Particle Denoising Diffusion Sampler

Angus PhillipsHai-Dang DauMichael John HutchinsonValentin De BortoliGeorge DeligiannidisArnaud Doucet

article2024ICML56 citations

Introduces the Particle Denoising Diffusion Sampler, an iterative particle method combining guided diffusions with a novel score matching loss to achieve asymptotically consistent sampling and normalizing constant estimation for complex, unnormalized target distributions.

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Generating samples from complex, unnormalized probability distributions and computing their normalizing constants are fundamental challenges across scientific computing, statistics, and machine learning. Traditional methods like annealing often degrade when exploring multimodal landscapes, while standard diffusion-based generative models rely heavily on available data samples and introduce persistent approximation errors when adapted to general sampling. The article addresses these bottlenecks by developing the Particle Denoising Diffusion Sampler (PDDS), a methodology designed to reliably sample from unnormalized target densities and deliver consistent, unbiased normalizing constant estimates.

The approach integrates guided denoising diffusions with Sequential Monte Carlo (SMC), a particle-based filtering framework. Rather than relying on static or heuristic approximations to reverse the noising diffusion process, PDDS formulates the time-reversed process as an iterative particle scheme. To correct for intermediate drift errors, the methodology introduces a Novel Score Matching (NSM) loss function to train neural network potential approximations. This novel loss eliminates the variance blow-up that standard score matching encounters as time steps approach zero, allowing particles to be systematically reweighted, resampled, and optionally perturbed using standard Markov Chain Monte Carlo moves.

The evaluation demonstrates several critical findings across synthetic benchmarks, Bayesian logistic regression, and high-dimensional models scaling up to 1,600 dimensions. First, PDDS, especially when combined with optional Markov Chain Monte Carlo steps, consistently matches or exceeds the performance of state-of-the-art baselines like CRAFT, DDS, and Path Integral Samplers in estimating normalizing constants. Second, on multimodal problems—such as a 20-dimensional mixture of 40 separated Gaussian components—PDDS substantially outperforms flow-based methods by preventing mode collapse and significantly reducing transport error. Third, theoretical analysis confirms that PDDS provides asymptotically consistent estimates and establishes that sorted stratified resampling prevents particle degeneration as discretization steps become infinitely fine.

These results provide practitioners with a scalable, mathematically rigorous framework that reduces computational risk and improves fidelity in high-dimensional probabilistic inference. Unlike flow-based methods that require extensive, problem-specific re-training to adjust time-step resolutions, PDDS uses standard, task-agnostic network architectures that seamlessly refine when time steps are subdivided. Decision-makers should consider adopting PDDS for complex posterior inference and explore pilot implementations where capturing separated modes is critical. Users must note that practical deployment uses a finite particle budget and relies on well-behaved initial variational approximations to maintain numerical stability, warranting caution and validation in unconstrained density settings.

CNRS Ecole Normale Supérieure University of Oxford diffusion probabilistic models gaussians matching algorithm Particle Denoising Diffusion Sampler diffusion models posterior sampling

arXiv: 2402.06320 angusphillips/particle_denoising_diffusion_sampler

Paper: Score-Based Generative Modeling through Stochastic Differential Equations, Yang Song et al. (2021). Its continuous-time SDE framework establishes the forward and reverse diffusion processes that PDDS turns into a particle-based sampling scheme.
Paper: Denoising Diffusion Probabilistic Models, Jonathan Ho et al. (2020). Its foundational denoising diffusion formulation clarifies the noise-corruption and learned-reversal setup that PDDS adapts for sampling unnormalized targets.

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Abstract

Denoising diffusion models have become ubiquitous for generative modeling. The core idea is to transport the data distribution to a Gaussian by using a diffusion. Approximate samples from the data distribution are then obtained by estimating the time-reversal of this diffusion using score matching ideas. We follow here a similar strategy to sample from unnormalized probability densities and compute their normalizing constants. However, the time-reversed diffusion is here simulated by using an original iterative particle scheme relying on a novel score matching loss. Contrary to standard denoising diffusion models, the resulting Particle Denoising Diffusion Sampler (PDDS) provides asymptotically consistent estimates under mild assumptions. We demonstrate PDDS on multimodal and high dimensional sampling tasks.

Topics

diffusion probabilistic models gaussians matching algorithm matching methods probability densities generative model

Keywords

Particle Denoising Diffusion Sampler diffusion models posterior sampling Monte Carlo sampling Feynman-Kac model stochastic differential equations importance sampling adaptive resampling

1. Introduction
2. Denoising Diffusions with Guidance
2.1. Noising and denoising diffusions
2.2. Denoising diffusions with guidance
2.3. Guidance approximation
3. Particle Denoising Diffusion Sampler
3.1. From continuous time to discrete time
3.2. From discrete time to particles
3.3. Algorithm settings
3.4. Theoretical results
4. Learning Potentials via Score Matching
4.1. Different score identities
4.2. Benefits of alternative score matching identity
4.3. Neural network parametrization
4.4. Training the neural network
4.5. Mechanisms behind potential improvement
5. Related Work
6. Experimental Results
6.1. Normalizing constant estimation
6.2. Sample quality
6.3. Iterations of potential approximation
7. Discussion
Impact Statement
Acknowledgements
References
Appendix
A. Particle Denoising Diffusion Sampler with Adaptive Resampling
B. Asymptotic Error Formulae for SMC
B.1. Chi-squared divergence
B.2. Generic Feynman-Kac formula and the associated SMC algorithm
B.3. Sorted stratified resampling schemes
B.4. Asymptotic error
C. Proofs
C.1. Proof of Proposition 2.1
C.2. Proof of Lemma 2.2
C.4. Proof of Proposition 3.3
C.3. Proof of Propositions 3.1 and 3.2
C.4.1. REGULARITY CONDITIONS FOR PROPOSITION 3.3
C.4.2. FORMAL PROOF
C.5. Proof of Proposition 4.1
C.6. Proof of Proposition 4.2
D. Experimental Details
D.1. Benchmarking targets
D.2. Algorithmic details and hyperparameter settings
D.2.1. SMC AND CRAFT SETTINGS
D.2.2. DDS SETTINGS
D.2.3. PIS SETTINGS
D.2.4. PDDS SETTINGS
D.3. Ablation studies
D.4. Additional results
D.5. Uncurated normalizing constant estimates

Knowls

Knowl 1 — Diffusion guidance rewrites target sampling in terms of a potential
model/method
Let the target density on $\mathbb{R}^d$ be $\pi(x)=\gamma(x)/Z$ , where $\gamma$ is evaluable and $Z=\int\gamma(x)\,dx$ is unknown. Use the standard Gaussian reference density $p_0=\mathcal{N}(0,I_d)$ and define $g_0(x)=\gamma(x)/p_0(x)$ . The forward Ornstein–Uhlenbeck diffusion is $dX_t=-\beta_tX_t\,dt+\sqrt{2\beta_t}\,dW_t$ , with $X_0\sim\pi$ , positive schedule $\beta_t$ , and $d$ -dimensional Brownian motion $W_t$ . Writing $\lambda_t=1-\exp(-2\int_0^t\beta_s\,ds)$ , its transition is $p(x_t\mid x_0)=\mathcal{N}(x_t;\sqrt{1-\lambda_t}x_0,\lambda_t I_d)$ . If $\pi_t$ is the time- $t$ marginal, define $g_t(x_t)=\int g_0(x_0)p(x_0\mid x_t)\,dx_0$ , where $p(x_0\mid x_t)$ is the conditional law under the same diffusion initialized instead at $p_0$ . Then $\pi_t(x_t)=p_0(x_t)g_t(x_t)/Z$ and $\nabla\log\pi_t(x_t)=-x_t+\nabla\log g_t(x_t)$ . Consequently, the exact reverse-time diffusion, which starts approximately from standard Gaussian noise at its initial end when $\int_0^T\beta_s\,ds$ is large, has drift $-\beta_{T-t}Y_t+2\beta_{T-t}\nabla\log g_{T-t}(Y_t)$ and noise scale $\sqrt{2\beta_{T-t}}$ . The unknown time-dependent potential $g_t$ is the obstacle to simulating this reverse process exactly.
Knowl 2 — Plug-in guidance can converge to the wrong Gaussian target
theoretical result
For the scalar target $\pi=\mathcal{N}(\mu,\sigma^2)$ , consider the common plug-in guidance approximation $\hat g_t(x)=g_0(\sqrt{1-\lambda_t}\,x)$ and simulate the resulting reverse diffusion from $Z^{(T)}_0\sim\mathcal{N}(0,1)$ . For any positive noise schedule with $\int_0^T\beta_s\,ds\to\infty$ , the terminal mean and variance converge to $\mu$ and $1$ if $\sigma=1$ . If $\sigma\ne1$ , they instead converge to $\mathbb{E}[Z^{(T)}_T]\to \frac{\mu}{1-\sigma^2}\bigl(1-\exp[-(1/\sigma^2-1)]\bigr)$ and $\operatorname{Var}(Z^{(T)}_T)\to\frac{1-\exp[-2(1/\sigma^2-1)]}{2(1/\sigma^2-1)}$ . Thus, even with no time-discretization error and an arbitrarily long diffusion, the approximation generally does not recover the target mean and variance. For a product of $d$ such Gaussian targets, the discrepancy accumulates across dimensions: the target-to-output Kullback–Leibler divergence grows linearly in $d$ , making importance-sampling correction require exponentially many samples for reasonable relative variance.
Knowl 3 — PDDS corrects approximate reverse diffusion with sequential particles
algorithm
Particle Denoising Diffusion Sampler (PDDS) targets $\pi(x)=\gamma(x)/Z$ by applying sequential Monte Carlo (SMC) to a discretized reverse diffusion. Choose $K$ steps on $[0,T]$ , let $\delta=T/K$ , and define $\alpha_k=1-\exp(-2\int_{(k-1)\delta}^{k\delta}\beta_s\,ds)$ . For each $k$ , choose an approximate potential $\hat g_k$ with $\hat g_0=g_0$ and $\hat g_K=1$ , and let $\hat\pi_k(x)\propto p_0(x)\hat g_k(x)$ ; this makes the initial particle law at $k=K$ standard Gaussian. Move particles backward from $k+1$ to $k$ using the Gaussian proposal $\hat p(x_k\mid x_{k+1})=\mathcal{N}(x_k;\sqrt{1-\alpha_{k+1}}x_{k+1}+\alpha_{k+1}\nabla\log\hat g_{k+1}(x_{k+1}),\alpha_{k+1}I_d)$ . Weight each proposed pair by $w_k=\frac{\hat g_k(x_k)p(x_k\mid x_{k+1})}{\hat g_{k+1}(x_{k+1})\hat p(x_k\mid x_{k+1})}$ , where $p(x_k\mid x_{k+1})$ is the exact backward transition of the reference diffusion. At each step, multiply the running normalizing-constant estimate by the mean incremental weight, normalize particle weights, and resample; optional Markov chain Monte Carlo moves must leave $\hat\pi_k$ invariant. The output is the empirical distribution of particles at $k=0$ and the product-of-mean-weights estimate of $Z$ . In the experiments, the authors used 2,000 particles, $T=1$ , a cosine noise schedule, and systematic resampling when effective sample size fell below 30% of the particle count; PDDS-MCMC additionally used 10 Metropolis-adjusted Langevin steps.
Knowl 4 — At fixed discretization, PDDS has unbiased normalizing-constant estimates and SMC limit theorems
theoretical result
For a fixed number $K$ of diffusion steps, suppose each incremental weight $w_k$ has finite second moment under the proposal pair law and the potential ratios satisfy $\int \pi_k(x)g_k(x)/\hat g_k(x)\,dx<\infty$ . Then the PDDS estimate $\hat Z_0^N$ is unbiased for $Z$ and has finite variance for finite particle count $N$ . With multinomial resampling at every step, $\sqrt{N}(\hat Z_0^N/Z-1)$ converges in distribution to a centered normal random variable. Its asymptotic variance is the sum of chi-squared discrepancies between the exact and approximate initial laws and the exact and approximate backward pair laws: $\sigma_K^2=\chi^2(\pi_K\|\hat\pi_K)+\sum_{k=0}^{K-1}\chi^2\!\left(\pi_{k+1}(x_{k+1})p(x_k\mid x_{k+1})\,\middle\|\,\hat\pi_{k+1}(x_{k+1})\hat p(x_k\mid x_{k+1})\right)$ . Here $\pi_k$ denotes the true noising marginal, $\hat\pi_k$ the normalized approximate intermediate density, and $p$ and $\hat p$ the exact and proposed reverse kernels; $\chi^2(P\|Q)=\int[(dP/dQ)-1]^2\,dQ$ . For every bounded test function $f$ , the particle estimate $N^{-1}\sum_{i=1}^N f(X_0^i)$ is also asymptotically normal around $\int f\,d\pi$ . The variance expression makes accurate potential and reverse-kernel approximations consequential: they reduce the chi-squared discrepancies that drive Monte Carlo error.
Knowl 5 — Sorted stratified resampling yields a finite continuous-time error bound
theoretical result
With sorted stratified resampling, PDDS admits a bound on the asymptotic relative error of its normalizing-constant estimate that remains controlled as the discretization is refined. For the continuous-time limit, assume $\beta_t\equiv1$ , regularity of the noising marginals (polynomial growth bounds on the first three spatial derivatives of their log densities and a uniform Gaussian-exponential moment bound), and constants $C_1,C_2$ such that $C_1^{-1}\le\bar g_t(x)/\tilde g_t(x)\le C_1$ and $\|\nabla\log\hat g_t(x)\|\le C_2(1+\|x\|)$ . Here $\bar g_t=g_t/Z$ and $\tilde g_t=\hat g_t/\int p_0(x)\hat g_t(x)\,dx$ are the normalized exact and approximate potentials, and $\hat\pi_T$ is the normalized approximate terminal law. If $\zeta_K^2$ denotes the sorted-stratified asymptotic error bound for $K$ steps, then $\limsup_{K\to\infty}\zeta_K^2\le\chi^2(\pi_T\|\hat\pi_T)+2\int_0^T\mathbb{E}_{X_t\sim\pi_t}\!\left[\frac{\bar g_t(X_t)}{\tilde g_t(X_t)}\|\nabla\log g_t(X_t)-\nabla\log\hat g_t(X_t)\|^2\right]dt$ . The bound identifies score error along the noising path, rather than only mismatch between terminal samples, as the main source of residual asymptotic error. Unlike repeated multinomial resampling, for which the paper notes the asymptotic variance can diverge as $K$ grows, sorted stratified resampling controls the resampling contribution under these assumptions.
Knowl 6 — Novel score matching uses target-score information to train diffusion potentials
equation
Let $X_0\sim\pi$ , and conditional on $X_0$ , let $X_k$ follow the forward diffusion transition $p(x_k\mid X_0)$ . Write $s_\theta(k,x)=\nabla_x\log\hat\pi_\theta(k,x)$ for a learned score, $\lambda_k$ for the diffusion noise variance parameter, and $\kappa_k=\sqrt{1-\lambda_k}$ . The standard denoising score matching (DSM) identity is $\nabla\log\pi_k(x_k)=\mathbb{E}[\nabla_{x_k}\log p(x_k\mid X_0)\mid X_k=x_k]$ . The paper's novel score matching (NSM) identity is $\nabla\log\pi_k(x_k)=\kappa_k\mathbb{E}[\nabla\log g_0(X_0)\mid X_k=x_k]-x_k$ , where $g_0=\gamma/p_0$ and $p_0=\mathcal{N}(0,I_d)$ . Corresponding regression objectives, with expectations over the joint law of $(X_0,X_k)$ , are $\ell_{\rm DSM}(\theta)=\sum_{k=1}^K\mathbb{E}\|s_\theta(k,X_k)-\nabla_{x_k}\log p(X_k\mid X_0)\|^2$ and $\ell_{\rm NSM}(\theta)=\sum_{k=1}^K\mathbb{E}\|s_\theta(k,X_k)+X_k-\kappa_k\nabla\log g_0(X_0)\|^2$ . Under the respective finite-moment conditions, expressive score models minimize either objective at the true marginal score. NSM is usable in this sampling setting because the target density and its gradient can be evaluated even when direct target samples are unavailable; PDDS particles provide approximate draws of $X_0$ for training.
Knowl 7 — NSM avoids the near-zero-time variance divergence of DSM in a Gaussian case
theoretical result
Consider a one-dimensional target $\pi=\mathcal{N}(\mu,\sigma^2)$ , constant diffusion rate $\beta_t=1$ , and a score network $s_\theta(t,x)$ that is continuously differentiable in its inputs and parameters and has at most polynomial growth. Assume also that $\mathbb{E}_\pi|\nabla_\theta s_\theta(0,X_0)|^2>0$ for every parameter value. For the DSM and NSM objectives defined by regression respectively to $\nabla_x\log p(X_t\mid X_0)$ and to $\kappa_t\nabla\log g_0(X_0)-X_t$ , the paper proves that the continuous-time DSM loss and the second moment of its unbiased stochastic-gradient estimator are infinite. In contrast, the NSM loss and the second moment of its unbiased stochastic-gradient estimator are finite. The divergence arises near time zero, where the DSM target contains a conditional-noise score whose variance grows without bound as the diffusion noise vanishes; the NSM target remains well behaved in this setting.
Knowl 8 — A constrained potential network supports iterative PDDS refinement
model/method
PDDS learns scalar potentials, not only vector-valued score fields, because its particle weights require potential values. The proposed parameterization uses a scalar network $r_\eta(k)$ and a vector network $N_\gamma(k,x)\in\mathbb{R}^d$ : $\log\hat g_\theta(k,x)=[r_\eta(k)-r_\eta(0)]\langle N_\gamma(k,x),x\rangle+[1-r_\eta(k)+r_\eta(0)]\log g_0(\sqrt{1-\lambda_k}\,x)$ , with $\theta=(\eta,\gamma)$ . It is anchored to the evaluable potential at time zero. To refine it, first run PDDS with a simple initial approximation; then repeatedly draw $X_0$ from the resulting particle approximation, sample $k$ uniformly from the discretized times, draw $X_k\sim p(\cdot\mid X_0)$ , and take a minibatch gradient step on either the DSM or NSM local loss. Use the updated network as the potential approximation in another PDDS run, and repeat. The SMC correction is essential to this feedback loop: training on the corrected particle output can improve on the distribution produced by the current approximate reverse diffusion, rather than merely fitting its errors. In the paper's three-mode mixture ablation, the simple approximation captured only one mode, while two potential-training iterations recovered the missed modes; removing SMC prevented the iterative process from making comparable progress. For the main experiments, the authors used 20 refinement rounds of 500 training updates each, batch size 300, Adam with initial learning rate $10^{-3}$ and exponential decay by a factor of 0.95 every 50 updates. Their diffusion-model networks used a 128-dimensional sinusoidal time embedding and 64-unit GeLU MLPs.
Knowl 9 — Benchmarks show strong normalizer and sample quality, especially on separated mixtures
empirical result
The benchmark comparisons plotted on pages 8–9 and 34–35 evaluate normalizing-constant estimates and sample quality against SMC, CRAFT, Path Integral Sampler (PIS), and Denoising Diffusion Sampler (DDS). The main targets include a six-mode two-dimensional Gaussian mixture, a 10-dimensional funnel, a 61-dimensional Sonar logistic-regression posterior, and a 1,600-dimensional log Gaussian Cox process; the experiments used 2,000 particles and a shared variational Gaussian reference where applicable. PDDS with MCMC had the best normalizing-constant bias and variance on the posterior targets except at four steps on Sonar, where CRAFT performed best. PDDS without MCMC was broadly comparable to CRAFT, and both PDDS variants outperformed PIS and DDS across the reported tasks. On the synthetic mixture, PDDS variants performed best, while CRAFT was strongest on the funnel. In a harder 40-component, highly separated Gaussian mixture tested at dimensions 1, 2, 5, 10, and 20, PDDS-MCMC substantially improved on CRAFT's normalizing-constant bias and variance, particularly at low step counts; it also achieved lower entropy-regularized Wasserstein-2 distance and recovered a greater proportion of the target modes. The plots communicate comparative trends across repeated training and sampling seeds; they do not provide a single exact numerical summary for these comparisons.
Knowl 10 — PDDS depends on a suitable reference potential and finite-particle safeguards
limitation
PDDS practically requires the initial ratio $g_0=\gamma/p_0$ to be a well-behaved potential, for example one with boundedness or suitable moments, so that particle weights and potential learning remain stable. The authors found a variational Gaussian approximation useful for reparameterizing the target in their experiments, but note that more sophisticated geometry-handling methods may be needed for harder targets. Although the method has asymptotic guarantees, practical runs use finitely many particles and therefore do not eliminate sampling error; the paper cautions against treating finite-particle output as automatically reliable for consequential decisions. Its uncurated normalizing-constant plots also show numerical overestimation outliers across all compared methods, attributed to numerical error and instability rather than to a problem unique to PDDS.

Coverage note — Detailed proofs and per-task baseline tuning grids are omitted because they support the central methods and findings rather than adding independent contributions.

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Citation

MLA

Phillips, A., et al. “Particle Denoising Diffusion Sampler”. arXiv, 2024, http://arxiv.org/abs/2402.06320v2.

APA

Phillips, A., Dau, H.-D., Hutchinson, M. J., Bortoli, V. D., Deligiannidis, G., & Doucet, A. (2024). Particle Denoising Diffusion Sampler. arXiv. http://arxiv.org/abs/2402.06320v2

Chicago

Phillips, A., H.-D. Dau, M. J. Hutchinson, V. D. Bortoli, G. Deligiannidis, and A. Doucet. 2024. “Particle Denoising Diffusion Sampler”. arXiv. http://arxiv.org/abs/2402.06320v2.

Harvard

Phillips, A. et al. (2024) “Particle Denoising Diffusion Sampler”, arXiv [Preprint]. Available at: http://arxiv.org/abs/2402.06320v2.

Vancouver

1. Phillips A, Dau H-D, Hutchinson MJ, Bortoli VD, Deligiannidis G, Doucet A (2024) Particle Denoising Diffusion Sampler. arXiv

BibTeX

@article{phillips2024particle,
  title = {Particle Denoising Diffusion Sampler},
  author = {Phillips, Angus and Dau, Hai-Dang and Hutchinson, Michael John and Bortoli, Valentin De and Deligiannidis, George and Doucet, Arnaud},
  year = {2024},
  journal = {arXiv},
  url = {http://arxiv.org/abs/2402.06320v2},
  eprint = {2402.06320}
}

Metadata:arXiv

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Particle Denoising Diffusion Sampler

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Knowls

Knowl 1 — Diffusion guidance rewrites target sampling in terms of a potential

Knowl 2 — Plug-in guidance can converge to the wrong Gaussian target

Knowl 3 — PDDS corrects approximate reverse diffusion with sequential particles

Knowl 4 — At fixed discretization, PDDS has unbiased normalizing-constant estimates and SMC limit theorems

Knowl 5 — Sorted stratified resampling yields a finite continuous-time error bound

Knowl 6 — Novel score matching uses target-score information to train diffusion potentials

Knowl 7 — NSM avoids the near-zero-time variance divergence of DSM in a Gaussian case

Knowl 8 — A constrained potential network supports iterative PDDS refinement

Knowl 9 — Benchmarks show strong normalizer and sample quality, especially on separated mixtures

Knowl 10 — PDDS depends on a suitable reference potential and finite-particle safeguards

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