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arxiv: 2510.03824 · v2 · pith:344FI2ZHnew · submitted 2025-10-04 · 💻 cs.LG · cs.AI· stat.ML

Proximal Diffusion Neural Sampler

Wei Guo , Jaemoo Choi , Yuchen Zhu , Molei Tao , Yongxin Chen This is my paper

Pith reviewed 2026-05-21 20:31 UTC · model grok-4.3

classification 💻 cs.LG cs.AIstat.ML
keywords diffusion modelsneural samplersmode collapseproximal optimizationstochastic optimal controlmolecular dynamicsdiscrete sampling
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The pith

Proximal Diffusion Neural Sampler decomposes training into proximal subproblems on path measures to reach multimodal targets without mode collapse.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper frames learning a diffusion neural sampler from an unnormalized distribution as a stochastic optimal control problem on path measures. It applies the proximal point method to break this into a sequence of simpler subproblems, each creating an intermediate distribution that moves closer to the target and encourages the sampler to visit all modes. The approach is made practical by replacing each proximal step with a proximal weighted denoising cross-entropy objective that works for both continuous and discrete variables. Experiments on molecular dynamics and statistical physics tasks show the staged path improves exploration compared with direct training.

Core claim

PDNS addresses mode collapse in multimodal targets by tackling the stochastic optimal control problem via proximal point method on the space of path measures, decomposing the learning process into simpler subproblems that create a path gradually approaching the desired distribution and promote thorough exploration across modes. For a practical and efficient realization, each proximal step is instantiated with a proximal weighted denoising cross-entropy (WDCE) objective.

What carries the argument

proximal point method on the space of path measures, which decomposes the overall stochastic control problem into a sequence of simpler subproblems whose solutions trace a path to the target distribution

Load-bearing premise

That each proximal step can be instantiated stably with the proximal weighted denoising cross-entropy objective for both continuous and discrete sampling without instabilities or tuning that prevents convergence to the full target.

What would settle it

Running PDNS on a mixture of Gaussians separated by high barriers and finding that generated samples still miss entire modes after the full sequence of proximal steps would show the staged path does not reliably prevent collapse.

Figures

Figures reproduced from arXiv: 2510.03824 by Jaemoo Choi, Molei Tao, Wei Guo, Yongxin Chen, Yuchen Zhu.

Figure 1
Figure 1. Figure 1: An example of mode collapse in WDCE-based method where the target distribution is [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Average 2-point correlations in both vertical and horizontal directions of samples from [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ablation studies on fixed γk for all stages k on MoS benchmark. We fix γk = ηk 1+ηk to a constant for all stages k and visualize the first four stages (k = 1, 2, 3, 4; left to right). Larger γk, i.e., weak proximal regularization, leads to rapid mode collapse whereas smaller γk preserves multimodal coverage. These results are consistent with the analysis in Sec. 3.1 [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Ablation on proximal step size ηk and the choice of the scheduler on MoS. We evaluate Sinkhorn (↓) and MMD (↓) across training epochs for multiple choices of proximal step size ηk and scheduling policy. γ denotes γk := ηk 1+ηk over stage k. The legend entry “γ = const” denotes runs with a fixed γk for all stages k. Note that 0 < γk ≤ 1; larger γk weakens the proximal effect and approaches the non-proximal … view at source ↗
Figure 5
Figure 5. Figure 5: Ablation on proximal step size ηk and the choice of the scheduler on LJ-13. We monitor γ and energy 2-Wasserstein distance (E(·)W2(↓)) across training epochs for multiple choices of proximal step size ηk and scheduling policy. γ denotes γk := ηk 1+ηk over stage k. The legend entry “γ = const” denotes runs with a fixed γk for all stages k. Note that 0 < γk ≤ 1; larger γk weakens the proximal effect and appr… view at source ↗
Figure 6
Figure 6. Figure 6: 2D kernel density estimates of ground-truth vs. generated samples, projected onto first [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Energy histograms for DW-4 and LJ-13. PDNS produces energy distributions that closely [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Visualization of non-cherry-picked samples from the learned [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Visualization of non-cherry-picked samples from the learned [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Visualization of non-cherry-picked samples from the learned [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Visualization of non-cherry-picked samples from the learned [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: An example of PNDS training for Potts model with [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
read the original abstract

The task of learning a diffusion-based neural sampler for drawing samples from an unnormalized target distribution can be viewed as a stochastic optimal control problem on path measures. However, the training of neural samplers can be challenging when the target distribution is multimodal with significant barriers separating the modes, potentially leading to mode collapse. We propose a framework named Proximal Diffusion Neural Sampler (PDNS) that addresses these challenges by tackling the stochastic optimal control problem via proximal point method on the space of path measures. PDNS decomposes the learning process into a series of simpler subproblems that create a path gradually approaching the desired distribution. This staged procedure traces a progressively refined path to the desired distribution and promotes thorough exploration across modes. For a practical and efficient realization, we instantiate each proximal step with a proximal weighted denoising cross-entropy (WDCE) objective. We demonstrate the effectiveness and robustness of PDNS through extensive experiments on both continuous and discrete sampling tasks, including challenging scenarios in molecular dynamics and statistical physics. Our code is available at https://github.com/AlexandreGUO2001/PDNS.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper frames training a diffusion-based neural sampler for unnormalized multimodal targets as a stochastic optimal control problem on path measures. It proposes Proximal Diffusion Neural Sampler (PDNS), which applies the proximal point method to decompose the problem into a sequence of simpler subproblems. Each subproblem is instantiated via a proximal weighted denoising cross-entropy (WDCE) objective that gradually refines a path toward the target distribution while promoting mode exploration. The method is evaluated on continuous and discrete sampling tasks, including molecular dynamics and statistical physics, with code released at https://github.com/AlexandreGUO2001/PDNS.

Significance. If the proximal decomposition yields stable subproblems that reliably improve mode coverage over standard diffusion samplers, the framework could provide a useful algorithmic template for sampling from complex distributions. The explicit code release supports reproducibility and is a clear strength.

major comments (2)
  1. [§3.2] §3.2 (proximal WDCE derivation): the weighting scheme derived from the previous iterate is presented as ensuring stable gradients and exploration, but no analysis or bounds are given on how the proximal parameter or weighting affects variance or bias as dimension or energy barriers increase; this directly bears on whether the staged path avoids collapse.
  2. [§4] §4 (experiments): the reported gains on multimodal targets are shown via qualitative samples and some metrics, but without ablations isolating the effect of the proximal steps versus the base WDCE objective or quantifying sensitivity to the proximal parameter, it is difficult to confirm that the decomposition itself drives the claimed robustness.
minor comments (2)
  1. [§2] Notation for the path-measure proximal operator is introduced without an explicit comparison to the standard KL-proximal operator used in related optimal-control literature.
  2. [Figure 3] Figure captions for the molecular-dynamics trajectories should include the specific barrier heights or temperatures used to allow direct comparison with prior samplers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (proximal WDCE derivation): the weighting scheme derived from the previous iterate is presented as ensuring stable gradients and exploration, but no analysis or bounds are given on how the proximal parameter or weighting affects variance or bias as dimension or energy barriers increase; this directly bears on whether the staged path avoids collapse.

    Authors: We acknowledge that Section 3.2 presents the proximal weighting scheme primarily through its derivation and motivation for stable gradients and gradual path refinement, without providing formal bounds on variance or bias as a function of dimension or energy barrier height. The manuscript argues for stability via the iterative proximal updates on path measures, supported by the empirical results on multimodal targets. In the revised manuscript we have added a paragraph in Section 3.2 discussing the practical role of the proximal parameter in controlling step size and its observed effect on exploration; a full theoretical analysis of bias-variance trade-offs under increasing dimension remains future work. revision: partial

  2. Referee: [§4] §4 (experiments): the reported gains on multimodal targets are shown via qualitative samples and some metrics, but without ablations isolating the effect of the proximal steps versus the base WDCE objective or quantifying sensitivity to the proximal parameter, it is difficult to confirm that the decomposition itself drives the claimed robustness.

    Authors: We agree that isolating the contribution of the proximal decomposition is important for validating the framework. The original experiments compare PDNS against standard diffusion samplers and other baselines on continuous and discrete tasks, but do not include an explicit non-proximal WDCE ablation or systematic sensitivity sweeps. We have added these ablations and sensitivity plots to the revised Section 4, showing that the staged proximal steps improve mode coverage relative to the base objective and that performance is robust across a range of proximal parameter values. revision: yes

Circularity Check

0 steps flagged

No circularity: PDNS derivation applies standard proximal point method to path-measure SOC without self-referential reduction

full rationale

The paper frames diffusion sampling as a stochastic optimal control problem on path measures and proposes to solve it via the proximal point method, which decomposes the task into a sequence of simpler subproblems each instantiated by a proximal WDCE objective. This construction is presented as a direct methodological extension rather than a redefinition of inputs or a fitted quantity renamed as prediction. No equations or steps in the provided text reduce the claimed path-measure decomposition or mode-exploration benefit to a tautology, self-citation chain, or ansatz smuggled from prior author work; the central procedure remains independently motivated by the proximal-point algorithm and is validated through separate experiments on continuous and discrete tasks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on viewing sampler training as a stochastic optimal control problem on path measures and assumes the proximal decomposition can be realized via WDCE; no explicit free parameters or invented entities are detailed in the abstract.

axioms (1)
  • domain assumption The task of learning a diffusion-based neural sampler can be formulated as a stochastic optimal control problem on path measures.
    Stated directly in the abstract as the starting point for the proximal approach.

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Reference graph

Works this paper leans on

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    employs annealed reference dynamics to provide stronger guidance during learning. CE-based Diffusion SamplersCross-entropy (CE) approaches replace forward-KL or relative- entropy training with a reverse-KL projection, typically of the form, which reduces to a weighted negative log-likelihood on trajectories. In the diffusion setting, this often appears as...

    work page 2024

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    Z T 0 1 2 ∥ut(Xt)−σ t∇logP ∗ T|t (XT |Xt)∥2dt # .(36) Finally, we obtain theweighted denoising cross entropyoptimization problem: u∗ = argmin u EX∼P ∗

    or adopts alternative training criteria such as action matching (Albergo & Vanden-Eijnden, 2025; Neklyudov et al., 2023). In practice, these approaches often inherit a significant computational 15 preprint cost from evaluating energies to form importance weights, motivating schemes that reduce or reuse weighting while retaining correctness. A.2 DISCRETEDI...

    work page 2025

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    SampleNtrajectories{X (i)}N i=1 ∼P v

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    Compute weights{w (i)}N i=1 by (31) for each corresponding trajectory{X (i)}N i=1

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  9. [9]

    Resample{X (i) 1 }N i=1 by following categorical distribution: { ˆX(i) 1 }N i=1 ∼Cat {ˆw(i)}N i=1,{X (i) 1 }N i=1 ,whereˆw (i) = w(i) PN i=1 w(i) .(40)

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    Z T 0 1 2 ∥uk−1 t (Xt)∥2dt+u k−1 t (Xt)·dW t −r(X 1) #! ,(45) and e(1−λk)r(XT ) dPref dPk−1 (X) = exp −

    Update the controlu θ :=uthrough ascore matchingloss with a resampled data. Remark.Particle Denosing Diffusion Sampler (PDDS) (Phillips et al., 2024) is one of the sampling method which leverages this Importance weighted CE method. C.2.3 THEORIES ONPROXIMALDIFFUSIONNEURALSAMPLERS In this section, we introduce the theory for PDNS written in SDE formulation...

    work page 2024

  11. [11]

    DrawNtrajectories{X (i)}N i=1 fromP θk−1

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  12. [12]

    Compute weights wθ∗ k ηk(X) = er(XT ) dPref dP¯θk−1 (X) ηk ηk+1 =(45);(51)

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    1 λ Eµλ(ex|XT ) X d:exd=M −logs θ(ex)d,X d T # , ∝E X∼P ¯θk−1 er(XT ) dPref dP¯θk−1 ηk ηk+1 Eλ∼Unif(0,1)

    resample{X (i) T }N i=1 by following categorical distribution: { ˜XT }N i=1 ∼Cat   ( wθk ηk(X(i)) PN i=1 wθk ηk(X(i)) )N i=1 ,{X (i) T }N i=1  .(52) Weighting-based AlgorithmAlternatively, using (50), we draw XT ∼P θk−1 T and and incorpo- rate wθ∗ k ηk directly in the objective, i.e., optimize a weighted loss where each sample contributes proportional...

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    global” means with respect toP ∗ and “local

    and take the output as the ground truth. E.3 ADDITIONALRESULTS Monitoring training procedure of PDNSIn Fig. 12, we demonstrate an example of the PNDS learning procedure for the Potts model with q= 4 states on a 16×16 lattice, at βcritical = 1.0986. In thek-th outer loop, we fit the target path measureP k. 30 preprint (a) PDNS (ours) (b) LEAPS (c) MH (d) G...

    work page