Interacting particle guidance for sampling reward-tilted generative priors

Adhithyan Kalaivanan; Zheng Zhao; Jens Sjölund; Fredrik Lindsten

cs.LGSep 29, 2026

Interacting particle guidance for sampling reward-tilted generative priors

Authors: Adhithyan Kalaivanan, Zheng Zhao, Jens Sjölund, Fredrik Lindsten

Organizations: Linköping University, Sweden · Uppsala University, Sweden

Abstract

Inference-time steering adapts pretrained diffusion and flow-based models to new tasks, e.g., to generate samples from a conditional distribution or samples with desired properties, without retraining. This can be formalized as sampling from a reward-tilted generative prior. As exact sampling from this distribution is intractable, guidance-based methods rely on approximations producing biased samples, and sequential Monte Carlo (SMC) methods correct for this bias using importance weights. However, while exact in the large particle limit, SMC suffers from weight degeneracy and particle collapse in practice. We propose interacting particle guidance (IPG), which replaces reweighting with transport. The particles interact through an additional drift, derived from the Feynman--Kac PDE to cancel the reweighting term, and remain unweighted. Choosing the drift in a reproducing kernel Hilbert space yields a closed-form solution that is cheap to compute, with negligible overhead compared to SMC. We demonstrate the method on Gaussian mixtures with known posteriors, and on high-dimensional image inpainting and protein structure inference tasks.

Figures & tables

Method	Resampling	ESS/ $N$ $\uparrow$	Mean error $\downarrow$	MMD $\downarrow$	SWD $\downarrow$
FKC	None	0.006 $\pm$ 0.002	7.593 $\pm$ 1.337	0.876 $\pm$ 0.165	0.682 $\pm$ 0.112
	Adaptive	0.771 $\pm$ 0.203	4.485 $\pm$ 0.872	0.184 $\pm$ 0.042	0.311 $\pm$ 0.055
	Every step	1.000 $\pm$ 0.000	3.685 $\pm$ 0.741	0.148 $\pm$ 0.021	0.260 $\pm$ 0.043
DriftLite	None	0.011 $\pm$ 0.004	5.691 $\pm$ 1.194	0.619 $\pm$ 0.139	0.509 $\pm$ 0.092
	Adaptive	0.864 $\pm$ 0.169	2.178 $\pm$ 0.357	0.097 $\pm$ 0.007	0.168 $\pm$ 0.021
	Every step	1.000 $\pm$ 0.000	2.483 $\pm$ 0.243	0.085 $\pm$ 0.004	0.182 $\pm$ 0.012

Table 1: Mean and standard deviation of the metrics on the GMM experiment. We use systematic resampling and “Adaptive” refers to resampling whenever ESS/

N

falls below

0.5

Figure 1: Samples on random 2-D slices of the 256-dimensional posterior.

Mask	Method	Avg.	Best	Avg.	Best	Avg.	Best	Cov. trace	LPIPS div.
		PSNR $\uparrow$		SSIM $\uparrow$		LPIPS $\downarrow$		Diversity $\uparrow$
Half	FKC	15.99	15.99	0.4534	0.4537	0.3352	0.3341	24.61	0.0041
	DriftLite	15.51	15.52	0.4392	0.4401	0.3466	0.3443	70.56	0.0125
	IPG	15.74	17.98	0.4427	0.4894	0.3455	0.3058	7990.21	0.2307
	IPG-CV	15.68	17.79	0.4418	0.4889	0.3462	0.3060	7742.43	0.2311
Box	FKC	24.53	24.53	0.6737	0.6739	0.1866	0.1859	14.02	0.0020

Table 2: Results on the image inpainting tasks with

N=16

particles. “Avg.” refers to the mean of the metrics computed between each of the

N

samples and the ground truth, and “Best” takes the best-of-

N

sample metrics before averaging across the test images.

Figure 2: DriftLite suffers from particle collapse, producing indistinguishable samples, while IPG-CV produces samples with perceptible differences.

Method	Avg.	Best	Avg.	Best	Rot. div. ( ∘ )	RMSD div. (Å)	Any designable $\uparrow$
	$\text{RMSD}_{\text{obs}}$ (Å) $\downarrow$		$\text{RMSD}_{\text{gt}}$ (Å) $\downarrow$		Diversity $\uparrow$
FKC	1.225	1.200	8.119	8.060	0.6	0.677	7/15
DriftLite	1.188	1.163	6.907	6.840	0.7	0.756	8/15
IPG	1.308	1.053	7.881	1.298	122.1	7.981	13/15
IPG-CV	1.290	1.048	7.543	1.232	126.9	7.800	15/15

Table 3: Results on the protein structure inference task with

N=16

particles. “Avg.” refers to the mean of the metrics computed between each of the

N

samples and the ground truth, and “Best” takes the best-of-

N

sample metrics for each test protein before averaging.

Figure 3: Samples produced on the backbone structure inference task for protein 30JI . They are intentionally not aligned with the ground truth to highlight the particle collapse in DriftLite. Samples from IPG-CV contain structures close to the ground truth, and have random orientations as expected.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Resampling	ESS/ $N$ $\uparrow$	Mean error $\downarrow$	MMD $\downarrow$	SWD $\downarrow$
FKC	None	0.006 $\pm$ 0.002	7.593 $\pm$ 1.337	0.876 $\pm$ 0.165	0.682 $\pm$ 0.112
FKC	Every step	1.000 $\pm$ 0.000	0.867 $\pm$ 0.070	0.184 $\pm$ 0.018	0.173 $\pm$ 0.011
DriftLite	None	0.011 $\pm$ 0.005	5.973 $\pm$ 1.857	0.658 $\pm$ 0.188	0.534 $\pm$ 0.136
DriftLite	Every step	1.000 $\pm$ 0.000	1.226 $\pm$ 0.117	0.124 $\pm$ 0.009	0.146 $\pm$ 0.007
IPG	None	–	0.850 $\pm$ 0.054	0.013 $\pm$ 0.003	0.093 $\pm$ 0.002
IPG-CV	None	–	0.850 $\pm$ 0.047	0.011 $\pm$ 0.002	0.092 $\pm$ 0.002

Appendix

Table 4: Mean and standard deviation of the metrics when overdamped Langevin is replaced with noisy SVGD on the GMM experiment.

Method	Resampling	ESS/ $N$ $\uparrow$	Mean error $\downarrow$	MMD $\downarrow$	SWD $\downarrow$
MLP	None	0.982 $\pm$ 0.003	0.649 $\pm$ 0.089	0.022 $\pm$ 0.002	0.089 $\pm$ 0.004
MLP-CV	None	0.983 $\pm$ 0.004	0.697 $\pm$ 0.035	0.019 $\pm$ 0.004	0.089 $\pm$ 0.001
Transformer	None	0.986 $\pm$ 0.002	0.675 $\pm$ 0.033	0.020 $\pm$ 0.002	0.088 $\pm$ 0.001
Transformer-CV	None	0.985 $\pm$ 0.001	0.687 $\pm$ 0.020	0.021 $\pm$ 0.003	0.089 $\pm$ 0.002

Appendix

Table 5: Mean and standard deviation of the metrics using neural network based corrective drifts.

PDB ID	Modeled	Deposited	Resolution (Å)
	Residue count
12FG	166	171	1.43
28HS	138	142	1.10
28RZ	186	186	2.50
30JI	173	173	1.75
30ZU	223	246	1.05

Appendix

Table 6: Target proteins used for the structure inference task. In some cases, the terminal residues are not resolved, resulting in a lower modeled residue count than the deposited count.

Figure 4: Distribution of pairwise diversity metrics for IPG and IPG-CV on 30JI . The bimodal RMSD div. indicates that structures close to reflections of the ground truth are occasionally sampled due to the imperfect generative prior.

Panel of Figure 5 — Figure 5: Samples produced by the methods along with the ground truth (GT) and observation (Obs) for the half-mask inpainting task.

Panel of Figure 6 — Figure 6: Samples produced by the methods along with the ground truth (GT) and observation (Obs) for the box-mask inpainting task.

Figure 7: Samples produced from observing random inter-residue distances for protein 30JI , visualized without Kabsch alignment to the ground truth (GT).

Figure 8: Samples produced from observing random inter-residue distances for protein 9XFS , visualized without Kabsch alignment to the ground truth (GT).

Inference-Time Guidance

Inference-time guidance steers pretrained generative models, language models, and policies toward desired outputs without requiring full retraining.

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Sampling Meets Interaction: Sequentially-Controlled Multi-Particle Flow-Maps for Efficient Inference-time Search

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Department of CSE, Washington University in St.Louis, USA · Department of Information Technology, Uppsala University, Sweden

Inference-Time Search Posterior Sampling

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While generative models have enabled training-free reward alignment, existing particle-based methods are fundamentally constrained by their propensity to local exploration within narrow regions of the underlying distribution, severely restricting sample diversity. This limitation becomes especially acute under tight reward-feedback budgets, where effective search demands broad, strategic exploration to uncover high-utility regions. To address this, we propose Sequentially-Controlled Interactive Multi-Particle Flow-Maps (IMPFM), a framework for feedback-efficient search. IMPFM progressively transports a group of interactive particles toward the target distribution, maintaining the broad coverage essential for heterogeneous preference alignment. IMPFM leverages a principled and efficient posterior sample-sharing mechanism across particles powered by flow maps. By correcting individual particle drift with the collective value gradient from the entire ensemble's posterior samples at each correction step, the framework maximizes sample utility to enable global exploration while actively mitigating reward over-optimization, typical of standard control frameworks. Paired with a principled exploration-exploitation reweighting mechanism involving multi-particle interaction, this sequentially corrected multi-particle dynamics explicitly preserves structural diversity and overcomes the weight degeneracy inherent to standard Sequential Monte Carlo (SMC) samplers. Crucially, we prove that the resulting sampling framework yields a multi-particle interaction-aware Feynman-Kac corrector that progressively steers the multi-particle system toward a KL-tilted target distribution, facilitating global exploration and preventing mode collapse. Extensive empirical evaluations and ablations across diverse search and alignment tasks confirm the efficacy of IMPFM over existing baselines.

Abstract

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