Title: Reverse Diffusion Sequential Monte Carlo Samplers
Venue: NeurIPS 2025
Accepted: yes
Keywords: sampling; sequential monte carlo; diffusion models; probabilistic methods;
Abstract: We propose a novel sequential Monte Carlo (SMC) method for sampling from unnormalized target distributions based on a reverse denoising diffusion process. While recent diffusion-based samplers simulate the reverse diffusion using approximate score functions, they can suffer from accumulating errors due to time discretization and imperfect score estimation. In this work, we introduce a principled SMC framework that formalizes diffusion-based samplers as proposals while systematically correcting for their biases. The core idea is to construct informative intermediate target distributions that progressively steer the sampling trajectory toward the final target distribution. 
Although ideal intermediate targets are intractable, we develop \emph{exact approximations} using quantities from the score estimation-based proposal, without requiring additional model training or inference overhead. The resulting sampler, termed \textit{\ourmethodfull}, enables consistent sampling and unbiased estimation of the target's normalization constant under mild conditions. We demonstrate the effectiveness of our method on a range of synthetic targets and real-world Bayesian inference problems.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel SMC method to sample from unnormalized target distributions. Utilizing the diffusion framework and building upon diffusion-based SMC methods, this paper designs informative intermediate targets to guide the resampling of the particles at each step, and the new sampler is called RDSMC. Asymptotic consistency to estimate the target distribution is established theoretically as the number of particles diverge. Numerical experiments are performed on synthetic and real-world data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. A new SMC algorithm, RDSMC, for sampling from unnormalized targets\n2. RDSMC enables estimation of the normalization constant\n3. The paper provides a theoretical consistency result for estimating the target and the normalization constant\n\nWeaknesses:\n1. It seems that in most reported experiments (especially with higher dimensional data), SLIPS (sometimes along with other baseline methods) is performing better than the proposed RDSMC.\n2. Also, especially for estimating the normalization constant, there are some other recent diffusion-based samplers that are not included for numerical comparison. For example, while not using information from the unnormalized densities, PDDS also estimates the normalization constant which should be numerically comparable.\n3. Overall, the benefits of using SMC against MCMC (e.g., RDMC) with diffusion-based sampling methods are still not clear to me. I am thinking of benefits other than the final distributional distance (e.g., computation, what is required for estimation, etc.).\n4. There are occasional typos that need to be fixed: \"an exception\" -> \"exceptions\" (line 43); comma -> multiplication (line 99)\n\nQUESTIONS:\nSome questions have been provided in the weakness section. My other questions are as follows.\n\n1. It is claimed that the density $q(u_t|x_t)$ need not be evaluated (line 137). However, that seems to be needed to compute the importance weights (in Alg. 2). Am I missing something?\n\n2. While in a different context, how is the design of the weighting functions similar to TDS?\n\nI would consider raising my score if my questions are properly addressed.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-20"}

comments: SUMMARY:
The authors developed a training-free diffusion sampler which treated diffusion-based sampler as the simulator within a sequential Monte Carlo framework. The authors proposed to resample the particles in each step to gradually steer the particles to the desired distributions and the issue of discretization bias and score estimation error can be greatly alleviated. The reverse diffusion SMC provides the unbiased estimate of the normalizing constant. Numerous methods show the promising potential of RDSMC compared to other diffusion samplers.

STRENGTHS AND WEAKNESSES:
Pros: 

1) using SMC (Line 3 in Alg 2) to estimate score function via denoising score identity seems to be an interesting idea. 

2) Eq(15) seems to have some potential alleviate the bias. I am not sure, need more evidence.

Cons:

1) I don't fully understand why the SMC setups can correct discretization bias and score estimation errors.

2) I don't understand the motivation to do training-free guidance in sampling setup, do we need a pretrained model in this case? 

3) Give more details on the motivation and/or derivation of eq (6) and eq(14).


* I didn't carefully check the proof.

QUESTIONS:
1. what is the key step to avoid discretization error/ score error? is it Eq(15)? more clarifications are needed.

2. Is this algo. more like Wu's twisted SMC or vanilla SMC.

LIMITATIONS:
Although I appreciate the elegance, I am not convinced if this is a good idea to use SMC to correct the discretization bias and score error. Samplers are well-studied in low-dim distributions, but SMC doesn't scale well unless there is some clever tricks such as Tweedie-like "look-ahead" design.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

comments: SUMMARY:
- The paper leverage the diffusion model paradigm to sample from a target distribution, which can be unnormalized.
- It introduces a training-free Sequential Monte Carlo (SMC) approach where the proposal are derived from a reverse diffusion kernel.
- The core idea is to augment the state space with an auxiliary variable that accounts for the randomness in the score estimation.

STRENGTHS AND WEAKNESSES:
## Strength

- The main novelty of the paper is the idea of augmenting the state space with an auxiliary variable to explicitly model the noise in the score estimation process.

## Weaknesses

**Major Theoretical Flaw**

The central methodology for score estimation appears to be fundamentally flawed:

- The proposed score estimation in Equation (5) is problematic. By examining the weight update in Equation (6), it appears the auxiliary variable $u$ is intended to act as a substitute for $x_0$ within the target density term.
- However, $u$ is sampled from the proposal distribution $N(x_t / \alpha_t, \sigma_t^2 / \alpha_t^2 I)$, which is a Gaussian with a very large variance, particularly in early steps. This means that samples of $u$ have a low to no chance of falling in regions where the target distribution has high mass.
- This turns the score calculation into an IS estimate where the proposal has a poor overlap with the target. Consequently, estimating the score in this way is likely as difficult as sampling from the target distribution directly, which defeats the purpose of the proposed method.

**Typos**

- In Line 96: "initilaizes" should be corrected to "initializes".
- There is a trailing comma in the denominator of the weight expression in Equation (6) that should be removed.

QUESTIONS:
.

LIMITATIONS:
.
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper combines the sequential Monte Carlo with the reverse diffusion path to provide a novel sampler to draw samples from an irregular target distribution. Different from typical sequential Monte Carlo, this paper introduces auxiliary variables $u_t$ to construct the proposal distribution $q(x_t|x_{t+1},u_{t+1})$ for simulating the transition in the reverse diffusion process and calculating the weights for the resampling. Different from RDMC, this paper will keep a sequence of particles in each update and use them to construct the proposal distribution for resampling. As a result, this paper shows that sequential Monte Carlo will be an unbiased sampler when the number of particles in the sequence approaches infinity and empirically shows the advantages of their methods.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength:**\n\n1. The problem expected to be solved (simulating reverse diffusion path to solve the sampling task) is interesting, and the core technique sequential Monte Carlo helps to approximate the ideal transition kernel $p_{0|t}$ by re-weighting the proposal with $w_i$ is practical since it has bypassed repetitive posterior sampling in previous work, e.g., RDMC.\n2. The core technique, i.e., introducing the auxiliary variables $u_t$ to couple sequential Monte Carlo and RDMC is original and effective, which helps to provide unbiased score estimation in expectation. Although the unbiasedness is conditioned on the infinite particle, it still makes a difference compared with RDMC. \n3. This paper is well written and makes it easy for readers to follow. \n\n**Weaknesses:**\n\n1. The theoretical results provided in this paper are limited. Some important theoretical properties for the proposed algorithm are unknown. For example, can the analysis be extended to the finite particle $N$ case? Will the algorithm converge? What kinds of convergence (high probability or in expectation) will be achieved? What is the size of particles required to be preserved to guarantee the convergence? Since the rejection sampling in high dimensions will bring the dimension curse, while this curse is preserved in the sequential Monte Carlo? \n2. The significance of the advantages of reverse diffusion SMC is limited. In the 2-d case, the improvement of TV distance is limited. Under the 10-d case, neither sliced KSD nor logZ estimation will have advantages. Although the algorithm is elegant and insightful, its effectiveness may be limited.\n3. Authors may add an independent table to clarify the notation, which will improve the readability of this paper. Currently, the notations are usually hidden in different parts of the paper.\n\nQUESTIONS:\nPlease refer to the weaknesses part.\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-04"}