen observations, and better adapts to varying stochastic conditions compared to the baseline.

Alexander DeRieux, Walid Saad

Bradley Department of Electrical and Computer Engineering, Virginia Tech Institute for Advanced Computing, Alexandria, VA 22305 USA

Distributional RL Quantum Generative Modeling Quantum Machine Learning Quantum RL

Jun 3, 2026cs.LG

Reinforcement Learning from Rich Feedback with Distributional DAgger

Reasoning models have advanced rapidly, but the dominant reinforcement learning from verifiable rewards (RLVR) recipe remains surprisingly narrow: sample many responses and reward each with a single bit indicating whether the final answer is correct. Yet many settings provide rich feedback, including execution traces, tool outputs, expert corrections, and model self-evaluations. We study how to use such feedback through a distributional variant of the classic imitation learning algorithm DAgger, where the learner has local access to an expert distribution on states visited by the current policy. This yields a simple forward cross-entropy objective that admits a blackbox expert and whose sequence-level gradient {conduct rich credit assignment by propagating} future expert-student disagreement back to earlier decisions. We show that prior RL with self-distillation objectives based on reverse KL or Jensen-Shannon fail to guarantee monotonic policy improvement: even when the expert has higher reward, their updates may increase probability on worse actions. In contrast, we show that forward cross-entropy admits monotonic policy improvement and enjoys guarantees on regret. We further show that our objective optimizes a lower bound on teacher-weighted likelihood of success, leading to improved Pass@N. Empirically, our approach, DistIL, improves over RLVR and RL with self-distillation baselines across a variety of domains: scientific reasoning, coding, and solving hard mathematical problems.

Rishabh Agrawal, Jacob Fein-Ashley, Paria Rashidinejad

University of Southern California

Distributional RL RL for Language Model Reasoning Credit Assignment in RL Imitation Learning Reinforcement Learning with Verifiable Rewards Policy Learning

May 29, 2026cs.LG

Multivariate Distributional Reinforcement Learning Using Sliced Divergences

Distributional reinforcement learning (DRL) models the full return distribution rather than expectations, but extending it to multivariate settings remains challenging. Many common metrics do not naturally generalize beyond one dimension or lose computational tractability, and the multivariate case introduces additional difficulties such as general matrix discounting, for which no contraction results are available. We introduce Sliced Distributional Reinforcement Learning (SDRL), which lifts tractable one-dimensional divergences to multivariate return distributions via projections. We prove Bellman contraction for uniform slicing under shared scalar discounting, and introduce a maximum-slicing variant with contraction under general dense discount matrices. SDRL supports a broad class of base divergences; we analyze Wasserstein, Cramér, and Maximum Mean Discrepancy (MMD), and characterize which SDRL variants suit the standard single-sample Bellman update used in distributional RL. We evaluate SDRL on a toy chain problem and a gridworld image-based environment as well as a subset of Atari games.

Baptiste Debes, Tinne Tuytelaars

Department of Electrical Engineering (ESAT), KU Leuven, Leuven, Belgium.

Distributional RL

May 28, 2026cs.LG

On Distributional Reinforcement Learning in Chaotic Dynamical Systems

Chaotic dynamical systems pose a fundamental challenge for Reinforcement Learning (RL): exponential sensitivity to initial conditions induces high-variance bootstrap targets and poorly conditioned gradient updates. Chaotic dynamics arise across scientific and engineering domains, from fluid flows and climate systems to multi-agent systems, where reliable learning is highly desirable. Standard RL methods optimise expected returns through scalar value functions, implicitly averaging over diverging trajectories and entangling trajectory level instability with the learning objective. We show that under mild statistical stability assumptions, the return distribution evolves more regularly than individual trajectories when measured under the

1

-Wasserstein metric, yielding a smoother distributional Bellman objective. By aligning optimisation with this measure level structure, distributional RL provides better conditioned learning. We offer a principled explanation for the advantages of distributional methods in chaotic systems and the geometries of RL objectives under chaos.

James Rudd-Jones, Mirco Musolesi, María Pérez-Ortiz

Centre for Artificial Intelligence Department of Computer Science University College London London, UK · Department of Computer Science and Engineering, University of Bologna Bologna, Italy

Distributional RL Reinforcement Learning Chaotic Dynamical Systems

May 20, 2026cs.LG

Reinforcement Learning over Predictive Distributions for LLM Regression

Large language models (LLMs) have emerged as flexible regressors capable of predicting real-valued quantities from heterogeneous inputs. Yet most LLM regression objectives optimize predictions independently, often yielding poor calibration. We introduce Distribution-Aware Reward (DAR), an on-policy reinforcement learning objective that instead jointly evaluates the empirical predictive distribution formed by multiple predictions for the same input. To translate this distribution-level objective into rollout-level rewards, we assign each prediction credit based on its leave-one-out contribution to the quality of the overall predictive distribution. This encourages predictions that are well-centered and appropriately dispersed around the target. We evaluate on three regression settings: a synthetic task probing interpolation and extrapolation, and two real-world scientific tasks involving code and molecular data. Across tasks, DAR produces better-calibrated uncertainty estimates while consistently reducing prediction error and improving ranking quality over supervised fine-tuning and pointwise reinforcement learning. Together, these results highlight the benefits of distribution-aware training for LLM regression.

Jungsoo Park, Hyungjoo Chae, Ethan Mendes +4

Georgia Institute of Technology · Allen Institute for AI

Distributional RL RL for Language Models Language Model Calibration

May 11, 2026cs.LG

Quotient-Categorical Representations for Bellman-Compatible Average-Reward Distributional Reinforcement Learning

Average-reward reinforcement learning requires estimating the gain and the bias, which is defined only up to an additive constant. This makes direct distributional analogues ill-posed on the real line. We introduce a quotient-space formulation in which state-indexed bias laws are identified up to a common translation, together with a categorical parameterization that respects this symmetry. On this quotient-categorical space, we define a projected average-reward distributional operator and show that it is well-defined, non-expansive in a coordinate Cramér metric, and admits fixed points. We then study sampled recursions whose mean-field maps are asynchronous relaxations of this operator. In an idealized centered-reward setting, a one-state temporal-difference update enjoys almost sure convergence together with finite-iteration residual bounds under both i.i.d. and Markovian sampling. When the gain is unknown, we augment the recursion with an online gain estimator, and prove non-expansiveness and Markovian convergence of the resulting coupled scheme. Finally, we show that synchronous exact updates are gain-independent at the quotient-law level, isolating a structural contrast between ideal quotient distributions and practical fixed-grid categorical representations.

Ege C. Kaya, Aliasghar Pourghani, Vijay Gupta +1

Elmore Family School of Electrical and Computer Engineering Purdue University West Lafayette, IN 47906

Distributional RL Reinforcement Learning Average-Reward RL Temporal-Difference Learning

May 8, 2026cs.LG

Quantile-Coupled Flow Matching for Distributional Reinforcement Learning

Unlike standard expected-return Reinforcement Learning (RL), Distributional RL (DRL) models the full return distribution, making it better-suited for uncertainty-aware and risk-sensitive decision-making. Conditional Flow Matching (CFM) critics have recently attracted attention for modelling continuous, multi-modal return distributions. Despite this interest, there remains a substantial metric mismatch: DRL theory relies on the distributional Bellman operator being contractive in the

p

-Wasserstein distance, yet existing CFM critics are trained with arbitrary source-target couplings, so their flow-matching losses are not Wasserstein-aligned surrogates for matching Bellman target return distributions. In this work, we address this mismatch by proposing FlowIQN, a CFM critic that sorts source and Bellman target samples within each mini-batch to approximate the monotone optimal transport coupling, replacing arbitrary pairings with quantile-aligned flow paths. We prove that the loss of our quantile-coupled CFM critic yields a Wasserstein-aligned approximate projection compatible with the foundations of DRL. To our knowledge, FlowIQN is the first flow-matching distributional critic with an explicit Wasserstein-aligned projection guarantee. We further extend FlowIQN with shortcut models for efficient inference. Empirical results show that FlowIQN improves Wasserstein return-distribution accuracy over other CFM critics. It also yields competitive performance on offline RL benchmarks across multiple policy extraction methods, providing a theoretically grounded CFM critic that is readily compatible with DRL pipelines. Code: https://github.com/ori-goals/flowIQN.

Michael Groom, Victor-Alexandru Darvariu, Lars Kunze +2

Oxford Robotics Institute, University of Oxford, Oxford, United Kingdom · Bristol Robotics Laboratory, University of the West of England, Bristol, United Kingdom · Dyson Institute of Engineering & Technology, Malmesbury, United Kingdom

Distributional RL Conditional Flow Matching Optimal Transport

May 7, 2026cs.LG

A Sharp Finite-Iteration Theory for Asynchronous Categorical Distributional Temporal-Difference Learning

We study finite-iteration behavior of asynchronous categorical distributional temporal-difference methods, covering scalar categorical TD (CTD) in the Cramér geometry and multivariate signed-categorical TD (MTD) in the maximum mean discrepancy (MMD) geometry. We establish high-probability guarantees under i.i.d. sampling and under a continuing Markovian trajectory. For CTD, the Cramér sample complexity to the true return law has leading term

\tilde O(ρ_{\min}^{-1}(1-γ)^{-2}\varepsilon^{-2})

in the i.i.d. and

\tilde O(μ_{\min}^{-1}(1-γ)^{-2}\varepsilon^{-2}+μ_{\min}^{-1}t_{\mathrm{mix}})

in the Markovian setting, without using variance reduction or data dropping. For MTD, the MMD sample complexity has leading terms

\tilde O(ρ_{\min}^{-1}(1-γ)^{-1-c}\varepsilon^{-2})

and

\tilde O(μ_{\min}^{-1}(1-γ)^{-1-c}\varepsilon^{-2}+μ_{\min}^{-1}t_{\mathrm{mix}})

, where

c\in(0,2)

is the homogeneity exponent of the MMD kernel, and they reduce to the CTD rates when

c=1

. These are, to our knowledge, the first finite-iteration rates for MTD. For undiscounted fixed-horizon policy evaluation, the same analysis applies to fixed-horizon versions of CTD and MTD under i.i.d. and episodic sampling, and the rates match the discounted ones with the effective horizon

(1-γ)^{-1}

replaced by the horizon

H

in the leading term. Matching minimax lower bounds show that the leading terms, and the implied

1

-Wasserstein rates, are optimal under i.i.d., Markovian, and episodic sampling. Together, these results provide a unified non-asymptotic analysis of asynchronous categorical distributional TD across scalar, multivariate, discounted, and fixed-horizon settings, with rates that are minimax optimal in their leading terms.

Ege C. Kaya, Abolfazl Hashemi

Elmore Family School of Electrical and Computer Engineering Purdue University West Lafayette, IN 47906

Distributional RL Reinforcement Learning Stochastic Approximation Temporal-Difference Learning

May 7, 2026cs.LG

Path-Coupled Bellman Flows for Distributional Reinforcement Learning

Distributional reinforcement learning (DRL) models the full return distribution, but existing finite-support or quantile-based methods rely on projections, while recent flow-based approaches can suffer from \emph{boundary mismatch} at the flow source or from \emph{high-variance} bootstrapping when current and successor noises are independent. We propose Path-Coupled Bellman Flows (PCBF), a continuous-time DRL method that learns return distributions with flow matching using \textbf{source-consistent Bellman-coupled paths}: the current path starts from the required base prior at

t{=}0

, reaches the Bellman target at

t{=}1

, and maintains a pathwise affine relation to the successor flow at intermediate times (without requiring time-

t

marginals to satisfy a distributional Bellman fixed point for all

t

). PCBF couples current and successor return flows through shared base noise and uses a

λ

-parameterized control-variate target:

λ{=}0

recovers an unbiased sample Bellman target, while

λ{>}0

trades controlled bias for variance reduction. Experiments on analytically tractable MRPs, OGBench, and D4RL show improved distributional fidelity and training stability, and competitive offline RL performance.

Boyang Xu, Qing Zou, Siqin Yang +1

School of Computing and Augmented Intelligence, Arizona State University, Tempe, AZ, USA.

Distributional RL Flow Matching

May 7, 2026cs.LG

Revisiting Adam for Streaming Reinforcement Learning

Learning from a sequence of interactions, as soon as observations are perceived and acted upon, without explicitly storing them, holds the promise of simpler, more efficient and adaptive algorithms. For over a decade, however, deep reinforcement learning walked the contrary path, augmenting agents with replay buffers or parallel sampling routines, in an effort to tame learning instability. Recently, this topic has been revisited by Elsayed et al. (2024), focusing on update computation through eligibility traces and modifications to the optimisation routine, resulting in the StreamQ algorithm. In this work we take a step back, investigating the efficacy of established updates, such as those implemented by DQN and C51 within this online setting. Not only do we find that they perform well, but through analysing how the optimisation algorithm generally, and Adam in particular, interacts with these updates, we contend that two properties are essential for robust performance: i) the derivative of the objective is to be bounded and ii) weight updates are variance-adjusted. Rigorous and exhaustive experimentation demonstrates that C51, which exhibits both characteristics, is competitive with StreamQ across a subset of 55 Atari games. Using these insights, we derive a variance-adjusted algorithm based on eligibility traces, termed Adaptive Q

(λ)

, which approaches double the human baseline on the same subset, surpassing existing methods by all performance metrics.

Florin Gogianu, Adrian Catalin Lutu, Razvan Pascanu

Bitdefender, Bucharest, Romania · Mila, Montreal, Canada

Distributional RL Reinforcement Learning Deep Q-Learning Adaptive Gradient Methods

May 5, 2026cs.LG

Quantile Geometry Regularization for Distributional Reinforcement Learning

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.

Zhaofan Zhang, Minghao Yang, Rufeng Chen +2

Information Hub, AI Thrust Hong Kong University of Science and Technology (Guangzhou) Guangzhou, Guangdong, China

Distributional RL Quantile Regression Risk-Sensitive RL Distributionally Robust Optimization Distributionally Robust RL

May 3, 2026cs.LG

Towards Efficient and Expressive Offline RL via Flow-Anchored Noise-conditioned Q-Learning

We propose Flow-Anchored Noise-conditioned Q-Learning (FAN), a highly efficient and high-performing offline reinforcement learning (RL) algorithm. Recent work has shown that expressive flow policies and distributional critics improve offline RL performance, but at a high computational cost. Specifically, flow policies require iterative sampling to produce a single action, and distributional critics require computation over multiple samples (e.g., quantiles) to estimate value. To address these inefficiencies while maintaining high performance, we introduce FAN. Our method employs a behavior regularization technique that uses a single flow policy iteration and requires a single Gaussian noise sample for distributional critics. Our theoretical analysis of convergence and performance bounds demonstrates that these simplifications not only improve efficiency but also lead to superior task performance. Experiments on robotic manipulation and locomotion tasks demonstrate that FAN achieves state-of-the-art performance while significantly reducing both training and inference runtimes. We release our code at https://github.com/brianlsy98/FAN.

Sungyoung Lee, Dohyeong Kim, Eshan Balachandar +2

The University of Texas at Austin, Austin, TX, USA · Independent Researcher, Seoul, South Korea.

Distributional RL Q-Learning Robotic RL Offline RL Policy Learning

Apr 26, 2026cs.LG

Distributional Reinforcement Learning via the Cramér Distance

This paper explores the application of the Soft Actor-Critic (SAC) algorithm within a Distributional Reinforcement Learning setting and introduces an implementation of such algorithm named Cramér-based Distributional Soft Actor-Critic (C-DSAC). The novel approach employs distributional reinforcement learning to represent state-action values, and minimizes the squared Cramér distance for learning the distribution. Empirical results across various robotic benchmarks indicate that our algorithm surpasses the performance of baseline SAC and contemporary distributional methods, with the performance advantage becoming increasingly pronounced in high-complexity environments. To explain the efficiency of the new approach, we conduct an analysis showing that its superior performance is partly due to \textit{confidence-driven} Q-value updates: High-variance target distributions (low confidence in target) lead to more conservative model updates, thereby attenuating the impact of overestimated values. This work deepens the understanding of distributional reinforcement learning, offering insights into the algorithmic mechanisms governing convergence and value estimation.

Vanya Aziz, Ivo Nowak, E. M. T Hendrix

HAW Hamburg · de Málaga

Distributional RL Robotic RL Soft Actor-Critic

Apr 22, 2026cs.LG

Distributional Value Estimation Without Target Networks for Robust Quality-Diversity

Quality-Diversity (QD) algorithms excel at discovering diverse repertoires of skills, but are hindered by poor sample efficiency and often require tens of millions of environment steps to solve complex locomotion tasks. Recent advances in Reinforcement Learning (RL) have shown that high Update-to-Data (UTD) ratios accelerate Actor-Critic learning. While effective, standard high-UTD algorithms typically utilise target networks to stabilise training. This requirement introduces a significant computational bottleneck, rendering them impractical for resource-intensive Quality-Diversity (QD) tasks where sample efficiency and rapid population adaptation are critical. In this paper, we introduce QDHUAC, a sample-efficient, target-free and distributional QD-RL algorithm that provides dense and low-variance gradient signals, which enables high-UTD training for Dominated Novelty Search whilst requiring an order of magnitude fewer environment steps. We demonstrate that our method enables stable training at high UTD ratios, achieving competitive coverage and fitness on high-dimensional Brax environments with an order of magnitude fewer samples than baselines. Our results suggest that combining target-free distributional critics with dominance-based selection is a key enabler for the next generation of sample-efficient evolutionary RL algorithms.

Behrad Koohy, Jamie Bayne

Luffy.AI Culham, Oxfordshire, United Kingdom

Distributional RL Reinforcement Learning Quality-Diversity Optimization Sample-Efficient RL

Apr 20, 2026stat.ML

Distributional Off-Policy Evaluation with Deep Quantile Process Regression

This paper investigates the off-policy evaluation (OPE) problem from a distributional perspective. Rather than focusing solely on the expectation of the total return, as in most existing OPE methods, we aim to estimate the entire return distribution. To this end, we introduce a quantile-based approach for OPE using deep quantile process regression, presenting a novel algorithm called Deep Quantile Process regression-based Off-Policy Evaluation (DQPOPE). We provide new theoretical insights into the deep quantile process regression technique, extending existing approaches that estimate discrete quantiles to estimate a continuous quantile function. A key contribution of our work is the rigorous sample complexity analysis for distributional OPE with deep neural networks, bridging theoretical analysis with practical algorithmic implementations. We show that DQPOPE achieves statistical advantages by estimating the full return distribution using the same sample size required to estimate a single policy value using conventional methods. Empirical studies further show that DQPOPE provides significantly more precise and robust policy value estimates than standard methods, thereby enhancing the practical applicability and effectiveness of distributional reinforcement learning approaches.

Qi Kuang, Chao Wang, Yuling Jiao +1

School of Statistics and Data Science, and Philosophy and Social Sciences Laboratory of Data Science in Finance and Economics at the Ministry of Education,2026 Jiangxi University of Finance and Economics · School of Statistics and Data Science, Shanghai University of Finance and EconomicsApr · School of Artificial Intelligence, and Hubei Key Laboratory of Computational Science, 24 Wuhan University

Distributional RL Quantile Regression Off-Policy Evaluation Sample Complexity

May 23, 2024cs.LG

On the Approximation and Convergence of Distributional Policy Gradient Algorithms for Risk-Sensitive Reinforcement Learning

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.

Xian Yu, Minheng Xiao, Lei Ying

Department of Integrated Systems Engineering, The Ohio State University, Columbus, OH, USA · Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA

Stochastic Optimization Distributional RL Risk-Sensitive RL Policy Gradient Methods