Offline Deep Q* Estimation with Diffusion Models

In offline RL, estimating the optimal action-value function $Q^*$ can be formulated as solving the optimal Bellman equation based solely on offline observations. A fundamental challenge is that the reward function and transition kernel are unknown, so the optimal Bellman operator is not directly observable from data. To address this issue, we propose a novel framework that decouples operator estimation from value function learning. In this approach, we first formulate conditional diffusion models to estimate the reward law and transition kernel, which induces a data-driven approximation of the optimal Bellman operator. We then plug these estimators into the Bellman equation and obtain a deep estimator of $Q^*$ by minimizing the empirical Bellman residual over a neural network function class. Theoretically, we first establish sharp nonasymptotic convergence rates for learning the optimal Bellman operator through an end-to-end analysis of conditional diffusion estimation in total variation distance. We then establish the oracle value-stage rate $\widetilde{\mathcal O}\bigl(n^{-\frac{2β}{d_x+d_a+2β}}\bigr)$ for the excess Bellman residual risk. Finally, under a concentrability condition, we translate this residual bound into an $L^2$ convergence rate of $\widetilde{\mathcal O}\bigl(n^{-\fracβ{d_x+d_a+2β}}\bigr)$ for the resulting deep estimator of $Q^*$, where $d_x$ and $d_a$ denote the dimensions of the state and action spaces, respectively, and $β$ denotes the Hölder smoothness index of $Q^*$. Importantly, our theoretical analysis does not rely on completeness assumptions commonly used in deep RL theory. Extensive numerical experiments demonstrate the effectiveness of the proposed method and its strong empirical performance.

Revisiting TD Target Aggregation under Uncertainty in Q-Learning

Deep Q-Networks (DQNs) learn value functions through bootstrapped temporal-difference updates, where future returns are approximated using a greedy maximization over next-state action values. While effective, this aggregation rule is inherently sensitive to estimation noise: when Q-values are uncertain, the maximization operator deterministically favors the largest estimate, regardless of its reliability, leading to amplified errors through bootstrapping. In this work, we propose the \textbf{S}uccessor Rollout \textbf{A}ggregation \textbf{D}eep \textbf{Q}-Network (SADQ), a simple modification to Q-learning that regularizes how the TD target is formed. SADQ uses one-step rollout predictions from a learned dynamics model to guide the comparison among candidate next-state actions, introducing additional structure into the aggregation step without altering the underlying learning framework. The resulting mixed Bellman update attenuates unreliable maxima while preserving the standard fixed point under diminishing model error. We provide theoretical analysis showing that SADQ reduces bootstrap-induced overestimation in a pointwise manner. Empirically, SADQ consistently improves training stability across classical control tasks, real-world vector-based environments, and Atari benchmarks when compared to strong DQN variants.

Spectral Analysis of Dueling Q-Learning

Q-learning is a fundamental algorithm in reinforcement learning (RL) for solving discounted Markov decision processes (MDPs) when the transition kernel is unknown. The deep Q-network (DQN) extends Q-learning by using a deep neural network for Q-function approximation, which makes Q-learning applicable to more practical high-dimensional problems. Dueling Q-learning decomposes the Q-function into a value function and an advantage function and learns the two components jointly, which can improve learning efficiency. However, the theoretical understanding of dueling Q-learning is still limited. Recent work has initiated an analysis of tabular dueling Q-learning, but existing guarantees focus on a regularized formulation and leave the pure tabular update less completely understood. This paper strengthens that line of analysis by adding a direct interpretation of the centered tabular decomposition and by establishing convergence guarantees for the unregularized, unprojected constant step-size recursion. In particular, we derive an exact switching linear system representation for deterministic dueling Q-learning and a finite-time error bound in expectation for the sampled stochastic version. The analysis clarifies how the value and advantage updates act as different gains on the action-common (value function) and action-differential (advantage function) components of the Q-function.

Mean Field Reinforcement Learning

This monograph provides an introduction to mean field reinforcement learning through the lens of Markov decision processes arising from large-population stochastic control with mean field interactions and common noise. Starting from the connection between multi-agent reinforcement learning and mean field control, it develops the probabilistic, mathematical, and control-theoretic framework needed to formulate representative-agent learning problems, analyze their relationship with finite-population systems, and study both general and linear-quadratic models. The presentation includes dynamic programming principles, propagation-of-chaos limits, and theoretical analyses of tabular Q-learning and policy-gradient methods. It also discusses numerical implementations, including tabular schemes and deep reinforcement learning methods such as deep deterministic policy gradient. The goal is to give readers a coherent bridge between mean field control theory and reinforcement learning methodology, emphasizing the mathematical structure of the problems and the design of tractable learning approaches for large stochastic populations.

NASDAQ: Normalized Observation Space Dynamics-Augmented Q-Learning

Augmenting model-free reinforcement learning (RL) with representations learned through observation dynamics prediction (observation-predictive RL) can improve sample efficiency and performance, with minor modifications and limited additional computation. However, this approach still struggles in challenging tasks with low-dimensional observations. In this paper, we identify a key factor behind this problem: unbalanced reconstruction losses across observation dimensions, where dimensions with larger value ranges dominate the loss. This encourages the agent to neglect dimensions with relatively small ranges, leading to degraded performance. To address this issue, we propose a novel normalization method tailored to online RL, which normalizes low-dimensional observations and balances the resulting losses and gradients. Beyond balancing reconstruction losses, observation normalization enables dynamics prediction to be performed in a normalized observation space, thereby providing a unified treatment of low- and high-dimensional inputs (e.g., physical states and images). Building on this idea, we further introduce Normalized Observation Space Dynamics-Augmented Q-learning (NASDAQ), a framework for observation-predictive RL applicable across diverse domains. NASDAQ learns state-action representations by coupling value learning with two auxiliary tasks: short-term value prediction and next normalized observation prediction. Extensive experiments demonstrate that NASDAQ achieves competitive or superior performance compared with state-of-the-art model-based and self-predictive RL methods, while requiring significantly less training wall-time.

Inverting the Bellman Equation: From $Q$-Values to World Models

Model-based and model-free reinforcement learning are traditionally viewed as separate paradigms: instead of learning a model of the transition kernel $P$, model-free agents typically estimate value functions tied to a specific policy and reward. In this paper, we challenge this dichotomy by proving that value-based agents trained on a sufficiently rich set of reward functions, e.g. using goal-conditioned RL, implicitly encode a unique and accurate world model. To extract this model in practice, we introduce \textit{$P$-learning}, an inverse analogue to $Q$-learning that samples from an agent's $Q$-values, policies and rewards to decode its internal model of the environment. We then provide sufficient conditions on the type and number of goals for which agents encode the true kernel $P$, covering both stochastic and deterministic MDPs over finite or continuous state spaces. Even when our assumptions are violated, we empirically demonstrate that agents trained on a handful of reward functions encode accurate dynamics in $\texttt{Reacher}$, $\texttt{MountainCar}$ and stochastic variants of $\texttt{FourRooms}$. Surprisingly, we find that policies trained exclusively on a \texttt{Reacher} agent's implicit world model are quasi-optimal on out-of-distribution, velocity-based goals despite position-only training -- suggesting that agents contain hidden generalisation capabilities and providing a new lens into the connection between model-based, model-free, and goal-conditioned RL.

Learning Fair Pareto-Optimal Policies in Multi-Objective Reinforcement Learning

Fairness is an important aspect of decision-making in multi-objective reinforcement learning (MORL), where policies must ensure both optimality and equity across multiple, potentially conflicting objectives. While single-policy MORL methods can learn fair policies for fixed user preferences using welfare functions such as the generalized Gini welfare function (GGF), they fail to provide the diverse set of policies necessary for dynamic or unknown user preferences. To address this limitation, we formalize the fair optimization problem in multi-policy MORL, where the goal is to learn a set of Pareto-optimal policies that ensure fairness across all possible user preferences. Our key technical contributions are threefold: (1) We show that for concave, piecewise-linear welfare functions (e.g., GGF), fair policies remain in the convex coverage set (CCS), which is an approximated Pareto front for linear scalarization. (2) We demonstrate that non-stationary policies, augmented with accrued reward histories, and stochastic policies improve fairness by dynamically adapting to historical inequities. (3) We propose three novel algorithms, which include integrating GGF with multi-policy multi-objective Q-Learning (MOQL), state-augmented multi-policy MOQL for learning non-statoinary policies, and its novel extension for learning stochastic policies. We evaluate our algorithms across various domains and compare our methods against the state-of-the-art MORL baselines. The empirical results show that our methods learn a set of fair policies that accommodate different user preferences.

Deep Q-Learning on Hölder Spaces

We study the operator-theoretic core of Q-learning in continuous-time stochastic control with continuous states and actions. In value-based reinforcement learning, each Q-learning or DQN update is built from a Bellman optimality target; our analysis isolates this target in a diffusion setting and studies its regularity and approximation complexity. Under uniform ellipticity and Hölder-regular coefficients, we show that a Bellman update maps bounded inputs into an anisotropic regularity class, smoothing the state variable while leaving only Lipschitz dependence on the action variable. This yields a compact family of Bellman iterates and motivates a tensor-product DeepONet architecture adapted to the mixed regularity of the problem. We then derive explicit approximation and resource bounds, together with a stiffness--complexity trade-off as the time step $δ\to 0$. The resulting theory makes a direct contribution to Q-learning theory at the level of Bellman target regularity and approximation in continuous stochastic control. At the same time, we do not claim a full convergence theorem for practical sampled Q-learning with exploration, replay, and stochastic gradient updates.

COP-Q: Safety-First Reinforcement Learning for Robot Control via Cholesky-Ordered Projection

Safe robot control requires maximizing return while satisfying safety constraints. In off-policy safe reinforcement learning, reward and safety Q-values are commonly learned by separate critic ensembles, with uncertainty handled independently for each objective. This objective-wise treatment neglects inter-objective correlation and can lead to overly conservative value estimates, thereby reducing sample efficiency. To address this issue, we propose Cholesky-Ordered Projection Q-learning (COP-Q), a safety-first method that incorporates inter-objective covariance into vector-valued Q-value estimation. COP-Q constructs a generalized confidence bound in the joint Q-value space and uses Cholesky factorization to encode objective priority in a sequential form. This preserves conservatism on safety while adaptively reducing excessive conservatism on the reward objective. The resulting estimate is used in both temporal-difference target computation and actor optimization. COP-Q incurs minimal computational overhead and is readily compatible with most existing deep Q-learning frameworks. Experiments on robot locomotion in Brax and safe navigation in Safety-Gymnasium, covering both hard- and soft-safety settings, demonstrate that COP-Q achieves strong safety performance together with competitive or improved sample efficiency relative to representative baselines.