Diffusion Value Function (DVF)

• DVF is a reinforcement learning paradigm that uses diffusion processes to embed environment structure into value function estimation.
• It leverages techniques like graph neural networks, denoising rollouts, and spectral feature mappings to improve credit assignment and sample efficiency.
• DVF methodologies are underpinned by rigorous theoretical foundations, including Bellman fixed-point equations and contraction properties, ensuring scalable policy optimization.

The Diffusion Value Function (DVF) is a set of advanced methodologies in reinforcement learning (RL) and generative modeling that leverage the structure of diffusion processes for value estimation, policy optimization, and environment modeling. DVF approaches have emerged independently in multi-agent RL on graphs, value-predictive diffusion models for control, efficient spectral learning in RL, and the dense fine-tuning of generative diffusion models. Across these domains, DVF variants explicitly embed environment topology or trajectory structure into value computations, often yielding scalable and theoretically grounded alternatives to standard value function approximations or reward assignment schemes.

1. Definitions and Core Mathematical Structures

The Diffusion Value Function is context-dependent and has received distinct formalizations:

• Multi-Agent RL on Graphs: DVF assigns to each agent a value by “diffusing” rewards over a known influence graph. For a set of agents $\mathcal{V}=\{1,\ldots,n\}$ interacting on a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with adjacency matrix $A$, the DVF for policy $\pi$ is defined as

$$V_D^\pi(S) = \mathbb{E}_\pi\left[ \sum_{t=0}^\infty \Gamma^{t+1} R^t \mid S^0 = S \right], \quad \text{where} \quad \Gamma = \gamma A D^{-1}$$

and $R^t$ is the vector of local agent rewards (Rashwan et al., 16 Jan 2026).

• Diffusion-Model-Based Value Function Estimation: DVF refers to estimating the expected value by simulating future states directly from a conditional diffusion model:

$$V^\pi(s_t) \approx \frac{1}{n}\sum_{i=1}^n r_\psi\!\bigl(s_{t+\Delta t,i},\,\pi(s_{t+\Delta t,i})\bigr), \quad s_{t+\Delta t,i} \sim \rho_\theta(s_{t+\Delta t} \mid s_t, \Delta t, \phi(\pi))$$

where rollouts are performed via a denoising diffusion process conditioned on state, timestep, and policy representation (Mazoure et al., 2023).

• Spectral Representation (Diff-SR): The DVF is not computed by explicit sampling but rather via a learned spectral feature map $\phi_\theta(s,a)$ that encodes the necessary information for value function approximation in a linear parameterization:

$Q^\pi(s,a) = \phi_\theta(s,a)^\top \xi^\pi$

This map is trained via a diffusion-based score-matching objective (Shribak et al., 2024).

• Process Value Function in Diffusion Model Fine-Tuning: For diffusion-based generative models, DVF predicts the expected cumulative reward from any intermediate noisy state $x_{T-t}$:

$$V_\phi(x_{T-t}, c) = \mathbb{E}[r(x_0, c) \mid x_{T-t}]$$

enabling dense reward-based learning throughout the denoising trajectory (Dai et al., 21 May 2025).

2. Theoretical Foundations and Properties

Each DVF variant is underpinned by distinct theoretical results:

• Graph-Structured DVF: The value function $V_D$ satisfies a vector-valued Bellman fixed-point equation:

$$V_D(S) = \Gamma\,\mathbb{E}_\pi[R^t + V_D(S^{t+1}) \mid S^t = S],$$

with the corresponding Bellman operator $\mathcal{T}^\pi$ a contraction under the sup–$\ell_1$ norm, ensuring existence and uniqueness of the fixed point even for infinite-horizon settings. Averaging over agent components recovers the standard global value function:

$n^{-1}\mathbf{1}^\top V_D^\pi(S) = V^\pi(S)$

(Rashwan et al., 16 Jan 2026).

• Policy Alignment: If all agents' DVFs are improved under a new policy, the global value increases, providing a strong form of alignment between local improvement and global return (Rashwan et al., 16 Jan 2026).
• Spectral Sufficiency: In Diff-SR, spectral properties guarantee that the feature representation $\phi_\theta$ is expressive enough to represent any $Q^\pi$, and the underlying Bellman operator is a contraction for TD-based learning (Shribak et al., 2024).
• Diffusion Process Value Function: The process DVF satisfies a fitted Bellman recursion along the reverse diffusion chain:

$$V(s_t) = \begin{cases} \mathbb{E}_{s_{t+1}\sim p_\theta(\cdot|x_{T-t},c)}[V(s_{t+1})], & t < T-1 \ r(x_0, c), & t = T-1 \end{cases}$$

(Dai et al., 21 May 2025)

3. Algorithmic Implementation and Learning Procedures

A variety of practical algorithms leverage DVF for scalable value learning and policy improvement:

• GNN-Based Critic for Graph MARL: The DVF is estimated with a graph neural network (GNN) parameterizing $V_{D,\phi}(S)$, trained by minimizing the squared diffusion TD-error:

$$\delta^t = \Gamma [R^t + V_{D,\phi}(S^{t+1})] - V_{D,\phi}(S^t)$$

This supports both centralized training with decentralized execution and parameter sharing across agents (Rashwan et al., 16 Jan 2026).

• Conditional Diffusion Model for Value Estimation: DVF is realized using a conditional denoising diffusion process, where sampling from $\rho_\theta$ directly implements (discount-weighted) state visitation measure estimation. Value is estimated by Monte-Carlo averaging over sampled futures scored by a reward predictor (Mazoure et al., 2023).
• Spectral Representation (Diff-SR): The feature extractor $\phi_\theta$ is optimized via a one-step score-matching loss, with no multi-step diffusion rollouts required at inference. Standard off-policy, linear-function-approximation actor–critic architectures consume these features for policy optimization (Shribak et al., 2024).
• Value-Guided Diffusion Fine-Tuning (VARD): The system first pretrains a process value function $V_\phi(x_{T-t},c)$ via regression to Monte-Carlo returns along the diffusion trajectory, then optimizes the generation policy using a combination of value maximization and KL penalty:

$$\mathcal{L}_{\rm VARD}(\theta) = -\mathbb{E}[V_\phi(x_{T-t}, c)] + \eta \mathbb{E}[\| x_{T-t} - x^0_{T-t} \|^2]$$

where $x^0_{T-t}$ are anchor samples from the pretrained model (Dai et al., 21 May 2025).

4. Applications and Empirical Evaluation

DVF-based methods have demonstrated efficacy and scalability in several domains:

Context	Problem Class	Empirical Summary
Graph-based MARL (DVF, DA2C)	Multi-agent credit assignment, GMDPs	Consistently outperforms local/global critics by up to 11% mean reward across benchmarks such as distributed coloring, power allocation, and firefighting (Rashwan et al., 16 Jan 2026).
Conditional diffusion for control	Offline RL, policy evaluation	Achieves high correlation (≳0.9) with true returns, especially on low-quality/behavioral-cloning RL datasets. Efficient zero-shot evaluation for arbitrary policies (Mazoure et al., 2023).
Spectral RL (Diff-SR)	MDPs/POMDPs with high-dimensional states	Achieves state-of-the-art or competitive returns on Gym-MuJoCo and Meta-World, with 3–4× wall-clock speedup over competing diffusion RL approaches (Shribak et al., 2024).
Diffusion model fine-tuning (VARD)	Reward-guided generative model editing	Substantially improves reward metrics in protein design, image aesthetics, and compressibility, with high sample efficiency and robustness to non-differentiable rewards (Dai et al., 21 May 2025).

5. Significance, Limitations, and Broader Implications

DVF and its derivatives provide principled mechanisms to inject structural information, either from explicit environment graphs, policy-conditional state visitations, or the Markovian structure of diffusion generative processes, into value function learning and optimization. This structure yields:

• Improved Credit Assignment: Agent-wise value factoring aligns reward propagation with actual influence structures, leading to more informative learning signals (Rashwan et al., 16 Jan 2026).
• Sample-Efficient Control: Direct diffusion sampling or spectral representations avoid compounding prediction errors and support learning from minimal action/reward labels (Mazoure et al., 2023, Shribak et al., 2024).
• Efficient Policy Optimization: Use of process value functions enables dense, stable RL-style fine-tuning of diffusion models, overcoming the sparsity and instability of prior methods for both differentiable and non-differentiable rewards (Dai et al., 21 May 2025).

However, certain limitations are noted:

• In MARL, DVF estimation depends on accurate knowledge of the influence graph and may be ill-behaved if the graph is misspecified (Rashwan et al., 16 Jan 2026).
• In diffusion-based value estimation, rollout-based methods are sensitive to noise schedules and require careful policy conditioning (Mazoure et al., 2023).
• Spectral methods necessitate the expressiveness of the feature map, and their statistical sufficiency depends on properties of the environment kernel (Shribak et al., 2024).
• For process DVF in VARD, value regression relies on diverse, representative samples along the backward diffusion path, and the method's stability may depend on KL regularization tuning (Dai et al., 21 May 2025).

A plausible implication is a trend towards integrating explicit structural priors—be it graph topology, transition kernels, or denoising chains—within value learning to improve scalability, robustness, and credit assignment in both classic RL and modern generative modeling.

6. Connections to Related Frameworks

DVF methods intersect with several established and emergent paradigms:

• Factored Value Functions: DVF generalizes factored critics that decompose global value using physical/locality priors, extending value factorization to infinite-horizon, graph-structured settings (Rashwan et al., 16 Jan 2026).
• Score-Based Learning and Energy-Based Modeling: The spectral DVF leverages random-Fourier expansions of exponential-family models, tightly connecting diffusion score learning to energy-based policy learning and classical spectral RL (Shribak et al., 2024).
• Process Value Functions in Diffusion Models: DVF's role in generative model fine-tuning aligns with advances in RL for non-differentiable objectives and dense reward propagation along trajectory-like generation processes (Dai et al., 21 May 2025).

The DVF construct thus represents a versatile and theoretically grounded principle for scalable value estimation and RL, adaptable across MARL, generative modeling, and representation learning.