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arxiv: 2605.17850 · v1 · pith:YQZURXGUnew · submitted 2026-05-18 · 📊 stat.ML · cs.CV· cs.LG· cs.NA· math.NA· math.PR

Simple Approximation and Derivative Free Inference-Time Scaling for Diffusion Models via Sequential Monte Carlo on Path Measures

Chenyang Wang , Weizhong Wang , Yinuo Ren , Jose Blanchet , Yiping Lu This is my paper

Pith reviewed 2026-05-20 01:08 UTC · model grok-4.3

classification 📊 stat.ML cs.CVcs.LGcs.NAmath.NAmath.PR
keywords diffusion modelssequential Monte CarloGirsanov theoreminference-time scalingderivative-free methodspath measuresimportance resamplingunbiased estimation
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The pith

URGE uses Girsanov path weights for unbiased, derivative-free resampling in diffusion model inference.

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

The paper presents URGE, a new algorithm for improving diffusion model samples at inference time without derivatives or biases. It applies sequential Monte Carlo by assigning multiplicative weights to entire trajectories using a Girsanov change of measure and resampling periodically. This avoids the repeated score evaluations required by prior guidance methods. The authors prove an equivalence between their path-wise weighting and traditional particle-wise approaches, showing both yield the same unbiased final distribution. Experiments demonstrate superior performance on benchmarks with simpler implementation.

Core claim

By leveraging the Girsanov theorem on path measures, one can attach a simple multiplicative weight to each simulated trajectory and resample them, achieving inference-time scaling that is derivative-free and unbiased. The key equivalence shows that the Girsanov path weight, through a backward conditional expectation, recovers the particle-level weights from previous work, ensuring identical terminal laws.

What carries the argument

The Girsanov path weight as a multiplicative factor on trajectories that enables unbiased importance resampling via sequential Monte Carlo on path measures.

Load-bearing premise

The underlying diffusion process must allow the Girsanov theorem to define a valid change of measure between the reference and guided dynamics.

What would settle it

Running a controlled simulation where the terminal distribution from URGE is compared to the exact target distribution or to a particle-wise SMC implementation and observing significant bias or mismatch would disprove the unbiasedness claim.

Figures

Figures reproduced from arXiv: 2605.17850 by Chenyang Wang, Jose Blanchet, Weizhong Wang, Yinuo Ren, Yiping Lu.

Figure 1
Figure 1. Figure 1: Step-wise resampling corrects suboptimal guided gener￾ation toward the reward-tilted distribution. (a) In Naive Guidance, blue trajectories follow the guided diffusion dX G t = [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: URGE achieves better performance compared with base￾line, while exhibiting monotonic and larger reward increase with N (unlike FK-Steering). With URGE, a smaller model (SD v1.5) can attain higher reward than an XL model [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative comparison of inference-time scaling strategy on the Gaussian Mixture example. In our main experiments, we use K = 40 components and Dim = 30 dimensions as the default setting. To further validate the robustness of URGE, we additionally conduct experiments with K = 80, Dim = 30 and K = 40, Dim = 60, respectively. We run experiments under three random seeds and report the averaged results, as ta… view at source ↗
Figure 5
Figure 5. Figure 5: Performance metrics versus discretization steps on Gaussian Mixture Model. for the prompts used. The selected prompts contain exactly two colored entities. On colored two-object prompts, URGE also performs strongly. The generated samples adhere more strictly to the specified color attributes and spatial ordering, demonstrating enhanced compositional fidelity. Similar to additional experiments on other task… view at source ↗
Figure 6
Figure 6. Figure 6: Inference time versus number of particles for the Gaussian Deblurring task [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Inference time for the inverse problems tasks on FFHQ-256 and ImageNet-256. Results report runtimes where AFDPS-ODE is run with k = 5 particles while all other methods use k = 10 particles. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Inference time for the text-to-image tasks on 20 prompts. FKG refers to FK-Steering with gradient guidance. Base is ran with only 1 particle. Prompt a photo of ... ... a blue laptop and a brown bear. ... a purple elephant and a brown sports ball. ... a white dining table and a red car. ... a blue cell phone and a green apple. Base Model SDv1.5 Base Model SDXL FK￾Steering SDv1.5, N = 4 URGE SDv1.5 N = 4 [P… view at source ↗
Figure 10
Figure 10. Figure 10: Additional visual examples for the Box inpainting problem and the Gaussian deblurring problem on FFHQ [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Additional visual examples for the Motion deblurring problem and the Super resolution problem on FFHQ. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
read the original abstract

iffusion-based generative models increasingly rely on inference-time guidance, adding a drift term or reweighting mixture of experts, to improve sample quality on task-specific objectives. However, most existing techniques require repeated score or gradient evaluations, introducing bias, high computational overhead, or both. We introduce \texttt{URGE}, Unbiased Resampling via Girsanov Estimation, a derivative-free inference-time scaling algorithm that performs path-wise importance reweighting via a Girsanov change of measure. Instead of computing gradient-based particle weights in previous work, \texttt{URGE} attaches a simple multiplicative weight to each simulated trajectory and periodically resamples. No score, no Hessian, and no PDE evaluation is required. We establish an equivalence between path-wise and particle-wise SMC: the Girsanov path weight admits a backward conditional expectation that recovers the previous particle-level weights, guaranteeing that both schemes produce the same unbiased terminal law. Empirically, \texttt{URGE} outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks, achieving better generation quality, while being significantly simpler to implement and fully gradient-free.

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

1 major / 2 minor

Summary. The paper introduces URGE, an inference-time scaling method for diffusion models that performs path-wise importance reweighting via a Girsanov change of measure on path measures. It claims this approach is derivative-free (no score, Hessian, or PDE evaluations), establishes an equivalence between path-wise and particle-wise SMC via a backward conditional expectation that recovers particle weights, and guarantees the same unbiased terminal law. Empirical results show outperformance over existing guidance baselines on synthetic tests and diffusion benchmarks.

Significance. If the central equivalence and unbiasedness claims hold under the discrete-time discretizations actually used in implementations, the method provides a simpler, fully gradient-free alternative to gradient-based guidance techniques, reducing computational overhead while improving generation quality. The self-contained equivalence result (without reducing to fitted quantities) would be a useful theoretical contribution for SMC applications in generative modeling.

major comments (1)
  1. [Theoretical analysis of equivalence (around the Girsanov path weight and backward conditional expectation)] The equivalence result (abstract and theoretical analysis) relies on a continuous-time Girsanov Radon-Nikodym derivative and backward conditional expectation to recover exact particle weights. However, all reported experiments and algorithms use finite-step Euler-Maruyama discretization. In discrete time the weight is a product of transition densities rather than an exponential martingale, and the conditional expectation does not necessarily commute with the discretization operator. No separate discrete-time proof, commutation argument, or error bound is provided to confirm that the unbiased terminal marginal is preserved exactly in the implemented algorithm.
minor comments (2)
  1. [Abstract] The abstract refers to 'synthetic tests and diffusion-model benchmarks' without naming the specific datasets, metrics (e.g., FID, precision/recall), or number of diffusion steps; these details should be stated explicitly for reproducibility.
  2. [Method section] Notation for the path measure and the Girsanov weight should be introduced with a clear definition of the underlying probability spaces before the equivalence statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the potential value of URGE as a derivative-free inference-time scaling approach. We address the major comment on the theoretical equivalence below and outline the revisions we will make.

read point-by-point responses

  1. Referee: The equivalence result (abstract and theoretical analysis) relies on a continuous-time Girsanov Radon-Nikodym derivative and backward conditional expectation to recover exact particle weights. However, all reported experiments and algorithms use finite-step Euler-Maruyama discretization. In discrete time the weight is a product of transition densities rather than an exponential martingale, and the conditional expectation does not necessarily commute with the discretization operator. No separate discrete-time proof, commutation argument, or error bound is provided to confirm that the unbiased terminal marginal is preserved exactly in the implemented algorithm.

    Authors: We thank the referee for this precise observation. The core equivalence is established in continuous time via the Girsanov Radon-Nikodym derivative on path space and the tower property of conditional expectations, which recovers the particle weights exactly. In the discrete Euler-Maruyama setting actually implemented, the path weight is indeed the product of per-step transition Radon-Nikodym factors. Because the backward conditional expectation is applied recursively at each discrete time index using the same transition kernels that define the discretization, the equivalence between path-wise and particle-wise weights continues to hold exactly for the finite-step process; the terminal marginal therefore remains unbiased under the implemented algorithm. Nevertheless, we agree that an explicit discrete-time statement strengthens the presentation. We will add a new subsection (and corresponding appendix) that states the equivalence directly for the Euler-discretized process, provides the recursive commutation argument, and includes a brief discussion of the approximation error to the continuous-time limit. These additions will be self-contained and will not alter the existing continuous-time analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence claim is a derived mathematical property

full rationale

The paper's central claim is an equivalence between path-wise and particle-wise SMC established via the Girsanov path weight admitting a backward conditional expectation that recovers particle weights, yielding the same unbiased terminal law. This is presented as a consequence of the change-of-measure construction rather than a self-definition or a fitted quantity renamed as a prediction. No equations in the abstract reduce the terminal law to its inputs by construction, and the method is described as derivative-free without invoking self-citations as load-bearing uniqueness theorems. The derivation remains self-contained against standard SMC and Girsanov theory; any discretization concerns affect correctness but do not indicate circularity in the claimed chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal structural assumptions needed for the stated claims to hold; no explicit free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The underlying diffusion process admits a Girsanov change of measure between the reference and guided path measures.
    Required for the path-wise importance weights to be well-defined and for the backward conditional expectation equivalence to recover particle weights.

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