Proximal Diffusion Models: Theory & Practice

• Proximal Diffusion Models (ProxDM) are generative frameworks that use proximal operators to replace traditional score estimation, enhancing sampling efficiency.
• They enable larger implicit update steps and robust handling of nonsmooth or heterogeneous data in both continuous and discrete state spaces.
• ProxDM integrates advances from convex optimization to achieve faster convergence and principled constraint management in diffusion-based modeling.

Proximal Diffusion Models (ProxDM) represent a class of generative modeling frameworks that leverage proximal operators—mathematical constructs central to implicit optimization methods—in the discretization, inference, and learning strategies of diffusion-based generative models. By substituting or augmenting the traditional score-based approaches with proximal mappings in either continuous or discrete state spaces, ProxDMs offer theoretical and empirical advances in sampling efficiency, generalization, robustness to model and data heterogeneity, and principled handling of domain constraints.

1. Concept and Theoretical Underpinnings

Proximal Diffusion Models (ProxDM) fundamentally depart from the dominant score-based paradigm of diffusion models, which revolve around learning the score function (i.e., the gradient of the log-density, $\nabla \log p(x)$) and utilizing it to define discretizations of reverse-time stochastic differential equations (SDEs) for sampling. While the forward discretization (e.g., Euler–Maruyama) of these SDEs necessitates small step sizes and large numbers of iterations, the ProxDM framework proposes a backward (implicit) discretization: each reverse step is expressed as a proximal map of the log-density or its noisy analog (Fang et al., 11 Jul 2025).

Formally, a single ProxDM sampling step can be written as

$$X_{k-1} = \operatorname{prox}_{-\lambda_k \log p_{t_{k-1}}}(\text{adjusted-noisy-input}),$$

where the proximal map

$$\operatorname{prox}_{f}(y) = \arg\min_x\, \left\{ f(x) + \frac{1}{2\lambda}\|x - y\|^2 \right\}$$

is applied to the noisy latent. Rather than learning the score $\nabla \log p_{t_{k-1}}(x)$, ProxDMs train a neural network to approximate the proximal operator directly via proximal matching. This enables implicit, larger-step updates and more global moves along the data probability manifold.

Theoretical results establish that such backward/proximal discretizations yield improved convergence rates. Specifically, under technical smoothness and moment assumptions, the ProxDM backward method achieves KL-divergence error $\varepsilon$ in $\widetilde{O}(d/\sqrt{\varepsilon})$ steps, compared with the $\widetilde{O}(d/\varepsilon)$ complexity of standard forward (score-based) discretizations (Fang et al., 11 Jul 2025). This improvement arises from the stability and larger-step capacity of proximal methods, paralleling their role in convex optimization.

2. Algorithmic Variants and Connections

Two main variants of ProxDM are considered:

• Fully Backward (Implicit) Discretization: Both drift/denoise and noise terms are evaluated at the future state, requiring a restriction on the step size but enabling maximal theoretical efficiency per iteration. The update can be expressed as:

$$X_{k-1} = \operatorname{prox}_{-\frac{2\gamma_k}{2-\gamma_k} \log p_{t_{k-1}}}\left( \frac{2}{2-\gamma_k} (X_k + \sqrt{\gamma_k} z_k) \right).$$

• Hybrid Discretization: The drift is split between old and new states, allowing for larger step sizes but at a modest theoretical cost, with the update:

$$X_{k-1} = \operatorname{prox}_{-\gamma_k \log p_{t_{k-1}}}\left( (1+\frac{1}{2}\gamma_k) X_k + \sqrt{\gamma_k} z_k \right).$$

These strategies are directly inspired by the backward Euler and proximal point algorithms from convex optimization, and their analysis relies on the properties of implicit discretizations in SDEs, as well as advances in proximal Langevin sampling (Ehrhardt et al., 2023, Klatzer et al., 2023).

3. Advantages and Practical Implications

The primary benefits of ProxDM—supported by both theory and empirical evidence—are as follows:

• Sampling Speed: High-quality samples are obtained in orders of magnitude fewer steps than with classical score-based samplers. For example, on MNIST and CIFAR-10, ProxDM methods achieve low Fréchet Inception Distance (FID) scores in as few as 10 iterations, whereas conventional methods require hundreds or thousands (Fang et al., 11 Jul 2025).
• Robustness to Nonsmoothness: The reliance on proximal operators rather than explicit gradient evaluation relaxes differentiability requirements. ProxDMs naturally address data distributions with jumps, discontinuities, or support on low-dimensional manifolds.
• Optimization Analogies: The core ProxDM step directly mirrors proximal point or implicit optimisation steps, suggesting the transference of acceleration, adaptive step-size, or variable splitting methods from convex optimization (Klatzer et al., 2023).
• Stability and Parameter Insensitivity: Implicit methods are less prone to instability and diverging trajectories due to poor hyperparameter choice. This proved particularly beneficial in experiments involving few-step sampling and noisy initialization (Fang et al., 11 Jul 2025).
• Constraint Handling: By encoding domain constraints or data priors as terms within the proximal operator, ProxDM enables the seamless inclusion of constraints, such as total variation for imaging or token-level restrictions in discrete spaces (Austin et al., 2021, Ehrhardt et al., 2023).

4. Connections with Structured and Discrete Diffusion Models

ProxDM generalizes naturally to discrete and structured state-spaces. In Discrete Denoising Diffusion Probabilistic Models (D3PMs), the forward process consists of Markov chains with structured transition matrices, and the reverse process can incorporate proximity operators or constraints specific to categorical variables, token embedding spaces, or absorbing states (e.g., mask tokens in language applications) (Austin et al., 2021).

Structured transition matrices $Q_t$ in D3PMs—for instance, those encoding locality or semantic proximity—can be viewed as proximal operations enforcing desired dynamics or priors over the discrete state space. Therefore, ProxDM frameworks subsume and extend discrete diffusion models by facilitating hybrid or domain-aware denoising pathways, illustrating the flexibility of the proximal approach.

5. Applications and Empirical Results

ProxDM has been successfully applied to diverse domains:

• Image and Text Generation: ProxDM achieves sharper and more realistic samples at reduced computational budgets, particularly in low-iteration regimes, outperforming conventional methods in both FID and Wasserstein distance (Fang et al., 11 Jul 2025).
• Image Restoration and Inverse Problems: By integrating data fidelity via measurement-consistent proximal operations, ProxDM-based approaches have improved perceptual and distortion metrics for super-resolution and inpainting, mitigating over-smoothing and reducing error accumulation (Wu et al., 25 Feb 2024).
• Bayesian Imaging and MCMC: The stochastic relaxed proximal-point and inexact proximal Langevin samplers, which employ similar backward/proximal discretizations, achieve accelerated posterior sampling with bounded bias in high-dimensional, non-smooth Bayesian inference (Ehrhardt et al., 2023, Klatzer et al., 2023).
• Federated and Heterogeneous Settings: Proximal terms incorporated into client objectives in federated diffusion model training (FedDM-prox) stabilize convergence and model quality under non-IID data distributions (Vora et al., 20 Jul 2024).
• Reward-guided Generation and RL: Proximal updates are effectively combined with reward difference prediction (PRDP) and with Proximal Policy Optimization (PPO), enabling stable reward finetuning and improved exploration via virtual trajectory generation (Deng et al., 13 Feb 2024, Tianci et al., 2 Sep 2024).

6. Open Directions and Limitations

Several avenues remain open for future research:

• Approximation Effects: Theoretical guarantees currently assume access to exact proximal operators. Quantifying the effect of network-based approximation errors or developing architectures closer to exact proximal mapping remains a priority.
• Optimized Sampling Schedules: The selection of time and regularization level sampling strategies in the training and sampling of ProxDM is currently heuristic; analytical or optimal methods could further enhance sample efficiency.
• Extensions to Other SDEs and ODEs: Adapting the ProxDM methodology to variance-exploding SDEs, probability flow ODEs, and other dynamical systems is an active area.
• Integration of Optimization Techniques: The close analogy to optimization theory implies possible gains from adopting variable splitting, adaptive step size, or acceleration methods.
• Empirical Scaling: While results are promising, further demonstrations are needed in large-scale, high-dimensional, or application-specific settings such as high-resolution image synthesis or molecular modeling.

7. Summary Table: Forward Score vs. Backward Proximal Discretization

Aspect	Score-Based (Forward)	ProxDM (Backward/Proximal)
Principal Update	$x_{k-1} \gets x_k + h\,s_\theta(x_k)$	$$x_{k-1} = \operatorname{prox}_{-\lambda_k \log p_{t_{k-1}}}(\dots)$$
Step Size Constraints	Small, many steps	Larger, fewer steps
Sample Efficiency	$\widetilde{O}(d/\varepsilon)$	$\widetilde{O}(d/\sqrt{\varepsilon})$
Differentiability	Requires score differentiability	Can handle nonsmooth log-density
Empirical Sensitivity	Can be unstable, sensitive	Increased robustness

• "Beyond Scores: Proximal Diffusion Models" (Fang et al., 11 Jul 2025)
• "Structured Denoising Diffusion Models in Discrete State-Spaces" (Austin et al., 2021)
• "Proximal Langevin Sampling With Inexact Proximal Mapping" (Ehrhardt et al., 2023)
• "Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling" (Klatzer et al., 2023)
• "Diffusion Posterior Proximal Sampling for Image Restoration" (Wu et al., 25 Feb 2024)
• "Wasserstein proximal operators describe score-based generative models and resolve memorization" (Zhang et al., 9 Feb 2024)
• "An Efficient Membership Inference Attack for the Diffusion Model by Proximal Initialization" (Kong et al., 2023)
• "The Uncanny Valley: A Comprehensive Analysis of Diffusion Models" (Ghanem et al., 20 Feb 2024)
• "FedDM: Enhancing Communication Efficiency and Handling Data Heterogeneity in Federated Diffusion Models" (Vora et al., 20 Jul 2024)
• "PRDP: Proximal Reward Difference Prediction for Large-Scale Reward Finetuning of Diffusion Models" (Deng et al., 13 Feb 2024)
• "Enhancing Sample Efficiency and Exploration in Reinforcement Learning through the Integration of Diffusion Models and Proximal Policy Optimization" (Tianci et al., 2 Sep 2024)

Proximal Diffusion Models thus unify and extend denoising-based, discrete-state, regularized, and constraint-aware generative modeling in both theory and practice, offering a rapidly expanding toolkit for principled, efficient, and robust high-dimensional generative modeling.