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Offline Deep Q* Estimation with Diffusion Models

Xiaohong Chen, Yuling Jiao, Lican Kang, Jerry Zhijian Yang, Chen Zhong

arXiv:2608.14401v1stat.MLcs.LG

TL;DR

Offline RL must estimate Q* from fixed data even though the reward law and transition kernel are unknown. The paper estimates those conditional objects with diffusion models, constructs an approximate optimal Bellman operator, and learns Q* from its residual. It obtains oracle-stage rates for Bellman residual and value estimation, while avoiding Bellman completeness but leaving broader settings as future directions.

  • Problem

    Offline RL must estimate Q* from previously collected data because repeated online interaction may be costly, unsafe, or infeasible, while recursive fitted methods can propagate approximation and optimization errors.

  • Method

    The framework estimates one-step conditional reward laws and transition kernels with diffusion models, plugs them into the optimal Bellman operator, and learns Q* over a neural-network class.

  • Results

    The estimator attains excess Bellman residual rate O(n^-2β/(d_x+d_a+2β)) and value-function error rate O(n^-β/(d_x+d_a+2β)), matching the oracle value-stage convergence rate.

  • Takeaways & Limitations

    Separating environment-law estimation from value learning yields a deep Q* estimation framework whose analysis avoids Bellman completeness assumptions.

  • Takeaways & Limitations

    The conclusion identifies extension to high-dimensional locomotion and partially observable settings as more challenging directions.

Abstract

from arXiv · show

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.

1 Introduction

Offline RL must estimate Q* from fixed data because the reward and transition mechanisms are unknown and online interaction may be unavailable. The paper separates environment-law estimation from value learning, using conditional diffusion models to approximate the optimal Bellman operator and deriving guarantees without Bellman completeness.

  • Motivation: Offline RL estimates value functions from limited, potentially distribution-shifted data when repeated online interaction may be costly, unsafe, or infeasible.The setting relies on previously collected data, often generated by an unknown behavior policy.
  • Existing approaches: Classical fitted-value methods repeatedly regress Bellman targets, but recursive bootstrapping can propagate approximation and optimization errors and max-based targets may induce overestimation bias.These methods include FQI, DQN, and DFQI.
  • Existing approaches: Model-based offline RL estimates rewards or transitions, but its performance depends heavily on the quality and stability of learned environment models, especially in high-dimensional continuous settings.Existing methods commonly use learned models for rollouts, uncertainty penalties, or conservative policy optimization.
  • Proposed framework: The proposed framework first estimates conditional reward laws and transition kernels with diffusion models, then plugs them into the optimal Bellman operator and learns Q* by minimizing empirical Bellman residuals.This separates reward and transition-law estimation from value-function learning rather than using learned models for policy rollouts.
  • Theory: The analysis derives convergence rates for diffusion-based operator learning and an oracle excess Bellman residual rate of ~O(n^-2β/(d_x+d_a+2β)).The rates are obtained through an end-to-end analysis of conditional diffusion estimation and error propagation through the Bellman operator.
  • Theory: Under concentrability, the resulting estimator achieves L2 rate ~O(n^-β/(d_x+d_a+2β)) without requiring Bellman completeness of the whole function class.The framework instead reduces the problem to approximating Q* for the optimal Bellman operator.

2 Preliminaries

The paper formulates offline reinforcement learning with discounted Markov decision processes and continuous state-action spaces. It also reviews conditional diffusion models, ReLU neural networks, and notation used in the analysis.

  • Markov Decision Processes: An MDP comprises state and action spaces, a transition kernel, a reward distribution, and a discount factor γ ∈ [0, 1).The transition kernel gives the next-state distribution for each state-action pair.
  • Data and Assumptions: The analysis assumes continuous state-action spaces, specifically X × A = [0, 1]^dx × [0, 1]^da, with i.i.d. observations under a behavior policy.Three independent batch datasets are used, with Sr and Sx for operator learning and D for value-function learning.
  • Markov Decision Processes: The optimal action-value function Q* is associated with a maximizing policy and satisfies the fixed-point equation Q* = T*Q*.The paper defines Q* through the optimal Bellman equation.
  • Diffusion Models: Diffusion models perturb data with Gaussian noise toward a tractable prior and reverse the process using a score function to generate target-distribution samples.Conditional diffusion models make the forward and reverse processes depend on a covariate, and score matching estimates the conditional score.
  • Deep Neural Networks: A ReLU DNN composes affine transformations with element-wise ReLU activations, parameterized by network depth, layer dimensions, and weights.The preliminaries define width, size, weight bounds, and uniform function bounds for these networks.

3 Method

The method estimates the unknown optimal Bellman operator from offline data using conditional diffusion models for rewards and transitions, then learns Q* by minimizing Bellman residuals. This decouples operator estimation from value-function learning while accommodating complex conditional laws.

  • Optimal Bellman Operator Learning: The unknown reward law and transition kernel make the optimal Bellman operator unobservable, motivating direct conditional-distribution estimation from offline data.The framework first learns these distributions and then constructs a data-driven Bellman operator.
  • Optimal Bellman Operator Learning: Conditional diffusion models estimate reward and transition laws, producing approximations bR(·|x, a) and bP(·|x, a).The reward and transition models are learned from separate batch datasets and can generate samples for a given state-action pair.
  • Value Function Learning: The final estimator bQ is obtained by empirical Bellman residual minimization over a uniformly bounded ReLU DNN class.The empirical optimal Bellman operator is a γ-contraction, and bQ is its associated empirical risk minimizer.
  • Optimal Bellman Operator Learning: The estimated reward and transition laws are incorporated into an empirical optimal Bellman operator, which supplies the target for value learning.The empirical operator uses generated reward and next-state samples for candidate Q-functions.
  • Method Design: Conditional diffusion models support flexible modeling of continuous and potentially multimodal reward and transition distributions without adding model-generated rollout trajectories.The paper emphasizes that this flexibility is used directly in constructing the Bellman operator.

4 Theoretical Analysis

The theoretical analysis separates errors from learning the Bellman operator and the value function, then combines them into convergence guarantees for the proposed estimator. It also establishes value-function bounds under concentrability and avoids requiring Bellman completeness.

  • Error decomposition: The method learns reward and transition laws with conditional diffusion models, then estimates the value function by minimizing Bellman residuals under the learned operator.The final error depends jointly on reward and transition estimation accuracy and value-function learning error.
  • Error decomposition: The error decomposition isolates operator discrepancies from statistical and approximation errors in the value-learning step.This decomposition enables separate bounds for optimal Bellman operator learning and value-function learning before combining them.
  • Bellman operator learning: The diffusion analysis accounts for reverse-process time discretization and achieves the minimax-optimal nonparametric rate up to logarithmic factors.The result is tailored to the discrete-time regime rather than assuming tractable integrals over a continuous time interval.
  • Assumptions and implications: The analysis avoids Bellman completeness by controlling approximation error directly through approximation of Q* within the neural-network class.Bellman completeness of the whole function class is not required.
  • Value-function learning: The value-stage analysis establishes an oracle excess-risk rate of approximately n^(-2β/(d_x+d_a+2β)) up to logarithmic factors.The rate concerns excess Bellman residual risk for the value-learning stage.
  • Value-function learning: Under a concentrability condition, controlling the Bellman residual yields an L2 value-function error rate of approximately n^(-β/(d_x+d_a+2β)) up to logarithmic factors.The bound depends on the mismatch constant between data, target, and successor comparison distributions.

5 Numerical Experiments

Experiments evaluate the proposed method on synthetic continuous-control benchmarks and a real-world sepsis-treatment task against MOPO, DFQI, and MABO. Across these settings, the method achieves the strongest reported evaluation performance, including coordinated control and simulated clinical treatment.

  • Experimental Setup: The evaluation covers Gymnasium’s Pendulum and Swimmer benchmarks plus the real-world MIMIC-III sepsis treatment task.The study compares the proposed method with MOPO, DFQI, and MABO.
  • Experimental Setup: Each Gymnasium offline dataset uses 200 random-policy episodes, while the proposed method uses 20,000 samples for diffusion training and 20,000 for Q* estimation.All methods are trained for 300 epochs, evaluated over 50 episodes, and results are averaged over three seeds.
  • Pendulum: The Pendulum results show that all methods improve during training, but the proposed method achieves the best evaluation reward.DFQI has competitive training behavior but lower evaluation performance, while MABO and MOPO obtain lower final evaluation rewards.
  • Swimmer: The proposed method achieves a Swimmer evaluation reward of 71.90, compared with 39.33 from the best baseline, MABO.The method maintains a relatively high training reward and performs best on the more challenging coordinated multi-joint control task.
  • Real-World Sepsis Treatment: The proposed method obtains the highest simulated sepsis-treatment reward, −5.33, improving over the clinician policy’s −5.59 and outperforming MOPO, MABO, and DFQI.Because reward is negative future SOFA, the reported improvement suggests lower predicted organ dysfunction in subsequent decision windows.

6 Conclusion

The paper develops an offline deep Q* framework that estimates reward and transition laws with conditional diffusion models before learning values by Bellman residual minimization. Its analysis separates operator estimation from value learning, avoids Bellman completeness, and derives rates supported by numerical experiments.

  • 6 Conclusion: The framework estimates reward laws and transition kernels from offline data, then incorporates the resulting Bellman operator estimator into deep Q* learning.This separates conditional generative modeling from value learning.
  • 6 Conclusion: The separation enables analysis of how optimal Bellman operator estimation errors propagate to value estimation without Bellman completeness assumptions.The paper explicitly identifies this assumption as unnecessary for its theoretical analysis.
  • 6 Conclusion: The analysis establishes convergence rates for reward-law and transition-kernel estimation, Monte Carlo approximation, excess Bellman residual risk, and value-function estimation.The conclusion attributes these rates to the proposed framework's theoretical analysis.
  • 6 Conclusion: O(n^-2β/(d_x+d_w+2β)) is the stated excess Bellman residual risk rate.The displayed notation uses d_x and d_w in the supplied conclusion passage.
  • 6 Conclusion: O(n^-β/(d_x+d_w+2β)) is the stated value-function estimation rate, matching the oracle value-stage convergence rate of a nonparametric least-squares estimator.Numerical experiments support the theoretical findings and practical effectiveness.
  • 6 Conclusion: Future work includes broader offline policy optimization and more challenging settings such as high-dimensional locomotion, partial observability, and large-scale world modeling.These settings are identified as directions for future investigation.

A.6 Error Bound for Approximating ∇log pt(x|w) with ReLU Neu-

This section constructs ReLU neural networks to approximate the diffusion score function across time intervals. The approximation combines localized density analysis, polynomial approximations, clipping, and switching networks while preserving stated complexity orders.

  • A.6 Error Bound for Approximating ∇log p_t(x|w) with ReLU Neural Networks: The construction approximates ∇log p_t(x|w) with ReLU networks over separate time intervals.The networks are later assembled through switching weights.
  • A.6 Error Bound for Approximating ∇log p_t(x|w) with ReLU Neural Networks: The constructed networks have depth O(log^4 N), width O(N^(d_x+d_w) log^7 N), and size O(N^(d_x+d_w) log^9 N).These complexity orders are repeated for the score-function approximators.
  • A.6 Error Bound for Approximating ∇log p_t(x|w) with ReLU Neural Networks: The approximation analysis decomposes errors into multiple terms and uses local polynomial, exponential-function, and neural-network approximations.The supplied proof passages describe these as successive approximation components.
  • A.6 Error Bound for Approximating ∇log p_t(x|w) with ReLU Neural Networks: Input clipping preserves the estimates on the relevant region and provides the required whole-space clipping identity.The clipping layer does not change the stated network-complexity orders.
  • A.6 Error Bound for Approximating ∇log p_t(x|w) with ReLU Neural Networks: Switching-weight products add only O(log N) depth and size per coordinate, which remains dominated by the original network complexities.The assembled network covers the separate time intervals.

A.7 Statistical Error of the Score Function

This section analyzes the statistical error of the score-function estimator over a constrained ReLU network class. The proof combines Lipschitz and covering-number bounds with concentration inequalities to control empirical estimation error.

  • A.7 Statistical Error of the Score Function: The statistical analysis begins by establishing Lipschitz continuity for the auxiliary loss over the restricted ReLU network class.The class is explicitly restricted to networks in G.
  • A.7 Statistical Error of the Score Function: Covering-number evaluation controls the complexity of the neural network class used in the statistical error bound.The supplied passages introduce covering-number evaluation as a distinct proof step.
  • A.7 Statistical Error of the Score Function: Bernstein’s inequality is used to control centered empirical loss deviations, while Hoeffding’s inequality bounds a later approximation term.The proof separates statistical and approximation errors.
  • A.7 Statistical Error of the Score Function: The resulting bounds retain logarithmic factors involving N, n_x, and the confidence parameter δ.The supplied expressions include terms such as log N, log n_x, and log(1/δ).
  • A.7 Statistical Error of the Score Function: The analysis decomposes the error into three terms and bounds them individually before combining the results.The passages identify statistical and approximation components within this decomposition.

A.8 Error Bound for the Score Function

This section combines the preceding approximation and statistical analyses to obtain the final score-function error bound. The resulting expression has a nonparametric rate in the diffusion approximation index and logarithmic dependence on sample size.

  • A.8 Error Bound for the Score Function: The theorem proof combines the preceding approximation and statistical bounds to derive the final score-function error.The supplied passages explicitly introduce the final statistical-error bound after substituting the component estimates.
  • A.8 Error Bound for the Score Function: The resulting rate includes a factor of order n_x^(-2α/(d_x+d_w+2α)) multiplied by logarithmic terms.The supplied expressions show log n_x and log^18 n_x factors in the final bounds.

B.1 Proof of Lemma 4.4

The proof bounds the discrepancy between the true reverse process and its continuous-time Euler–Maruyama interpolation using path-law comparisons, score-error control, and total variation arguments.

  • The proof compares the path laws of the true reverse process and the Euler–Maruyama interpolation.The two processes are initialized from the same distribution and share diffusion coefficient 2I_dx.
  • Pinsker’s inequality converts the resulting relative-entropy control into a total variation bound for the terminal distributions.The terminal distributions are identified as p_T1 and the discretized distribution ep_T1.
  • Gaussian convolution identities, Lipschitz bounds, moment estimates, and the Burkholder–Davis–Gundy inequality control the intermediate process terms.The proof applies these tools to bound drift differences, state increments, and score residuals.
  • The discretization argument requires the maximum time-step size to be below a deterministic threshold Δ_ε.For sufficiently small Δ, higher-order terms involving Δ^2, Δ^3, and Δ^8 are absorbed by the leading term.
  • The analysis controls score approximation and temporal variation through integrated error quantities E and V_Δ.These quantities measure score error over time and mean-square variation between discretization times.

B.3 Proof of Lemma 4.6

This proof bounds a boundary contribution by using the density assumptions and moment estimates associated with the terminal distribution.

  • The proof decomposes the bound into terms controlled by Assumption 4.1 and the terminal density p_T1.The boundary term is treated separately from the other contributions.
  • The factorization 1 − μ_T1^d_x = (1 − μ_T1)(1 + μ_T1 + ··· + μ_T1^{d_x−1}) is used in the estimate.This rewrites the dimension-dependent expression into a product involving 1 − μ_T1.
  • Combining the preceding inequalities completes the proof of the lemma.

B.4 Proofs of Theorem 4.2 and Corollary 4.1

The proofs combine transition and reward-law estimation bounds with empirical-process and neural-network approximation arguments to establish the operator and value-stage guarantees.

  • Theorem 4.2 and Corollary 4.1: Theorem 4.2 is obtained by applying Lemmas 4.4–4.6 to three error terms and replacing the estimated transition law with the true transition law.
  • Theorem 4.2 and Corollary 4.1: The reward-law error bound follows by repeating the transition-kernel argument with the next-state variable replaced by the reward variable.
  • Value-stage analysis: The value-stage proof controls statistical error using covering numbers, Hoeffding inequalities, and offset Rademacher complexity.The analysis uses ε-covers of the function class and bounds the associated empirical fluctuations.
  • Value-stage analysis: The approximation analysis uses a ReLU deep neural network class whose size, depth, width, and weight bounds determine the covering-number estimate.

C.2 Proof of Theorem 4.4

The proof establishes Theorem 4.4 by first showing that the empirical optimal Bellman operator is a γ-contraction and then combining statistical and approximation bounds.

  • The empirical optimal Bellman operator bT* is shown to be a γ-contraction under the supremum norm.
  • The contraction argument considers arbitrary measurable functions Q1 and Q2 bounded between 0 and R_max.
  • Using the same samples at each state-action pair allows the operator comparison to share the reward and transition samples.
  • The theorem follows by combining the statistical error bound with the approximation error bound.

D Main Results

The section develops an error decomposition for the learned Bellman operator and derives value-estimation guarantees. Under the stated conditions, the analysis yields rates for Bellman residuals and L2 estimation while avoiding Bellman completeness assumptions.

  • Error decomposition: The proof decomposes the excess Bellman residual into operator-estimation and value-learning errors.The decomposition is stated for the learned estimator and is combined with subsequent theorem bounds.
  • L2 control: Under the density-ratio and positive-measure conditions, the resulting bound includes √Cδ/(1−γ) times the Bellman-operator error plus γδ/(1−γ).These conditions support the comparison-kernel argument used to relate residual and L2 errors.
  • Value-stage rate: n^{-2β/(d_x+d_a+2β)} is obtained for the value-stage rate through the choice m ≲ n^{-2β/(d_x+d_a+2β)}.The rate appears in the theorem proof alongside the network-size choice.

E Auxiliary Results

The auxiliary section collects technical probability, information-theoretic, diffusion, and neural-network approximation results used in the analysis. It also records a rate expression for the combined estimation problem.

  • Rate bound: The auxiliary results include a combined rate of n^{-max{α/(d_x+d_a+2α), β/(d_x+d_a+2β)}}.This expression appears as the final rate obtained after combining the preceding bounds.
  • Probability tools: The section states probability and information-theoretic tools including Hoeffding, Bernstein, KL-divergence, mutual-information, and randomized-estimator bounds.These are listed as auxiliary lemmas for concentration and lower-bound arguments.
  • Diffusion tools: Diffusion-related lemmas connect path measures and terminal distributions and provide score-based identities used in conditional diffusion analysis.The listed results include a conditional density setting and Gaussian convolution score identity.
  • Neural-network tools: Neural-network lemmas provide concatenation, parallelization, identity, product, reciprocal, square-root, clipping, and switching constructions.The constructions specify how ReLU networks can be composed or used to approximate elementary operations.
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