Discrete Diffusion Inference-Time Control with Nested Sequential Monte Carlo

Lohithsai Yadala Chanchu; Hany Abdulsamad; Christian A. Naesseth

stat.MLAug 20, 2026

Discrete Diffusion Inference-Time Control with Nested Sequential Monte Carlo

Authors: Lohithsai Yadala Chanchu, Hany Abdulsamad, Christian A. Naesseth

Organizations: University of Amsterdam

Abstract

We study inference-time control for text generation in discrete diffusion language models, where the goal is to steer sampling toward sequence-level rewards without retraining. Prior work in this domain has focused on particle-based methods such as best-of- $n$ sampling and bootstrap sequential Monte Carlo, which may suffer from overoptimism and weight degeneracy, respectively. We address these limitations using \emph{nested} sequential Monte Carlo methods. We formulate nested SMC (NSMC) and fully-adapted nested SMC (FA-NSMC) for Feynman--Kac steering, identifying and correcting errors in prior formulations that lead to biased final estimates. We evaluate these methods on toxicity and fluency steering tasks, showing that NSMC and FA-NSMC consistently outperform best-of- $n$ and bootstrap SMC.

Figures & tables

Panel of Figure 1 — Figure 1: Illustration of different particle-based steering strategies. (Left) Best-of- $n$ : independent proposals with selection based on exponentiated rewards. (Middle) Nested sequential importance sampling: particles propagate sequentially with weight updates but no resampling. (Right) Nested SMC: particle filtering approach where each outer particle spawns an inner SMC, approximating the locally optimal proposal distribution for improved effective sample size.

Panel of Figure 2 — Algorithm 1 Nested Proposal

Panel of Figure 3 — Algorithm 2 Nested SMC

Table 4

Table 5

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 7 — Algorithm 3 Bootstrap SMC

Panel of Figure 8 — Algorithm 4 Fully-Adapted Nested SMC

Figure 3: Intermediate rewards under toxicity steering increase over time, with higher population size $M$ , yielding better rewards but diminishing returns and a largely stable ranking across time steps.

Method	Distinct-1 $\uparrow$	Distinct-2 $\uparrow$
BoN	0.29	0.74
SMC	0.26	0.71
NSMC	0.26	0.70
FA-NSMC	0.26	0.71

Appendix

Table 5: Output diversity on the toxicity task (Distinct-

n

; higher is more diverse) with

N=8

K=4

\lambda=10

$x_{0}$ samples	Particles	Inner particles	Toxic acc	Ext toxic acc
4	4	4	0.13	0.13
4	4	8	0.17	0.17
4	8	4	0.03	0.01
4	8	8	0.20	0.16
16	4	4	0.16	0.13
16	4	8	0.26	0.20

Appendix

Table 6: Toxicity rates for biased NSMC implementation according to Uehara et al. (2025) averaged over 10 repetitions (

\lambda=10

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Open paper AskPDF arXiv

Many inference-time tasks for pretrained discrete diffusion models and diffusion language models reduce to drawing samples from a tilted version of the pretrained distribution. Feynman-Kac sequential Monte Carlo (SMC) makes this correction exact in principle, but its prescribed weights routinely degenerate when the proposal dynamics are misaligned with the tilt, capping the practical gains from additional particles. We introduce FluxLite, a lightweight, training-free proposal-control framework for discrete diffusion. On the sparse directed graph of pretrained reverse rates, any sparse jump-rate perturbation can be exactly compensated by a

q_t

-weighted graph-divergence term in the Feynman-Kac potential; the target path is therefore preserved while the residual reweighting variance becomes a local convex objective. We instantiate this principle as two practical samplers: a one-hop local reallocation rule (HEU) and a small nonnegative quadratic program over pretrained-rate bases (D-VCG). We further prove population stability under the standard score-entropy training loss, identifying a tilted-path coverage factor that governs robustness to score error, together with finite-particle convergence for a fixed controlled Feynman-Kac recursion. Empirically, FluxLite improves over standard Feynman-Kac SMC baselines by up to two orders of magnitude in terminal KL on an analytically tractable finite-state CTMC benchmark, and reduces row-correlation MSE on 2D Ising sampling by 5-7x in geometric mean and up to 55x at peak.

Abstract

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