用于加速离散扩散中序贯蒙特卡洛引导的价值驱动调度裁剪

# 1Introduction
Source: [https://arxiv.org/html/2610.20669](https://arxiv.org/html/2610.20669)
Value\-Informed Schedule Trimming for Accelerated Sequential Monte Carlo Guidance in Discrete DiffusionJaehyeon Kim1,∗, Ziseok Lee1,∗, Siyoung Choi1,∗, Kyungsu Kim1,†1Seoul National University ∗equal contribution,†corresponding author:\{kyskim\}@snu\.ac\.krReward\-guided discrete diffusion enables test\-time alignment with downstream objectives, but applying Sequential Monte Carlo \(SMC\) guidance at every denoising step is computationally expensive\. Active guidance steps require additional reward evaluations and, for gradient\-based proposals, value\-gradient computations\. Sparse guidance reduces this cost by guiding only a subset of timesteps, but commonly used schedules are fixed heuristics whose effectiveness can vary substantially across tasks\. We propose VISTA, a training\-free method that automatically selects guided timesteps under a fixed budget\. VISTA formulates sparse guidance as approximation of the ideal full\-guidance probability path and derives an estimable upper bound onℓ1\\ell\_\{1\}path deviation\. Using value statistics collected from a small number of full\-guidance warmup generations, VISTA optimizes the resulting empirical surrogate exactly via dynamic programming\. The warmup generations can be retained as model outputs, and after schedule selection VISTA has the same per\-generation guidance complexity as other sparse schedules with the same budget\. Across text, biological sequence, and image generation tasks, VISTA discovers task\-dependent schedules and outperforms hand\-designed sparse schedules in most tested settings, while approaching or exceeding full\-guidance performance on several task metrics\. In toxic\-text generation, for example, VISTA attains a 0\.99 unique\-toxic ratio compared with 1\.00 for full guidance while reducing post\-selection per\-generation latency from 204\.23 s to 38\.16 s, a 5\.35×\\timesspeedup\.Lab:[https://aibl\.snu\.ac\.kr](https://aibl.snu.ac.kr/)![[Uncaptioned image]](https://arxiv.org/html/2610.20669v1/aibl_logo.png)

## 1Introduction

Discrete diffusion models\[[2](https://arxiv.org/html/2610.20669#bib.bib5)\]have emerged as a powerful framework for generating structured objects such as text, biological sequences, and discrete images\. In these domains, practitioners often prioritize not only the likelihood under a pretrained model but also alignment with external objectives, such as safety in language, biological activity in DNA, or constraint satisfaction\. Consequently, a primary goal is to sample from a reward\-tilted distribution that steers the pretrained model toward higher expected rewards while preserving its generative capabilities\.

Several reward\-tilted guidance techniques for discrete domains have recently been introduced\[[25](https://arxiv.org/html/2610.20669#bib.bib16),[30](https://arxiv.org/html/2610.20669#bib.bib4)\]\. However, these independent sampling methods—which generate samples along separate trajectories—often rely on derivative\-based approximations that can introduce significant error and bias\[[19](https://arxiv.org/html/2610.20669#bib.bib8)\]\. To address this limitation, Sequential Monte Carlo \(SMC\) methods have been proposed\. SMC begins by generating multiple candidate particles using a proposal policy\. To align the particles with the reward\-tilted distribution, it employs a resampling mechanism: particles with high value functions are duplicated, while those with low values are discarded \(see Figure[1](https://arxiv.org/html/2610.20669#S4.F1)\-\(b\)\)\. Under the asymptotic conditions used in the SMC analysis, this framework recovers the exact reward\-tilted distribution at guided timesteps\[[19](https://arxiv.org/html/2610.20669#bib.bib8),[28](https://arxiv.org/html/2610.20669#bib.bib1),[24](https://arxiv.org/html/2610.20669#bib.bib2),[9](https://arxiv.org/html/2610.20669#bib.bib9),[17](https://arxiv.org/html/2610.20669#bib.bib14),[22](https://arxiv.org/html/2610.20669#bib.bib17)\]\.

Despite their theoretical elegance, the computational cost of SMC guidance remains a bottleneck in practical applications\. Each guided denoising step requires not only a pretrained diffusion transition but also an evaluation by an external reward model; moreover, gradient\-based proposals often necessitate additional backward passes\. Applying this correction at every timestepttcan dominate the total wall\-clock time\. Consequently, sparse guidance—where only a subset ofT′<TT^\{\\prime\}<Ttimesteps is guided—provides a natural way to improve the efficiency of inference\-time alignment, whereTTdenotes the total number of diffusion steps\. This approach can reduce the total reward and gradient computation to a fraction ofT′/TT^\{\\prime\}/T\(see Table[2](https://arxiv.org/html/2610.20669#S4.T2)for a detailed breakdown\)\.

However, a central question remains under\-explored: which steps should be guided? Existing sparse guidance schedules are typically heuristic\-based and fixed, for example by restricting guidance to a prescribed interval\[[18](https://arxiv.org/html/2610.20669#bib.bib27),[14](https://arxiv.org/html/2610.20669#bib.bib6),[38](https://arxiv.org/html/2610.20669#bib.bib7)\]or applying it at uniformly spaced timesteps\[[31](https://arxiv.org/html/2610.20669#bib.bib22),[12](https://arxiv.org/html/2610.20669#bib.bib38),[16](https://arxiv.org/html/2610.20669#bib.bib39)\]\. Such fixed schedules can be suboptimal because the effective timing of guidance depends on the task and reward landscape\. As we show empirically, some tasks favor guidance concentrated in the early denoising stages, whereas others benefit from combining early guidance with corrections distributed throughout the reverse process\. Ultimately, a schedule that is effective for one reward landscape can be suboptimal for another\.

To mitigate this issue, we propose VISTA\-SMC \(Value\-Informed Schedule Trimming for Accelerated Sequential Monte Carlo\), a training\-free, plug\-and\-play procedure for identifying a budgeted schedule of guided steps\. By minimizing a tractable surrogate for the discrepancy between the full\-guidance trajectory and the reduced\-guidance path, VISTA characterizes the guidance dynamics of a given task and identifies a highly effective schedule tailored to its specific reward landscape\.

Specifically, VISTA addresses sparse SMC guidance as a full\-path approximation problem: it chooses the guided\-step set \(i\.e\.,𝒢\\mathcal\{G\}\) so that the reduced\-guidance path stays close to the ideal full\-guidance path \(i\.e\.,\{πt∗\}t=0T\\\{\\pi\_\{t\}^\{\*\}\\\}\_\{t=0\}^\{T\}\)\. Directly optimizing this gap is infeasible because the ground\-truth full\-guidance marginals are not available as explicit distributions over the discrete state space\. Our main observation is that the gap admits a tractable upper bound expressed through value expectations along the full\-guidance path\. These expectations can be estimated from a small number of warmup full\-guidance runs; the resulting empirical budgeted objective is then solved exactly by dynamic programming\. The selected schedule is reused for all later generations with the same model, reward, budget, and proposal family, so the warmup cost is amortized and the per\-generation comparison to sparse heuristics is fair\.

Our contributions are ordered by conceptual importance:

1. 1\.Automatic budgeted guidance scheduling\.We introduce a training\-free scheduler that automatically selects a task\-specific subset of SMC guidance timesteps under a fixed budget, using full\-guidance value statistics instead of a predefined temporal pattern\.
2. 2\.Full\-guidance gap formulation\.We view sparse SMC guidance as approximating the ideal full\-guidance probability path, not only the final marginal\. This is crucial: final\-only correction may be asymptotically valid yet can lead to diversity collapse with finite particles\.
3. 3\.Tractable gap minimization without ground\-truth marginals\.To our knowledge, VISTA is the first sparse\-SMC scheduler to target the gap to full\-guidance marginals without accessing those intractable marginals\. We prove an estimable upper bound onℓ1\\ell\_\{1\}path deviation and optimize the resulting surrogate under a fixed budget\.
4. 4\.Practical schedule construction\.VISTA uses a few initial full\-guidance generations to estimate a task\-specific schedule\. These generations are retained as generated outputs rather than discarded preprocessing\. Afterward, VISTA has the same per\-generation complexity as heuristic sparse schedules with the same budget, although heuristics can operate sparsely from the first generation\.
5. 5\.Task\-specific empirical schedules\.On Binary MNIST, toxic text, and DNA sequence generation, VISTA finds reward\-dependent schedules and outperforms hand\-designed sparse baselines in most tested settings at the same budget\. It also yields large speedups over full guidance, including a5\.35×5\.35\\timespost\-selection speedup in the toxic\-text SMC\-grad experiment atT′=15T^\{\\prime\}=15, while achieving a unique\-toxic ratio of0\.990\.99compared with1\.001\.00under full guidance\.

## 2Related Work

##### Discrete and Masked Diffusion\.

Diffusion models define a forward process that transforms a data distribution into a tractable prior and learn reverse transitions for generation\. While continuous diffusion models typically use Gaussian perturbations and score\-based reverse dynamics\[[13](https://arxiv.org/html/2610.20669#bib.bib11),[33](https://arxiv.org/html/2610.20669#bib.bib12)\], discrete diffusion models define Markov transitions over categorical or masked states\[[2](https://arxiv.org/html/2610.20669#bib.bib5),[29](https://arxiv.org/html/2610.20669#bib.bib3)\]\. Masked diffusion language models instantiate this idea by using a special \[MASK\] token as the high\-noise state and iteratively unmasking or denoising tokens\. Our work focuses on inference\-time reward guidance for such reverse processes, with a particular emphasis on addressing the challenges of discrete state spaces\.

##### Sequential Monte Carlo Methods\.

SMC samplers propagate particles through a proposal policy and use resampling to correct bias, effectively aligning the particles with the target reward\-tilted distribution\[[7](https://arxiv.org/html/2610.20669#bib.bib31),[24](https://arxiv.org/html/2610.20669#bib.bib2),[6](https://arxiv.org/html/2610.20669#bib.bib32)\]\. Recent work has adapted SMC to diffusion inference for tasks such as continuous and discrete diffusion steering, test\-time alignment, inverse problems, and tree\-structured search\[[28](https://arxiv.org/html/2610.20669#bib.bib1),[32](https://arxiv.org/html/2610.20669#bib.bib26),[11](https://arxiv.org/html/2610.20669#bib.bib21),[1](https://arxiv.org/html/2610.20669#bib.bib33),[37](https://arxiv.org/html/2610.20669#bib.bib10),[15](https://arxiv.org/html/2610.20669#bib.bib15),[10](https://arxiv.org/html/2610.20669#bib.bib34),[5](https://arxiv.org/html/2610.20669#bib.bib35)\]\. While these methods primarily focus on*how*particles are weighted or proposed,VISTAis complementary: it automatically identifies the most critical timesteps for guidance, determining*when*these expensive corrections should be executed to maximize overall efficiency\.

##### Inference\-time Acceleration of Diffusion Guidance\.

Prior work has reduced the cost of diffusion guidance using hand\-designed temporal schedules, including restricting guidance to a prescribed interval\[[18](https://arxiv.org/html/2610.20669#bib.bib27),[14](https://arxiv.org/html/2610.20669#bib.bib6),[38](https://arxiv.org/html/2610.20669#bib.bib7)\]or uniformly spaced timesteps\[[31](https://arxiv.org/html/2610.20669#bib.bib22),[12](https://arxiv.org/html/2610.20669#bib.bib38),[16](https://arxiv.org/html/2610.20669#bib.bib39)\]\. Other approaches reduce guidance overhead through adaptive rules, caching, or multirate integration\[[4](https://arxiv.org/html/2610.20669#bib.bib18),[23](https://arxiv.org/html/2610.20669#bib.bib20),[20](https://arxiv.org/html/2610.20669#bib.bib19)\]\. These methods demonstrate that guidance need not be applied at every denoising step, but their schedules are either manually specified or developed primarily for continuous\-image guidance and network\-evaluation reuse\. In contrast,VISTAis specifically formulated for reward\-guided SMC in discrete state spaces and provides a theoretical bound on the probability\-path deviation\. Table[1](https://arxiv.org/html/2610.20669#S2.T1)summarizes these key distinctions\.

Table 1:Comparison ofVISTAwith existing guidance acceleration methods\. Discrete: Designed for discrete state spaces; Reward: Handles external reward functions; Theoretical Guarantee: Provides a rigorous theoretical justification for the acceleration schedule\.MethodDiscreteRewardTheoretical GuaranteeAdaptive Guidance\[[4](https://arxiv.org/html/2610.20669#bib.bib18)\]×××FasterCache\[[23](https://arxiv.org/html/2610.20669#bib.bib20)\]×××Tortoise\-and\-Hare\[[20](https://arxiv.org/html/2610.20669#bib.bib19)\]××✓Guidance Interval\[[18](https://arxiv.org/html/2610.20669#bib.bib27)\]×××VISTA\(Ours\)✓✓✓

## 3Preliminaries

### 3\.1Reference Reverse Process

Let𝒳\\mathcal\{X\}be a discrete state space\. A pretrained reverse diffusion model defines a reference path:

pT​\(xT\)​∏t=0T−1pt\|t\+1​\(xt\|xt\+1\),p\_\{T\}\(x\_\{T\}\)\\prod\_\{t=0\}^\{T\-1\}p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\),\(1\)with one\-step reverse kernelspt\|t\+1p\_\{t\|t\+1\}and marginalsptp\_\{t\}\. For0≤s≤t≤T0\\leq s\\leq t\\leq T, we denote byps\|tp\_\{s\|t\}the composed transition fromxtx\_\{t\}toxsx\_\{s\}\. In a masked diffusion model,xTx\_\{T\}is typically a fully masked sequence, andx0x\_\{0\}is a clean sequence\.

### 3\.2Reward\-Tilted Full\-Guidance Path

Given a reward functionr⁡\(x0\)r\(x\_\{0\}\)for a samplex0x\_\{0\}, our objective is to maximize the expected reward while regularizing the divergence from the pretrained model’s prior\. Specifically, we aim to find an optimal policy\{πt\}t=T0\\\{\\pi\_\{t\}\\\}\_\{t=T\}^\{0\}that maximizes:

𝔼\{πt\}t=T0\[r\(x0\)−α∑t=0T−1KL\(πt\|t\+1\(⋅\|xt\+1\)∥pt\|t\+1\(⋅\|xt\+1\)\)−αKL\(πT\(⋅\)∥pT\(⋅\)\)\],\\mathbb\{E\}\_\{\\\{\\pi\_\{t\}\\\}\_\{t=T\}^\{0\}\}\\left\[r\(x\_\{0\}\)\-\\alpha\\sum\_\{t=0\}^\{T\-1\}\\text\{KL\}\(\\pi\_\{t\|t\+1\}\(\\cdot\|x\_\{t\+1\}\)\\parallel p\_\{t\|t\+1\}\(\\cdot\|x\_\{t\+1\}\)\)\-\\alpha\\text\{KL\}\(\\pi\_\{T\}\(\\cdot\)\\parallel p\_\{T\}\(\\cdot\)\)\\right\],\(2\)
whereα\>0\\alpha\>0is a temperature parameter controlling the trade\-off between reward maximization and prior preservation\[[8](https://arxiv.org/html/2610.20669#bib.bib23),[34](https://arxiv.org/html/2610.20669#bib.bib24),[35](https://arxiv.org/html/2610.20669#bib.bib25)\]\. The closed\-form solution to this optimization problem is:

πt\|t\+1∗​\(xt\|xt\+1\)\\displaystyle\\pi\_\{t\|t\+1\}^\{\*\}\(x\_\{t\}\|x\_\{t\+1\}\)=Vt​\(xt\)Vt\+1​\(xt\+1\)​pt\|t\+1​\(xt\|xt\+1\),t∈\{0,…,T−1\}\\displaystyle=\\frac\{V\_\{t\}\(x\_\{t\}\)\}\{V\_\{t\+1\}\(x\_\{t\+1\}\)\}p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\),\\quad t\\in\\\{0,\\dots,T\-1\\\}\(3\)πt∗​\(xt\)\\displaystyle\\pi\_\{t\}^\{\*\}\(x\_\{t\}\)=1Z​Vt​\(xt\)​pt​\(xt\),\\displaystyle=\\frac\{1\}\{Z\}V\_\{t\}\(x\_\{t\}\)p\_\{t\}\(x\_\{t\}\),\(4\)
whereZ:=𝔼x0∼p0​\[R⁡\(x0\)\]Z:=\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\}\}\[R\(x\_\{0\}\)\]is the partition function\. Here,R⁡\(x0\):=exp⁡\(r⁡\(x0\)/α\)R\(x\_\{0\}\):=\\exp\(r\(x\_\{0\}\)/\\alpha\)represents the exponentiated reward, andVt​\(xt\)V\_\{t\}\(x\_\{t\}\)is the exponentiated value function defined as:

Vt\(xt\):=𝔼x0∼p0\|t\(⋅\|xt\)\[R\(x0\)\]\.V\_\{t\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\|t\}\(\\cdot\|x\_\{t\}\)\}\\left\[R\(x\_\{0\}\)\\right\]\.\(5\)Although the value functionvt​\(xt\):=α​log⁡Vt​\(xt\)v\_\{t\}\(x\_\{t\}\):=\\alpha\\log V\_\{t\}\(x\_\{t\}\)is standard in the literature\[[35](https://arxiv.org/html/2610.20669#bib.bib25),[36](https://arxiv.org/html/2610.20669#bib.bib13)\], the exponentiated notationVtV\_\{t\}simplifies our derivation\.

For a detailed derivation, we refer the reader to\[[35](https://arxiv.org/html/2610.20669#bib.bib25)\]or our Appendix[B\.1](https://arxiv.org/html/2610.20669#A2.SS1)\.

###### Definition 3\.1\.

We callπt∗\\pi\_\{t\}^\{\*\}thefull\-guidance path\. It is the gold\-standard path thatVISTAaims to approximate\.

### 3\.3SMC Proposal Families

Algorithm 1SMC guidance1:

xT\(i\)∼πT∗​\(xT\)\(i=1,2,⋯,N\)x\_\{T\}^\{\(i\)\}\\sim\\pi\_\{T\}^\{\*\}\(x\_\{T\}\)\\quad\(i=1,2,\\cdots,N\)
2:for

t=T−1,T−2,⋯,0t=T\-1,T\-2,\\cdots,0do

3:Propagation:

xt\(i\)​∼iid​qt\|t\+1​\(xt\|xt\+1\(i\)\)\(i=1,2,⋯,N\)x\_\{t\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}^\{\(i\)\}\)\\quad\(i=1,2,\\cdots,N\)
4:Resampling: Duplicate particles with high

VtV\_\{t\}and discard those with low

VtV\_\{t\}to ensure

xt\(i\)∼πt∗​\(xt\)x\_\{t\}^\{\(i\)\}\\sim\\pi\_\{t\}^\{\*\}\(x\_\{t\}\)
5:endfor

SMC guidance propagates particles through a proposal policyqt\|t\+1q\_\{t\|t\+1\}and utilizes the resampling to align the particle distribution with the target pathπt∗\\pi\_\{t\}^\{\*\}\(Algorithm[1](https://arxiv.org/html/2610.20669#alg1); see Algorithm[4](https://arxiv.org/html/2610.20669#alg4)for details\)\. Generally, the closerqt\|t\+1q\_\{t\|t\+1\}is toπt\|t\+1∗\\pi\_\{t\|t\+1\}^\{\*\}, the higher the sampling efficiency\. We evaluateVISTAon two proposal families established in recent discrete SMC literature\[[28](https://arxiv.org/html/2610.20669#bib.bib1)\]:

- •SMC\-baseuses the pretrained transition:qt\|t\+1​\(xt\|xt\+1\)=pt\|t\+1​\(xt\|xt\+1\)q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)=p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\.
- •SMC\-graduses a first\-order Taylor approximation of the value function to steer the proposal, yieldingqt\|t\+1\(xt\|xt\+1\)∝pt\|t\+1\(xt\|xt\+1\)exp\(1α∇vt\+1\(xt\+1\)⋅xt\)q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\\propto p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\\exp\(\\frac\{1\}\{\\alpha\}\\nabla v\_\{t\+1\}\(x\_\{t\+1\}\)\\cdot x\_\{t\}\)\.

VISTAis proposal\-agnostic; it does not modify the underlying proposal family\. Instead, it strategically selects which timesteps should incur the computational overhead of value estimation and resampling to maximize efficiency\.

## 4VISTA: Value\-Informed Schedule Trimming

In this section, we introduce Reduced SMC Guidance, a framework designed to accelerate the inference of guided diffusion models by strategically skipping guidance steps\.

### 4\.1Sparse guidance as full\-path approximation

Let𝒯=\{0,1,⋯,T−1\}\\mathcal\{T\}=\\\{0,1,\\cdots,T\-1\\\}denote the set of reverse timesteps\. A sparse schedule is a subset𝒢⊂𝒯\\mathcal\{G\}\\subset\\mathcal\{T\}with budget\|𝒢\|=T′\|\\mathcal\{G\}\|=T^\{\\prime\}\. Under this framework, reward guidance is applied if and only ift∈𝒢t\\in\\mathcal\{G\}, as detailed in Algorithm[2](https://arxiv.org/html/2610.20669#alg2)\.

Algorithm 2Diffusion Guidance Reduction1:

xT\(i\)∼πT∗​\(xT\)\(i=1,2,⋯,N\)x\_\{T\}^\{\(i\)\}\\sim\\pi\_\{T\}^\{\*\}\(x\_\{T\}\)\\quad\(i=1,2,\\cdots,N\)
2:for

t=T−1,T−2,⋯,0t=T\-1,T\-2,\\cdots,0do

3:

Guidance is ON←t∈𝒢\\text\{Guidance is ON\}\\leftarrow t\\in\\mathcal\{G\}
4:ifGuidance is ONthen

5:Propagation:

xt\(i\)​∼iid​qt\|t\+1​\(xt\|xt\+1\(i\)\)\(i=1,2,⋯,N\)x\_\{t\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}^\{\(i\)\}\)\\quad\(i=1,2,\\cdots,N\)
6:Resampling: Adjust particles

\{xt\(i\)\}\\\{x\_\{t\}^\{\(i\)\}\\\}to ensure

xt\(i\)∼πt∗​\(xt\)x\_\{t\}^\{\(i\)\}\\sim\\pi\_\{t\}^\{\*\}\(x\_\{t\}\)
7:else

8:Propagation:

xt\(i\)​∼iid​pt\|t\+1​\(xt\|xt\+1\(i\)\)\(i=1,2,⋯,N\)x\_\{t\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}^\{\(i\)\}\)\\quad\(i=1,2,\\cdots,N\)
9:endif

10:endfor

For anyt∈𝒯t\\in\\mathcal\{T\}, we define the most recent active step⌈t⌉\\lceil t\\rceilas:

⌈t⌉:=min⁡\{s∈𝒢∪\{T\}∣s≥t\}\.\\lceil t\\rceil:=\\min\\\{s\\in\\mathcal\{G\}\\cup\\\{T\\\}\\mid s\\geq t\\\}\.\(6\)
Ift∈𝒢t\\in\\mathcal\{G\}, then⌈t⌉=t\\lceil t\\rceil=tand the sampler performs a guided SMC correction\. Ift∉𝒢t\\notin\\mathcal\{G\}, it propagates from⌈t⌉\\lceil t\\rceiltottusing unguided reference transitions\. This observation leads to the following definition:

###### Definition 4\.1\.

For a sparse schedule𝒢\\mathcal\{G\}, thereduced\-guidance pathis

πt𝒢​\(xt\):=∑x⌈t⌉∈𝒳π⌈t⌉∗​\(x⌈t⌉\)​pt\|⌈t⌉​\(xt\|x⌈t⌉\)\.\\pi\_\{t\}^\{\\mathcal\{G\}\}\(x\_\{t\}\):=\\sum\_\{x\_\{\\lceil t\\rceil\\in\\mathcal\{X\}\}\}\\pi\_\{\\lceil t\\rceil\}^\{\*\}\(x\_\{\\lceil t\\rceil\}\)p\_\{t\|\\lceil t\\rceil\}\(x\_\{t\}\|x\_\{\\lceil t\\rceil\}\)\.\(7\)

A naive objective would be to find𝒢\\mathcal\{G\}such that only the final marginalπ0𝒢\\pi\_\{0\}^\{\\mathcal\{G\}\}matches the targetπ0∗\\pi\_\{0\}^\{\*\}\. Theoretically, if the number of particlesNNis infinite, even a minimal schedule like𝒢=\{0\}\\mathcal\{G\}=\\\{0\\\}is sufficient to recover the exact reward\-tilted distribution at the final step\. However, this ignores a critical practical challenge: in the finite\-NNregime, delaying guidance until the end causes the sampler to postpone all reward information\. This leads to diversity collapse and poor reward \(see Fig\.[1](https://arxiv.org/html/2610.20669#S4.F1)\)\. To mitigate this degradation in practical sampling, we instead aim to minimize the deviation between the full\-guidance and reduced\-guidance paths across all timesteps\.

Figure 1:Impact of guidance schedules on particle distributions\. A minimal schedule \(e\.g\.,𝒢=\{0\}\\mathcal\{G\}=\\\{0\\\}\) is asymptotically correct asN→∞N\\to\\infty, but with finite particles, it leads to diversity collapse through the duplication of a few terminal particles\. In contrast, full guidance \(𝒢=𝒯\\mathcal\{G\}=\\mathcal\{T\}\) maintains intermediate marginals close to the reward\-tilted path, ensuring both high reward and sample diversity\.###### Definition 4\.2\.

TheTotal Deviationfor a schedule𝒢\\mathcal\{G\}is defined as the averageℓ1\\ell\_\{1\}distance between the full\-guidance and reduced\-guidance paths:

𝐓𝐃⁡\(𝒢\):=1T​∑t=0T−1‖πt∗−πt𝒢‖1=1T​∑t=0T−1∑xt∈𝒳\|πt∗​\(xt\)−πt𝒢​\(xt\)\|\.\\mathbf\{TD\}\(\\mathcal\{G\}\):=\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\\|\\pi\_\{t\}^\{\*\}\-\\pi\_\{t\}^\{\\mathcal\{G\}\}\\\|\_\{1\}=\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\sum\_\{x\_\{t\}\\in\\mathcal\{X\}\}\|\\pi\_\{t\}^\{\*\}\(x\_\{t\}\)\-\\pi\_\{t\}^\{\\mathcal\{G\}\}\(x\_\{t\}\)\|\.\(8\)

This definition formalizes the core objective ofVISTA: schedule selection is transformed from a heuristic choice of intervals into a principled, budgeted optimization to minimize the gap between the sparse sampling path and the ideal full\-guidance probability path\.

### 4\.2A tractable upper bound

A major obstacle in directly optimizing𝐓𝐃⁡\(𝒢\)\\mathbf\{TD\}\(\\mathcal\{G\}\)is the intractability of the full\-guidance pathπt∗\\pi\_\{t\}^\{\*\}\. The following theorem provides a computable surrogate based on value expectations that can be estimated from a small number of warmup runs\.

###### Theorem 4\.3\.

Assume0<R⁡\(x0\)≤Rmax<∞0<R\(x\_\{0\}\)\\leq R\_\{\\max\}<\\infty\. For any schedule𝒢⊂𝒯\\mathcal\{G\}\\subset\\mathcal\{T\},

𝐓𝐃⁡\(𝒢\)≤𝐔𝐓𝐃⁡\(𝒢\):=2T​∑t∈𝒯∖𝒢\[1−1Rmax​𝔼x⌈t⌉∼π⌈t⌉∗​\[V⌈t⌉​\(x⌈t⌉\)\]\]\.\\mathbf\{TD\}\(\\mathcal\{G\}\)\\leq\\mathbf\{UTD\}\(\\mathcal\{G\}\):=\\frac\{2\}\{T\}\\sum\_\{t\\in\\mathcal\{T\}\\setminus\\mathcal\{G\}\}\[1\-\\frac\{1\}\{R\_\{\\text\{max\}\}\}\\mathbb\{E\}\_\{x\_\{\\lceil t\\rceil\}\\sim\\pi\_\{\\lceil t\\rceil\}^\{\*\}\}\[V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\]\]\.\(9\)

The proof of this inequality is provided in Section[B\.3](https://arxiv.org/html/2610.20669#A2.SS3)\.

Our goal is to identify a schedule𝒢\\mathcal\{G\}that minimizes𝐔𝐓𝐃\\mathbf\{UTD\}\. This approach—optimizing a tractable upper bound when the primary objective is intractable—is a well\-established strategy in machine learning\[[3](https://arxiv.org/html/2610.20669#bib.bib28),[27](https://arxiv.org/html/2610.20669#bib.bib29),[26](https://arxiv.org/html/2610.20669#bib.bib30)\]\.

For a fixed budget of guidance stepsT′=\|𝒢\|T^\{\\prime\}=\|\\mathcal\{G\}\|, minimizing𝐔𝐓𝐃\\mathbf\{UTD\}is equivalent to maximizing the following objective:

S⁡\(𝒢\):=∑t∈𝒯∖𝒢𝔼x⌈t⌉∼π⌈t⌉∗​\[V⌈t⌉​\(x⌈t⌉\)\]\.S\(\\mathcal\{G\}\):=\\sum\_\{t\\in\\mathcal\{T\}\\setminus\\mathcal\{G\}\}\\mathbb\{E\}\_\{x\_\{\\lceil t\\rceil\}\\sim\\pi^\{\*\}\_\{\\lceil t\\rceil\}\}\\left\[V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\\right\]\.\(10\)Intuitively, if the state at the most recent active stepx⌈t⌉x\_\{\\lceil t\\rceil\}already exhibits a high expected reward, subsequent unguided propagation is less likely to deviate significantly from the optimal path\.

Crucially, this optimization does not require knowledge of the full marginalsπt∗​\(xt\)\\pi\_\{t\}^\{\*\}\(x\_\{t\}\), nor the constantRmaxR\_\{\\max\}; it relies solely on the Monte Carlo estimation of𝔼xt∼πt∗​\[Vt​\(xt\)\]\\mathbb\{E\}\_\{x\_\{t\}\\sim\\pi^\{\*\}\_\{t\}\}\\left\[V\_\{t\}\(x\_\{t\}\)\\right\]\.

### 4\.3Description of the Method

VISTAestimates𝔼xt∼πt∗​\[Vt​\(xt\)\]\\mathbb\{E\}\_\{x\_\{t\}\\sim\\pi^\{\*\}\_\{t\}\}\\left\[V\_\{t\}\(x\_\{t\}\)\\right\]by runningMMfull\-guidance warmup simulations\. LetV^t\(i,m\)\\hat\{V\}\_\{t\}^\{\(i,m\)\}be the value estimate ofii\-th particle at timestepttduring themm\-th warmup run\. We approximate the mean value as:

𝔼xt∼πt∗​\[Vt​\(xt\)\]≈V^t:=1M​N​∑m=1M∑i=1NV^t\(i,m\)\.\\mathbb\{E\}\_\{x\_\{t\}\\sim\\pi^\{\*\}\_\{t\}\}\\left\[V\_\{t\}\(x\_\{t\}\)\\right\]\\approx\\hat\{V\}\_\{t\}:=\\frac\{1\}\{MN\}\\sum\_\{m=1\}^\{M\}\\sum\_\{i=1\}^\{N\}\\hat\{V\}\_\{t\}^\{\(i,m\)\}\.\(11\)The empirical schedule𝒢^\\hat\{\\mathcal\{G\}\}is then obtained by solving:

𝒢^∈argmax𝒢⊂𝒯:\|𝒢\|=T′S^\(𝒢\),S^\(𝒢\):=∑t∈𝒯∖𝒢V^⌈t⌉\\hat\{\\mathcal\{G\}\}\\in\\arg\\max\_\{\\mathcal\{G\}\\subset\\mathcal\{T\}:\|\\mathcal\{G\}\|=T^\{\\prime\}\}\\hat\{S\}\(\\mathcal\{G\}\),\\quad\\hat\{S\}\(\\mathcal\{G\}\):=\\sum\_\{t\\in\\mathcal\{T\}\\setminus\\mathcal\{G\}\}\\hat\{V\}\_\{\\lceil t\\rceil\}\(12\)
The optimization in Eq\. \([12](https://arxiv.org/html/2610.20669#S4.E12)\) is solved exactly by dynamic programming inO⁡\(T2​T′\)O\(T^\{2\}T^\{\\prime\}\)time \(see Algorithm[6](https://arxiv.org/html/2610.20669#alg6)\), which is negligible for the horizons used here\. Once𝒢^\\hat\{\\mathcal\{G\}\}is selected,VISTAutilizes this sparse schedule for all subsequent generations to accelerate inference\.

### 4\.4Computational Time Analysis

VISTArequiresMMinitial full\-guidance generations for schedule selection\. Once the schedule is determined, it has the same per\-generation complexity as other sparse guidance methods under the same guidance budget\. Table[2](https://arxiv.org/html/2610.20669#S4.T2)summarizes the resulting computational costs\.

Table 2:Per\-generation cost after schedule selection\. Here,ppis a pretrained reverse\-model evaluation,qqis an amortized proposal,rris a reward\-model evaluation,∇v\\nabla vis a value\-gradient evaluation,NNis the number of particles,TTis the diffusion horizon,T′T^\{\\prime\}is the guidance budget, andJJis the number of rollout samples for value estimation \(see Algorithm[5](https://arxiv.org/html/2610.20669#alg5)for details\)\.VariantFull GuidanceSparse GuidanceSMC\-baseN​T×p,N​T​J×rNT\\times p,\\ NTJ\\times rN​T×p,N​T′​J×rNT\\times p,\\ NT^\{\\prime\}J\\times rSMC\-gradN​T×p,N​T​J×r,N​T×∇vNT\\times p,\\ NTJ\\times r,\\ NT\\times\\nabla vN​T×p,N​T′​J×r,N​T′×∇vNT\\times p,\\ NT^\{\\prime\}J\\times r,\\ NT^\{\\prime\}\\times\\nabla v

## 5Experiments

To evaluate the effectiveness ofVISTA, we focus our main experiments on Masked Diffusion Models \(MDMs\), with experiments on other diffusion models provided in Appendix[C](https://arxiv.org/html/2610.20669#A3)\. We compareVISTAagainst full guidance and several sparse scheduling baselines\. Specifically,Interval\-kk\(k=1,…,5k=1,\\ldots,5\) allocates all guidance steps to thekk\-th equally sized segment of the reverse diffusion trajectory, where Interval\-1 corresponds to the earliest denoising stage and Interval\-5 to the latest\.Uniformdistributes the guidance steps evenly across the entire trajectory\.Top\-VVselects the timesteps with the largest estimated valuesV^t\\hat\{V\}\_\{t\}, whileTop\-Δ​V\\Delta Vselects the timesteps with the largest consecutive value differencesV^t−V^t\+1\\hat\{V\}\_\{t\}\-\\hat\{V\}\_\{t\+1\}\. Both value\-based baselines use the same warmup value statistics asVISTA\. Unless otherwise stated, all reported times correspond to the per\-generation cost after schedule selection; warmup costs are reported separately\. Comprehensive experimental details are available in Appendix[C](https://arxiv.org/html/2610.20669#A3)\.

### 5\.1Synthetic Experiment: Binary MNIST

We first evaluateVISTAon Binary MNIST using the MDM with the SMC\-grad proposal, based on the codebase of\[[28](https://arxiv.org/html/2610.20669#bib.bib1)\]\. The reward is defined asr⁡\(x0\)=log⁡p⁡\(c∣x0\)r\(x\_\{0\}\)=\\log p\(c\\mid x\_\{0\}\), whereccdenotes the condition “the digit is 4”\. We use a total horizon ofT=100T=100withα=1\.0\\alpha=1\.0,N=20N=20particles,J=1J=1rollout for value estimation, andM=3M=3full\-guidance warmup generations\. A samplex0x\_\{0\}is considered “valid” ifr⁡\(x0\)\>−0\.1r\(x\_\{0\}\)\>\-0\.1\. To measure diversity,Unique Validcounts the number of distinct valid samples, where samples with Jaccard similarity≥0\.85\\geq 0\.85are treated as identical\. Results are averaged over 50 independent seeds, yielding 1,000 generated samples per schedule\. The warmup phase took 24\.7 s\. For evaluation,𝐔𝐓𝐃\\mathbf\{UTD\}was computed withRmax=1R\_\{\\max\}=1, which is valid sinceR⁡\(x0\)=p⁡\(c∣x0\)≤1R\(x\_\{0\}\)=p\(c\\mid x\_\{0\}\)\\leq 1\. Figure[2](https://arxiv.org/html/2610.20669#S5.F2)summarizes the results, and complete numerical results, including the SMC\-base setting, are provided in Appendix[C\.2\.1](https://arxiv.org/html/2610.20669#A3.SS2.SSS1)\.

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/mnist_mdm_grad_time.png)

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/mnist_mdm_grad_unique_valid.png)

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/mnist_mdm_grad_utd.png)

Figure 2:Binary MNIST \(SMC\-grad\)\. Execution time, Unique Valid, and UTD as the guidance budgetT′T^\{\\prime\}varies\.
### 5\.2Toxic Text Generation

Toxic text generation is a high\-dimensional discrete task where the reward model must guide the sampler against a strong language prior\. The reward is defined asr⁡\(x0\)=log⁡p⁡\(c∣x0\)r\(x\_\{0\}\)=\\log p\(c\\mid x\_\{0\}\)for a target toxicity condition\. Without guidance, the base model satisfies the target toxicity condition in only0\.8%0\.8\\%of generations\. We reportUnique Toxic Ratio, defined as the fraction of generated samples that are both toxic and unique, where samples with normalized Levenshtein distance≤0\.05\\leq 0\.05are treated as identical\. We setT=100T=100,α=0\.2\\alpha=0\.2,N=8N=8,J=4J=4, andM=3M=3, and use the SMC\-grad proposal\. For each of 15 prompts, we perform two independent runs, yielding15×2×8=24015\\times 2\\times 8=240generated samples for each scheduling method\. The warmup phase required 639\.6 s\. Figure[3](https://arxiv.org/html/2610.20669#S5.F3)summarizes the results, and complete numerical results are provided in Appendix[C\.2\.2](https://arxiv.org/html/2610.20669#A3.SS2.SSS2)\. Our implementation is based on the codebase of\[[28](https://arxiv.org/html/2610.20669#bib.bib1)\]\.

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/toxic_execution_time.png)

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/toxic_unique_ratio.png)

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/toxic_dp_score.png)

Figure 3:Toxic Text \(SMC\-grad\)\. Execution time, Unique Toxic Ratio, andS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)as the guidance budgetT′T^\{\\prime\}varies\.
### 5\.3CIFAR\-10 Image Generation

We evaluate CIFAR\-10 generation experiment using the MDM, SMC\-base framework, based on the codebase of\[[30](https://arxiv.org/html/2610.20669#bib.bib4)\]\. The reward is defined asr⁡\(x0\)=log⁡p⁡\(c\|x0\)r\(x\_\{0\}\)=\\log p\(c\|x\_\{0\}\), whereccrepresents the condition “the image is an airplane”\. We useT=1,000T=1,000time steps,N=20N=20particles,α=1\.0\\alpha=1\.0,J=5J=5rollout for value estimation, andM=5M=5warmup simulations\. A samplex0x\_\{0\}is considered “valid” ifp⁡\(c\|x0\)\>0\.5p\(c\|x\_\{0\}\)\>0\.5\.Unique Validis number of valid samples, where samples with LPIPS distance≤0\.03\\leq 0\.03are considered identical\. The warmup phase took 217\.9 s\. Results are averaged over 50 runs, totaling1,0001,000generated samples per schedule\. The graph of experiment result is in Figure[4](https://arxiv.org/html/2610.20669#S5.F4)\.

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/cifar10_execution_time.png)

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/cifar10_unique_success.png)

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/cifar10_shat.png)

Figure 4:CIFAR\-10 \(SMC\-base\)\. Execution time, unique valid samples, andS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)as the guidance budgetT′T^\{\\prime\}varies\.
### 5\.4Biological Sequence Design

For DNA generation, we follow the experimental setup of\[[28](https://arxiv.org/html/2610.20669#bib.bib1)\]\. We use the rewardr⁡\(x0\)=log⁡p⁡\(c∣x0\)r\(x\_\{0\}\)=\\log p\(c\\mid x\_\{0\}\), wherep⁡\(c∣x0\)p\(c\\mid x\_\{0\}\)denotes the predicted ATAC accessibility of the generated sequence\. Following\[[39](https://arxiv.org/html/2610.20669#bib.bib37)\], a sequence is considered biologically active if its ATAC\-Acc score exceeds0\.50\.5\. We defineUnique Validas the number of distinct biologically active sequences, where two sequences with Hamming distance≤1\\leq 1are treated as identical\. We setT=128T=128,α=0\.01\\alpha=0\.01,N=8N=8,J=4J=4, andM=3M=3\. Results are averaged over 30 independent runs, yielding 240 generated sequences for each scheduling method\. The warmup phase took 49\.1 s\.

For Protein Fitness, we use the pretrained masked diffusion model from\[[29](https://arxiv.org/html/2610.20669#bib.bib3)\]together with a fitness oracle as the reward modelr⁡\(x0\)r\(x\_\{0\}\)\. A sequence is considered valid ifr⁡\(x0\)≥0\.5r\(x\_\{0\}\)\\geq 0\.5, andUnique Validcounts the number of distinct valid sequences, where two sequences with Hamming distance≤1\\leq 1are treated as identical\. We setT=128T=128,α=0\.3\\alpha=0\.3,N=32N=32,J=3J=3, andM=3M=3\. Results are averaged over 20 independent runs, yielding 640 generated sequences for each scheduling method\. the warmup phase took 19\.5 s\.

The main results are presented in Tables[4](https://arxiv.org/html/2610.20669#S5.T4)and[4](https://arxiv.org/html/2610.20669#S5.T4), with the corresponding schedules shown in Appendix[C\.2\.4](https://arxiv.org/html/2610.20669#A3.SS2.SSS4)\.

Table 3:DNA sequence generation \(SMC\-grad\): execution time, unique valid, andS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)\.T′T^\{\\prime\}PolicyTime \[s\]↓\\downarrowUnique Valid↑\\uparrowS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)↑\\uparrow128Full15\.34±0\.9715\.34\\pm 0\.975\.90±1\.35\\mathbf\{5\.90\\pm 1\.35\}–8Interval 12\.61±0\.632\.61\\pm 0\.630\.23±0\.420\.23\\pm 0\.421\.89×10−41\.89\\times 10^\{\-4\}Interval 22\.54±0\.542\.54\\pm 0\.540\.20±0\.480\.20\\pm 0\.482\.89×10−62\.89\\times 10^\{\-6\}Interval 32\.69±0\.662\.69\\pm 0\.660\.60±0\.760\.60\\pm 0\.763\.773\.77Interval 42\.87±0\.762\.87\\pm 0\.761\.03±1\.621\.03\\pm 1\.624\.574\.57Interval 52\.78±0\.752\.78\\pm 0\.751\.27±1\.151\.27\\pm 1\.151\.43×10−61\.43\\times 10^\{\-6\}Top\-VV2\.66±0\.732\.66\\pm 0\.731\.27±1\.151\.27\\pm 1\.151\.43×10−61\.43\\times 10^\{\-6\}Top\-Δ​V\\Delta V2\.94±0\.742\.94\\pm 0\.742\.30±1\.992\.30\\pm 1\.998\.958\.95Uniform2\.79±0\.662\.79\\pm 0\.660\.83±0\.520\.83\\pm 0\.527\.777\.77VISTA2\.84±0\.612\.84\\pm 0\.612\.73±1\.90\\mathbf\{2\.73\\pm 1\.90\}12\.40\\mathbf\{12\.40\}24Interval 15\.04±0\.875\.04\\pm 0\.870\.40±0\.610\.40\\pm 0\.612\.64×10−52\.64\\times 10^\{\-5\}Interval 25\.09±0\.705\.09\\pm 0\.700\.77±1\.020\.77\\pm 1\.020\.150\.15Interval 35\.09±0\.725\.09\\pm 0\.721\.50±1\.261\.50\\pm 1\.263\.373\.37Interval 44\.96±0\.684\.96\\pm 0\.683\.37±1\.853\.37\\pm 1\.852\.662\.66Interval 54\.93±0\.844\.93\\pm 0\.843\.20±1\.993\.20\\pm 1\.991\.24×10−61\.24\\times 10^\{\-6\}Top\-VV5\.05±0\.705\.05\\pm 0\.703\.07±1\.413\.07\\pm 1\.410\.190\.19Top\-Δ​V\\Delta V5\.63±0\.805\.63\\pm 0\.804\.93±1\.84\\mathbf\{4\.93\\pm 1\.84\}9\.589\.58Uniform5\.51±0\.805\.51\\pm 0\.802\.70±1\.392\.70\\pm 1\.397\.107\.10VISTA5\.29±0\.815\.29\\pm 0\.814\.17±1\.214\.17\\pm 1\.2112\.08\\mathbf\{12\.08\}

Table 4:Protein Fitness generation \(SMC\-base\): execution time, unique valid, andS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)\.T′T^\{\\prime\}PolicyTime \[s\]↓\\downarrowUnique Valid↑\\uparrowS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)↑\\uparrow128Full3\.123\.129\.00±2\.77\\mathbf\{9\.00\\pm 2\.77\}–16Interval 11\.531±0\.0031\.531\\pm 0\.00313\.20±3\.2413\.20\\pm 3\.243\.79×1033\.79\\times 10^\{3\}Interval 21\.530±0\.0041\.530\\pm 0\.00410\.65±4\.4410\.65\\pm 4\.442\.95×1032\.95\\times 10^\{3\}Interval 31\.530±0\.0021\.530\\pm 0\.0028\.10±2\.158\.10\\pm 2\.152\.07×1032\.07\\times 10^\{3\}Interval 41\.529±0\.0021\.529\\pm 0\.0026\.00±2\.206\.00\\pm 2\.201\.07×1031\.07\\times 10^\{3\}Interval 51\.517±0\.0031\.517\\pm 0\.0035\.35±2\.165\.35\\pm 2\.161\.12×1021\.12\\times 10^\{2\}Top\-VV1\.519±0\.0021\.519\\pm 0\.0028\.40±2\.688\.40\\pm 2\.683\.34×1033\.34\\times 10^\{3\}Top\-Δ​V\\Delta V1\.531±0\.0021\.531\\pm 0\.00212\.60±2\.5812\.60\\pm 2\.584\.23×1034\.23\\times 10^\{3\}Uniform1\.521±0\.0031\.521\\pm 0\.00310\.85±3\.7910\.85\\pm 3\.793\.83×1033\.83\\times 10^\{3\}VISTA1\.530±0\.0041\.530\\pm 0\.00414\.15±3\.80\\mathbf\{14\.15\\pm 3\.80\}4\.53×𝟏𝟎𝟑\\mathbf\{4\.53\\times 10^\{3\}\}32Interval 11\.777±0\.0031\.777\\pm 0\.00312\.60±4\.04\\mathbf\{12\.60\\pm 4\.04\}3\.57×1033\.57\\times 10^\{3\}Interval 21\.777±0\.0051\.777\\pm 0\.0059\.60±4\.589\.60\\pm 4\.582\.81×1032\.81\\times 10^\{3\}Interval 31\.776±0\.0041\.776\\pm 0\.0046\.80±3\.246\.80\\pm 3\.241\.82×1031\.82\\times 10^\{3\}Interval 41\.773±0\.0051\.773\\pm 0\.0056\.00±2\.106\.00\\pm 2\.109\.15×1029\.15\\times 10^\{2\}Interval 51\.760±0\.0061\.760\\pm 0\.0066\.10±2\.476\.10\\pm 2\.4796\.096\.0Top\-VV1\.762±0\.0031\.762\\pm 0\.0038\.55±2\.958\.55\\pm 2\.952\.62×1032\.62\\times 10^\{3\}Top\-Δ​V\\Delta V1\.774±0\.0081\.774\\pm 0\.00811\.25±3\.4911\.25\\pm 3\.493\.56×1033\.56\\times 10^\{3\}Uniform1\.768±0\.0041\.768\\pm 0\.00411\.80±2\.8211\.80\\pm 2\.823\.33×1033\.33\\times 10^\{3\}VISTA1\.776±0\.0021\.776\\pm 0\.00212\.00±3\.3212\.00\\pm 3\.323\.94×𝟏𝟎𝟑\\mathbf\{3\.94\\times 10^\{3\}\}

### 5\.5Discussion

The schedules selected byVISTAdiffer substantially across tasks, highlighting the importance of adapting guidance timing to the underlying model and reward function\. Binary MNIST places most guidance steps in the early part of the reverse process, whereas Toxic Text, DNA, and CIFAR\-10 distribute guidance more broadly across the trajectory\. These distinct patterns suggest that a single fixed temporal schedule is unlikely to be uniformly effective across tasks\.

Interestingly, sparse guidance can sometimes outperform full guidance on diversity\-sensitive metrics, as observed on CIFAR\-10\. This is consistent with the possibility that overly frequent correction can reduce particle diversity in finite\-particle SMC\. Overall, the results indicate that selectively allocating guidance steps can provide a favorable trade\-off between downstream performance and inference cost\.

Finally, the weighted extension in Appendix[C](https://arxiv.org/html/2610.20669#A3)shows that the same schedule\-selection principle can also be applied beyond masked diffusion through non\-uniform temporal weighting\.

## 6Limitation

VISTAhas three main limitations\. First,VISTArequires a small number of full\-guidance warmup generations before schedule optimization\. These generations can be retained as model outputs while simultaneously providing the value statistics used for schedule selection, but they still introduce an upfront computational cost\. Therefore, the benefit ofVISTAis most pronounced when the selected schedule is reused across multiple subsequent generations\.

Second, the UTD objective is a conservative upper bound on path deviation rather than a direct objective for downstream task performance\. As a result, the schedule that optimizesS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)does not necessarily maximize the downstream evaluation metric at every guidance budget\. Tighter surrogates that better capture finite\-particle effects and downstream sample quality are an important direction for future work\.

Third, the weighted extension for non\-masked diffusion requires a user\-specified temporal weighting functionλ⁡\(t\)\\lambda\(t\)\. Our experiments show that non\-uniform weighting can extendVISTAbeyond masked diffusion, but the appropriate choice ofλ⁡\(t\)\\lambda\(t\)may depend on the diffusion process and task\. Automatically learning or adapting these temporal weights remains an open problem\.

## 7Conclusion

We introducedVISTA, a training\-free scheduler for sparse reward\-guided SMC in discrete diffusion\. By minimizing an estimable upper bound on path deviation,VISTAidentifies task\-specific guidance schedules under a fixed computational budget\. Our experiments across image, text, and biological domains demonstrate that automatic schedule selection can retain strong downstream performance, and in several settings even outperform full guidance, while substantially reducing inference latency\. Ultimately, this work shows that for efficient discrete diffusion alignment, choosing*when*to guide is as important as designing*how*to guide\.

## Acknowledgments

This work was partly supported by the KHIDI grant funded by the Korean government \(MOHW\) \[No\.RS\-2025\-02307233, No\.RS\-2026\-25613012\], the NRF or IITP grants funded by the Korean government \(MSIT\) \[No\.05\-26\-04\-0094, No\.RS\-2026\-25472075, No\.RS\-2025\-02305581, No\.RS\-2025\-25442338, and No\.RS\-2021\-II211343\], the ITIP grant funded by the Korean government \(MOTIR\) \[No\.RS\-2026\-25549946\], the Research grant from SNU, and the Strategic Hub grant for International Research Collaboration of SNU\. Kyungsu Kim is affiliated with the School of Transdisciplinary Innovations, Department of Biomedical Science, Interdisciplinary Program in AI \(IPAI\) & Graduate School of AI, Medical Research Center, and AI Institute at SNU\.

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Appendix

## Appendix ALLM Usage Disclosure

We used a large\-language model \(LLM\) only for minor text editing, such as correcting typos and adjusting wording to a formal academic tone\. The LLM was not involved in research ideation, experimental design, implementation, or analysis\. All scientific content and claims were conceived and written by the authors, who take full responsibility for the entire paper\.

## Appendix BTheoretical Details

### B\.1Background

###### Lemma B\.1\.

Vt\(xt\)=𝔼xs∼ps\|t\(⋅\|xt\)\[Vs\(xs\)\]V\_\{t\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{x\_\{s\}\\sim p\_\{s\|t\}\(\\cdot\|x\_\{t\}\)\}\[V\_\{s\}\(x\_\{s\}\)\]\(13\)where0≤s<t≤T0\\leq s<t\\leq T\. Also,

𝔼xt∼pt​\[Vt​\(xt\)\]=𝔼x0∼p0​\[R⁡\(x0\)\]\.\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\}\}\[V\_\{t\}\(x\_\{t\}\)\]=\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\}\}\[R\(x\_\{0\}\)\]\.\(14\)

###### Proof\.

Vt\(xt\)=𝔼x0∼p0\|t\(⋅\|xt\)\[R\(x0\)\]V\_\{t\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\|t\}\(\\cdot\|x\_\{t\}\)\}\[R\(x\_\{0\}\)\]\(15\)comes from the definition of value function\.

𝔼xs∼ps\|t\(⋅\|xt\)\[Vs\(xs\)\]=𝔼xs∼ps\|t\(⋅\|xt\)\[𝔼x0∼p0\|s\(⋅\|xs\)\[R\(x0\)\]\]\\displaystyle\\mathbb\{E\}\_\{x\_\{s\}\\sim p\_\{s\|t\}\(\\cdot\|x\_\{t\}\)\}\[V\_\{s\}\(x\_\{s\}\)\]=\\mathbb\{E\}\_\{x\_\{s\}\\sim p\_\{s\|t\}\(\\cdot\|x\_\{t\}\)\}\[\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\|s\}\(\\cdot\|x\_\{s\}\)\}\[R\(x\_\{0\}\)\]\]\(16\)=\\displaystyle=𝔼x0∼p0\|t\(⋅\|xt\)\[R\(x0\)\]=Vt\(xt\)\.\\displaystyle\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\|t\}\(\\cdot\|x\_\{t\}\)\}\[R\(x\_\{0\}\)\]=V\_\{t\}\(x\_\{t\}\)\.\(17\)
𝔼xT∼pT\[VT\(xT\)\]=𝔼xT∼pT\[𝔼x0∼p0\|T\(⋅\|xT\)\[R\(x0\)\]\]=𝔼x0∼p0\[R\(x0\)\]\.\\displaystyle\\mathbb\{E\}\_\{x\_\{T\}\\sim p\_\{T\}\}\[V\_\{T\}\(x\_\{T\}\)\]=\\mathbb\{E\}\_\{x\_\{T\}\\sim p\_\{T\}\}\[\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\|T\}\(\\cdot\|x\_\{T\}\)\}\[R\(x\_\{0\}\)\]\]=\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\}\}\[R\(x\_\{0\}\)\]\.\(18\)∎

###### Theorem B\.2\.

\{πt\}t=T0\\\{\\pi\_\{t\}\\\}\_\{t=T\}^\{0\}that maximizes

𝔼\{πt\}t=T0\[r\(x0\)−α∑t=T−10KL\(πt\|t\+1\(⋅\|xt\+1\)\|\|pt\|t\+1\(⋅\|xt\+1\)\)−αKL\(πT\(⋅\)\|\|pT\(⋅\)\)\]\\mathbb\{E\}\_\{\\\{\\pi\_\{t\}\\\}\_\{t=T\}^\{0\}\}\[r\(x\_\{0\}\)\-\\alpha\\sum\_\{t=T\-1\}^\{0\}\\KL\(\\pi\_\{t\|t\+1\}\(\\cdot\|x\_\{t\+1\}\)\|\|p\_\{t\|t\+1\}\(\\cdot\|x\_\{t\+1\}\)\)\-\\alpha\\KL\(\\pi\_\{T\}\(\\cdot\)\|\|p\_\{T\}\(\\cdot\)\)\]\(19\)\(α\>0\\alpha\>0\) is

πt​\(xt\)=1Z​Vt​\(xt\)​pt​\(xt\)\.\(Z:=𝔼x0∼p0​\[R⁡\(x0\)\]\)\\pi\_\{t\}\(x\_\{t\}\)=\\frac\{1\}\{Z\}V\_\{t\}\(x\_\{t\}\)p\_\{t\}\(x\_\{t\}\)\.\\quad\(Z:=\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{0\}\}\[R\(x\_\{0\}\)\]\)\(20\)

###### Proof\.

We define the Markov process\{πt∗\}t=T0\\\{\\pi\_\{t\}^\{\*\}\\\}\_\{t=T\}^\{0\}as

πT∗​\(xT\)\\displaystyle\\pi\_\{T\}^\{\*\}\(x\_\{T\}\)=1Z​VT​\(xT\)​pT​\(xT\),\\displaystyle=\\frac\{1\}\{Z\}V\_\{T\}\(x\_\{T\}\)p\_\{T\}\(x\_\{T\}\),\(21\)πt\|t\+1∗​\(xt\|xt\+1\)\\displaystyle\\pi\_\{t\|t\+1\}^\{\*\}\(x\_\{t\}\|x\_\{t\+1\}\)=Vt​\(xt\)​pt\|t\+1​\(xt\|xt\+1\)Vt\+1​\(xt\+1\)\(t=0,1,⋯,T−1\)\.\\displaystyle=\\frac\{V\_\{t\}\(x\_\{t\}\)p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\}\{V\_\{t\+1\}\(x\_\{t\+1\}\)\}\\quad\(t=0,1,\\cdots,T\-1\)\.\(22\)\(These distributions are well\-defined\(normalized\) by Lemma[B\.1](https://arxiv.org/html/2610.20669#A2.Thmtheorem1)\.\)

We observe

𝔼\{πt\}t=T0\[r\(x0\)−α∑t=T−10KL\(πt\|t\+1\(⋅\|xt\+1\)\|\|pt\|t\+1\(⋅\|xt\+1\)\)−αKL\(πT\(⋅\)\|\|pT\(⋅\)\)\]\\displaystyle\\mathbb\{E\}\_\{\\\{\\pi\_\{t\}\\\}\_\{t=T\}^\{0\}\}\[r\(x\_\{0\}\)\-\\alpha\\sum\_\{t=T\-1\}^\{0\}\\KL\(\\pi\_\{t\|t\+1\}\(\\cdot\|x\_\{t\+1\}\)\|\|p\_\{t\|t\+1\}\(\\cdot\|x\_\{t\+1\}\)\)\-\\alpha\\KL\(\\pi\_\{T\}\(\\cdot\)\|\|p\_\{T\}\(\\cdot\)\)\]\(23\)=\\displaystyle=𝔼x0∼π0​\[r⁡\(x0\)\]−α​∑t=T−10𝔼xt,xt\+1∼πt,t\+1​\[log⁡πt\|t\+1​\(xt\|xt\+1\)pt\|t\+1​\(xt\|xt\+1\)\]−α​𝔼xT∼πT​\[log⁡πT​\(xT\)pT​\(xT\)\]\\displaystyle\\mathbb\{E\}\_\{x\_\{0\}\\sim\\pi\_\{0\}\}\[r\(x\_\{0\}\)\]\-\\alpha\\sum\_\{t=T\-1\}^\{0\}\\mathbb\{E\}\_\{x\_\{t\},x\_\{t\+1\}\\sim\\pi\_\{t,t\+1\}\}\[\\log\\frac\{\\pi\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\}\{p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\}\]\-\\alpha\\mathbb\{E\}\_\{x\_\{T\}\\sim\\pi\_\{T\}\}\[\\log\\frac\{\\pi\_\{T\}\(x\_\{T\}\)\}\{p\_\{T\}\(x\_\{T\}\)\}\]\(24\)=\\displaystyle=𝔼xT∼πT𝔼xT−1∼πT−1\|T\(⋅\|xT\)⋯𝔼x0∼π0\|1\(⋅\|x1\)\\displaystyle\\mathbb\{E\}\_\{x\_\{T\}\\sim\\pi\_\{T\}\}\\mathbb\{E\}\_\{x\_\{T\-1\}\\sim\\pi\_\{T\-1\|T\}\(\\cdot\|x\_\{T\}\)\}\\cdots\\mathbb\{E\}\_\{x\_\{0\}\\sim\\pi\_\{0\|1\}\(\\cdot\|x\_\{1\}\)\}\[r⁡\(x0\)−α​∑t=T−10log⁡πt\|t\+1​\(xt\|xt\+1\)pt\|t\+1​\(xt\|xt\+1\)−α​log⁡πT​\(xT\)pT​\(xT\)\]\.\\displaystyle\\quad\[r\(x\_\{0\}\)\-\\alpha\\sum\_\{t=T\-1\}^\{0\}\\log\\frac\{\\pi\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\}\{p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\}\-\\alpha\\log\\frac\{\\pi\_\{T\}\(x\_\{T\}\)\}\{p\_\{T\}\(x\_\{T\}\)\}\]\.\(25\)
If we use the recurrence relation

𝔼xs∼πs\|s\+1\(⋅\|xs\+1\)\[vs\(xs\)−αlogπs\|s\+1​\(xs\|xs\+1\)ps\|s\+1​\(xs\|xs\+1\)\]\\displaystyle\\mathbb\{E\}\_\{x\_\{s\}\\sim\\pi\_\{s\|s\+1\}\(\\cdot\|x\_\{s\+1\}\)\}\[v\_\{s\}\(x\_\{s\}\)\-\\alpha\\log\\frac\{\\pi\_\{s\|s\+1\}\(x\_\{s\}\|x\_\{s\+1\}\)\}\{p\_\{s\|s\+1\}\(x\_\{s\}\|x\_\{s\+1\}\)\}\]\(26\)=\\displaystyle=𝔼xs∼πs\|s\+1\(⋅\|xs\+1\)\[vs\+1\(xs\+1\)−αlogπs\|s\+1​\(xs\|xs\+1\)πs\|s\+1∗​\(xs\|xs\+1\)\]\\displaystyle\\mathbb\{E\}\_\{x\_\{s\}\\sim\\pi\_\{s\|s\+1\}\(\\cdot\|x\_\{s\+1\}\)\}\[v\_\{s\+1\}\(x\_\{s\+1\}\)\-\\alpha\\log\\frac\{\\pi\_\{s\|s\+1\}\(x\_\{s\}\|x\_\{s\+1\}\)\}\{\\pi\_\{s\|s\+1\}^\{\*\}\(x\_\{s\}\|x\_\{s\+1\}\)\}\]\(27\)=\\displaystyle=vs\+1\(xs\+1\)−αKL\(πs\|s\+1\(⋅\|xs\+1\)\|\|πs\|s\+1∗\(⋅\|xs\+1\)\)\\displaystyle v\_\{s\+1\}\(x\_\{s\+1\}\)\-\\alpha\\KL\(\\pi\_\{s\|s\+1\}\(\\cdot\|x\_\{s\+1\}\)\|\|\\pi\_\{s\|s\+1\}^\{\*\}\(\\cdot\|x\_\{s\+1\}\)\)\(28\)consequently, Eq\. \([25](https://arxiv.org/html/2610.20669#A2.E25)\) could be simplified into

𝔼xT∼πT\[vT\(xT\)−αlogπT​\(xT\)pT​\(xT\)\]−α∑t=T−10𝔼xt\+1∼πt\+1\[KL\(πt\|t\+1\(⋅\|xt\+1\)\|\|πt\|t\+1∗\(⋅\|xt\+1\)\)\]\\displaystyle\\mathbb\{E\}\_\{x\_\{T\}\\sim\\pi\_\{T\}\}\[v\_\{T\}\(x\_\{T\}\)\-\\alpha\\log\\frac\{\\pi\_\{T\}\(x\_\{T\}\)\}\{p\_\{T\}\(x\_\{T\}\)\}\]\-\\alpha\\sum\_\{t=T\-1\}^\{0\}\\mathbb\{E\}\_\{x\_\{t\+1\}\\sim\\pi\_\{t\+1\}\}\[\\KL\(\\pi\_\{t\|t\+1\}\(\\cdot\|x\_\{t\+1\}\)\|\|\\pi\_\{t\|t\+1\}^\{\*\}\(\\cdot\|x\_\{t\+1\}\)\)\]\(29\)=\\displaystyle=αlogZ−α∑t=T−10𝔼xt\+1∼πt\+1\[KL\(πt\|t\+1\(⋅\|xt\+1\)\|\|πt\|t\+1∗\(⋅\|xt\+1\)\)\]−αKL\(πT\(⋅\)\|\|πT∗\(⋅\)\)\.\\displaystyle\\alpha\\log Z\-\\alpha\\sum\_\{t=T\-1\}^\{0\}\\mathbb\{E\}\_\{x\_\{t\+1\}\\sim\\pi\_\{t\+1\}\}\[\\KL\(\\pi\_\{t\|t\+1\}\(\\cdot\|x\_\{t\+1\}\)\|\|\\pi\_\{t\|t\+1\}^\{\*\}\(\\cdot\|x\_\{t\+1\}\)\)\]\-\\alpha\\KL\(\\pi\_\{T\}\(\\cdot\)\|\|\\pi\_\{T\}^\{\*\}\(\\cdot\)\)\.\(30\)
We now easily see that this objective is maximized iffπT=πT∗\\pi\_\{T\}=\\pi\_\{T\}^\{\*\}andπt\|t\+1=πt\|t\+1∗\\pi\_\{t\|t\+1\}=\\pi\_\{t\|t\+1\}^\{\*\}for allt=0,1,⋯,T−1t=0,1,\\cdots,T\-1\.

The final step is proving

πt∗​\(xt\)=1Z​Vt​\(xt\)​pt​\(xt\)\.\\pi\_\{t\}^\{\*\}\(x\_\{t\}\)=\\frac\{1\}\{Z\}V\_\{t\}\(x\_\{t\}\)p\_\{t\}\(x\_\{t\}\)\.\(31\)This is satisfied att=Tt=T\. If

πt\+1∗​\(xt\+1\)=1Z​Vt\+1​\(xt\+1\)​pt​\(xt\+1\),\\pi\_\{t\+1\}^\{\*\}\(x\_\{t\+1\}\)=\\frac\{1\}\{Z\}V\_\{t\+1\}\(x\_\{t\+1\}\)p\_\{t\}\(x\_\{t\+1\}\),\(32\)then

πt∗​\(xt\)\\displaystyle\\pi\_\{t\}^\{\*\}\(x\_\{t\}\)=∑xt\+1πt\+1∗​\(xt\+1\)​πt\|t\+1∗​\(xt\|xt\+1\)\\displaystyle=\\sum\_\{x\_\{t\+1\}\}\\pi\_\{t\+1\}^\{\*\}\(x\_\{t\+1\}\)\\pi\_\{t\|t\+1\}^\{\*\}\(x\_\{t\}\|x\_\{t\+1\}\)\(33\)=∑xt\+1Vt\+1​\(xt\+1\)​pt\+1​\(xt\+1\)Z​Vt​\(xt\)​pt\|t\+1​\(xt\|xt\+1\)Vt\+1​\(xt\+1\)\\displaystyle=\\sum\_\{x\_\{t\+1\}\}\\frac\{V\_\{t\+1\}\(x\_\{t\+1\}\)p\_\{t\+1\}\(x\_\{t\+1\}\)\}\{Z\}\\frac\{V\_\{t\}\(x\_\{t\}\)p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\}\{V\_\{t\+1\}\(x\_\{t\+1\}\)\}\(34\)=1Z​Vt​\(xt\)​pt​\(xt\)\.\\displaystyle=\\frac\{1\}\{Z\}V\_\{t\}\(x\_\{t\}\)p\_\{t\}\(x\_\{t\}\)\.\(35\)Proof ends by induction\.

∎

Algorithm 3Multinomial resampling1:Input:

\{x\(i\)\}i=1N,\{w\(i\)\}i=1N\\\{x^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w^\{\(i\)\}\\\}\_\{i=1\}^\{N\}
2:for

k←1,2,⋯,Nk\\leftarrow 1,2,\\cdots,Ndo

3:

ik​∼iid​Cat​\(i:w\(i\)∑n=1Nw\(n\)\)i\_\{k\}\\overset\{\\text\{iid\}\}\{\\sim\}\\text\{Cat\}\(i:\\frac\{w^\{\(i\)\}\}\{\\sum\_\{n=1\}^\{N\}w^\{\(n\)\}\}\)
4:

y\(k\)←x\(ik\)y^\{\(k\)\}\\leftarrow x^\{\(i\_\{k\}\)\}
5:endfor

6:Output:

\{y\(k\)\}k=1N,\{1\}k=1N\\\{y^\{\(k\)\}\\\}\_\{k=1\}^\{N\},\\\{1\\\}\_\{k=1\}^\{N\}

### B\.2Reduced SMC algorithm

Algorithm[3](https://arxiv.org/html/2610.20669#alg3)defines the idealized multinomial resampling operator utilized in our analysis\. In practice, more efficient variants such as systematic, adaptive, or partial resampling may be employed; we refer the reader to\[[24](https://arxiv.org/html/2610.20669#bib.bib2)\]for a comprehensive review of these methods\.

###### Lemma B\.3\(Importance Sampling\)\.

Let\(x\(i\),x~\(i\)\)​∼iid​ρ​\(x,x~\)\(x^\{\(i\)\},\\tilde\{x\}^\{\(i\)\}\)\\overset\{\\text\{iid\}\}\{\\sim\}\\rho\(x,\\tilde\{x\}\), and letw⁡\(x,x~\)≥0w\(x,\\tilde\{x\}\)\\geq 0be a weight function such thatw\(i\)=w⁡\(x\(i\),x~\(i\)\)w^\{\(i\)\}=w\(x^\{\(i\)\},\\tilde\{x\}^\{\(i\)\}\)\. AsN→∞N\\rightarrow\\infty, the output particles\{y\(k\)\}k=1N\\\{y^\{\(k\)\}\\\}\_\{k=1\}^\{N\}of Algorithm[3](https://arxiv.org/html/2610.20669#alg3)satisfy:

y\(k\)​∼iid​π​\(x\)∝∑x~w⁡\(x,x~\)​ρ​\(x,x~\)\.y^\{\(k\)\}\\overset\{\\text\{iid\}\}\{\\sim\}\\pi\(x\)\\propto\\sum\_\{\\tilde\{x\}\}w\(x,\\tilde\{x\}\)\\rho\(x,\\tilde\{x\}\)\.\(36\)

###### Proof\.

For any test functionϕ⁡\(x\)\\phi\(x\),

𝔼x\(i\),x~\(i\)​∼iid​ρ​\(x,x~\)​\[ϕ⁡\(y\(k\)\)\]\\displaystyle\\mathbb\{E\}\_\{x^\{\(i\)\},\\tilde\{x\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}\\rho\(x,\\tilde\{x\}\)\}\[\\phi\(y^\{\(k\)\}\)\]=𝔼x\(i\),x~\(i\)​∼iid​ρ​\(x,x~\)​\[1N​∑i=1Nw⁡\(x\(i\),x~\(i\)\)​ϕ​\(x\(i\)\)1N​∑n=1Nw⁡\(x\(n\),x~\(n\)\)\]\.\\displaystyle=\\mathbb\{E\}\_\{x^\{\(i\)\},\\tilde\{x\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}\\rho\(x,\\tilde\{x\}\)\}\[\\frac\{\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}w\(x^\{\(i\)\},\\tilde\{x\}^\{\(i\)\}\)\\phi\(x^\{\(i\)\}\)\}\{\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}w\(x^\{\(n\)\},\\tilde\{x\}^\{\(n\)\}\)\}\]\.\(37\)
By the law of large numbers, asN→∞N\\rightarrow\\infty,

1N​∑n=1Nw⁡\(x\(n\),x~\(n\)\)\\displaystyle\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}w\(x^\{\(n\)\},\\tilde\{x\}^\{\(n\)\}\)→a\.s\.𝔼x,x~∼ρ​\[w⁡\(x,x~\)\]\\displaystyle\\xrightarrow\{\\text\{a\.s\.\}\}\\mathbb\{E\}\_\{x,\\tilde\{x\}\\sim\\rho\}\[w\(x,\\tilde\{x\}\)\]\(38\)1N​∑i=1Nw⁡\(x\(i\),x~\(i\)\)​ϕ​\(x\(i\)\)\\displaystyle\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}w\(x^\{\(i\)\},\\tilde\{x\}^\{\(i\)\}\)\\phi\(x^\{\(i\)\}\)→a\.s\.𝔼x,x~∼ρ​\[w⁡\(x,x~\)​ϕ​\(x\)\]=𝔼x,x~∼ρ​\[w⁡\(x,x~\)\]×𝔼x∼π​\[ϕ⁡\(x\)\]\.\\displaystyle\\xrightarrow\{\\text\{a\.s\.\}\}\\mathbb\{E\}\_\{x,\\tilde\{x\}\\sim\\rho\}\[w\(x,\\tilde\{x\}\)\\phi\(x\)\]=\\mathbb\{E\}\_\{x,\\tilde\{x\}\\sim\\rho\}\[w\(x,\\tilde\{x\}\)\]\\times\\mathbb\{E\}\_\{x\\sim\\pi\}\[\\phi\(x\)\]\.\(39\)
Thus, we get

𝔼x\(i\),x~\(i\)​∼iid​ρ​\(x,x~\)​\[ϕ⁡\(y\(k\)\)\]​→N→∞​𝔼x∼π​\[ϕ⁡\(x\)\],\\mathbb\{E\}\_\{x^\{\(i\)\},\\tilde\{x\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}\\rho\(x,\\tilde\{x\}\)\}\[\\phi\(y^\{\(k\)\}\)\]\\overset\{N\\rightarrow\\infty\}\{\\rightarrow\}\\mathbb\{E\}\_\{x\\sim\\pi\}\[\\phi\(x\)\],\(40\)soy\(k\)∼πy^\{\(k\)\}\\sim\\pi\.

Lastly, regarding the independence of\{y\(k\)\}k=1N\\\{y^\{\(k\)\}\\\}\_\{k=1\}^\{N\}, the multinomial resampling process draws eachy\(k\)y^\{\(k\)\}independently given the particle set\. AsN→∞N\\rightarrow\\infty, this process is equivalent to i\.i\.d\. sampling from the target distributionπ\\pi, as the empirical distribution converges to the trueπ\\pi\.

∎

Algorithm 4Full SMC guidance1:for

i←1,2,⋯,Ni\\leftarrow 1,2,\\cdots,Ndo

2:

xT\(i\)​∼iid​qT​\(xT\)x\_\{T\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}q\_\{T\}\(x\_\{T\}\)
3:

x0\(i,j\)​∼iid​p0\|T​\(x0\|xT\(i\)\)\(j=1,2,⋯,J\)x\_\{0\}^\{\(i,j\)\}\\overset\{\\text\{iid\}\}\{\\sim\}p\_\{0\|T\}\(x\_\{0\}\|x\_\{T\}^\{\(i\)\}\)\\quad\(j=1,2,\\cdots,J\)
4:

V^T\(i\)←1J​∑j=1JR⁡\(x0\(i,j\)\)\\hat\{V\}\_\{T\}^\{\(i\)\}\\leftarrow\\frac\{1\}\{J\}\\sum\_\{j=1\}^\{J\}R\(x\_\{0\}^\{\(i,j\)\}\)
5:

wT\(i\)←pT​\(xT\(i\)\)×V^T\(i\)/qT​\(xT\(i\)\)w\_\{T\}^\{\(i\)\}\\leftarrow p\_\{T\}\(x\_\{T\}^\{\(i\)\}\)\\times\\hat\{V\}\_\{T\}^\{\(i\)\}/q\_\{T\}\(x\_\{T\}^\{\(i\)\}\)
6:endfor

7:

\{xT\(i\)\}i=1N,\{wT\(i\)\}i=1N←Resample​\(\{xT\(i\)\}i=1N,\{wT\(i\)\}i=1N\)\\\{x\_\{T\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w\_\{T\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\\leftarrow\\text\{Resample\}\(\\\{x\_\{T\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w\_\{T\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\)
8:for

t←T−1,T−2,⋯,0t\\leftarrow T\-1,T\-2,\\cdots,0do

9:for

i←1,2,⋯,Ni\\leftarrow 1,2,\\cdots,Ndo

10:

xt\(i\)​∼iid​qt\|t\+1​\(xt\|xt\+1\(i\)\)x\_\{t\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}^\{\(i\)\}\)
11:

x0\(i,j\)​∼iid​p0\|t​\(x0\|xt\(i\)\)\(j=1,2,⋯,J\)x\_\{0\}^\{\(i,j\)\}\\overset\{\\text\{iid\}\}\{\\sim\}p\_\{0\|t\}\(x\_\{0\}\|x\_\{t\}^\{\(i\)\}\)\\quad\(j=1,2,\\cdots,J\)
12:

V^t\(i\)←1J​∑j=1JR⁡\(x0\(i,j\)\)\\hat\{V\}\_\{t\}^\{\(i\)\}\\leftarrow\\frac\{1\}\{J\}\\sum\_\{j=1\}^\{J\}R\(x\_\{0\}^\{\(i,j\)\}\)
13:

wt\(i\)←pt\|t\+1​\(xt\(i\)\|xt\+1\(i\)\)qt\|t\+1​\(xt\(i\)\|xt\+1\(i\)\)×V^t\(i\)V^⌈t\+1⌉\(i\)×w⌈t\+1⌉\(i\)w\_\{t\}^\{\(i\)\}\\leftarrow\\frac\{p\_\{t\|t\+1\}\(x\_\{t\}^\{\(i\)\}\|x\_\{t\+1\}^\{\(i\)\}\)\}\{q\_\{t\|t\+1\}\(x\_\{t\}^\{\(i\)\}\|x\_\{t\+1\}^\{\(i\)\}\)\}\\times\\frac\{\\hat\{V\}\_\{t\}^\{\(i\)\}\}\{\\hat\{V\}\_\{\\lceil t\+1\\rceil\}^\{\(i\)\}\}\\times w\_\{\\lceil t\+1\\rceil\}^\{\(i\)\}
14:endfor

15:

\{xt\(i\)\}i=1N,\{wt\(i\)\}i=1N←Resample​\(\{xt\(i\)\}i=1N,\{wt\(i\)\}i=1N\)\\\{x\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\\leftarrow\\text\{Resample\}\(\\\{x\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\)
16:endfor

Algorithm 5Sparse SMC guidance1:for

i←1,2,⋯,Ni\\leftarrow 1,2,\\cdots,Ndo

2:

xT\(i\)​∼iid​qT​\(xT\)x\_\{T\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}q\_\{T\}\(x\_\{T\}\)
3:

x0\(i,j\)​∼iid​p0\|T​\(x0\|xT\(i\)\)\(j=1,2,⋯,J\)x\_\{0\}^\{\(i,j\)\}\\overset\{\\text\{iid\}\}\{\\sim\}p\_\{0\|T\}\(x\_\{0\}\|x\_\{T\}^\{\(i\)\}\)\\quad\(j=1,2,\\cdots,J\)
4:

V^T\(i\)←1J​∑j=1JR⁡\(x0\(i,j\)\)\\hat\{V\}\_\{T\}^\{\(i\)\}\\leftarrow\\frac\{1\}\{J\}\\sum\_\{j=1\}^\{J\}R\(x\_\{0\}^\{\(i,j\)\}\)
5:

wT\(i\)←pT​\(xT\(i\)\)×V^T\(i\)/qT​\(xT\(i\)\)w\_\{T\}^\{\(i\)\}\\leftarrow p\_\{T\}\(x\_\{T\}^\{\(i\)\}\)\\times\\hat\{V\}\_\{T\}^\{\(i\)\}/q\_\{T\}\(x\_\{T\}^\{\(i\)\}\)
6:endfor

7:

\{xT\(i\)\}i=1N,\{wT\(i\)\}i=1N←Resample​\(\{xT\(i\)\}i=1N,\{wT\(i\)\}i=1N\)\\\{x\_\{T\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w\_\{T\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\\leftarrow\\text\{Resample\}\(\\\{x\_\{T\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w\_\{T\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\)
8:for

t←T−1,T−2,⋯,0t\\leftarrow T\-1,T\-2,\\cdots,0do

9:if

t∈𝒢t\\in\\mathcal\{G\}then

10:for

i←1,2,⋯,Ni\\leftarrow 1,2,\\cdots,Ndo

11:

xt\(i\)​∼iid​qt\|t\+1​\(xt\|xt\+1\(i\)\)x\_\{t\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}^\{\(i\)\}\)
12:

x0\(i,j\)​∼iid​p0\|t​\(x0\|xt\(i\)\)\(j=1,2,⋯,J\)x\_\{0\}^\{\(i,j\)\}\\overset\{\\text\{iid\}\}\{\\sim\}p\_\{0\|t\}\(x\_\{0\}\|x\_\{t\}^\{\(i\)\}\)\\quad\(j=1,2,\\cdots,J\)
13:

V^t\(i\)←1J​∑j=1JR⁡\(x0\(i,j\)\)\\hat\{V\}\_\{t\}^\{\(i\)\}\\leftarrow\\frac\{1\}\{J\}\\sum\_\{j=1\}^\{J\}R\(x\_\{0\}^\{\(i,j\)\}\)
14:

wt\(i\)←pt\|t\+1​\(xt\(i\)\|xt\+1\(i\)\)qt\|t\+1​\(xt\(i\)\|xt\+1\(i\)\)×V^t\(i\)V^⌈t\+1⌉\(i\)×w⌈t\+1⌉\(i\)w\_\{t\}^\{\(i\)\}\\leftarrow\\frac\{p\_\{t\|t\+1\}\(x\_\{t\}^\{\(i\)\}\|x\_\{t\+1\}^\{\(i\)\}\)\}\{q\_\{t\|t\+1\}\(x\_\{t\}^\{\(i\)\}\|x\_\{t\+1\}^\{\(i\)\}\)\}\\times\\frac\{\\hat\{V\}\_\{t\}^\{\(i\)\}\}\{\\hat\{V\}\_\{\\lceil t\+1\\rceil\}^\{\(i\)\}\}\\times w\_\{\\lceil t\+1\\rceil\}^\{\(i\)\}
15:endfor

16:

\{xt\(i\)\}i=1N,\{wt\(i\)\}i=1N←Resample​\(\{xt\(i\)\}i=1N,\{wt\(i\)\}i=1N\)\\\{x\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\\leftarrow\\text\{Resample\}\(\\\{x\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\},\\\{w\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\)
17:else

18:for

i←1,2,⋯,Ni\\leftarrow 1,2,\\cdots,Ndo

19:

xt\(i\)​∼iid​pt\|t\+1​\(xt\|xt\+1\(i\)\)x\_\{t\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}^\{\(i\)\}\)
20:endfor

21:endif

22:endfor

###### Theorem B\.4\.

SayN,J→∞N,J\\rightarrow\\infty\. At every guided timet∈𝒢∪\{T\}t\\in\\mathcal\{G\}\\cup\\\{T\\\}, the particles in Algorithm[5](https://arxiv.org/html/2610.20669#alg5)are distributed according to the ideal guided target distribution, i\.e\.,xt\(i\)​∼iid​πt∗\(i=1,2,⋯,N\)x\_\{t\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}\\pi\_\{t\}^\{\*\}\\quad\(i=1,2,\\cdots,N\)\.

###### Proof\.

SinceJ→∞J\\rightarrow\\infty, the Monte Carlo value estimateV^t\(i\)\\hat\{V\}\_\{t\}^\{\(i\)\}converges almost surely to the true value functionVt​\(xt\(i\)\)V\_\{t\}\(x\_\{t\}^\{\(i\)\}\)by the strong law of large numbers\.

Base Case: Att=Tt=T, the particles are sampled fromqTq\_\{T\}and weighted bywT∝pT​\(xT\)​VT​\(xT\)/qT​\(xT\)w\_\{T\}\\propto p\_\{T\}\(x\_\{T\}\)V\_\{T\}\(x\_\{T\}\)/q\_\{T\}\(x\_\{T\}\)\. By Lemma[B\.3](https://arxiv.org/html/2610.20669#A2.Thmtheorem3), after resampling at Line[7](https://arxiv.org/html/2610.20669#alg5.l7), we havexT\(i\)​∼iid​πT∗​\(xT\)∝pT​\(xT\)​VT​\(xT\)x\_\{T\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}\\pi\_\{T\}^\{\*\}\(x\_\{T\}\)\\propto p\_\{T\}\(x\_\{T\}\)V\_\{T\}\(x\_\{T\}\)\.

Inductive Step: Assume thatx⌈t\+1⌉\(i\)​∼iid​π⌈t\+1⌉∗​\(x⌈t\+1⌉\)x\_\{\\lceil t\+1\\rceil\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}\\pi\_\{\\lceil t\+1\\rceil\}^\{\*\}\(x\_\{\\lceil t\+1\\rceil\}\)\.

Then, just before the line[16](https://arxiv.org/html/2610.20669#alg5.l16),

\(xt\(i\),xt\+1\(i\),x⌈t\+1⌉\(i\)\)​∼iid​qt\|t\+1​\(xt\|xt\+1\)⋅pt\+1\|⌈t\+1⌉​\(xt\+1\|x⌈t\+1⌉\)⋅p⌈t\+1⌉​\(x⌈t\+1⌉\)​V⌈t\+1⌉​\(x⌈t\+1⌉\)Z\.\(x\_\{t\}^\{\(i\)\},x\_\{t\+1\}^\{\(i\)\},x\_\{\\lceil t\+1\\rceil\}^\{\(i\)\}\)\\overset\{\\text\{iid\}\}\{\\sim\}q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\\cdot p\_\{t\+1\|\\lceil t\+1\\rceil\}\(x\_\{t\+1\}\|x\_\{\\lceil t\+1\\rceil\}\)\\cdot\\frac\{p\_\{\\lceil t\+1\\rceil\}\(x\_\{\\lceil t\+1\\rceil\}\)V\_\{\\lceil t\+1\\rceil\}\(x\_\{\\lceil t\+1\\rceil\}\)\}\{Z\}\.\(41\)Just after the line[16](https://arxiv.org/html/2610.20669#alg5.l16),

\(xt\(i\),xt\+1\(i\),x⌈t\+1⌉\(i\)\)\\displaystyle\(x\_\{t\}^\{\(i\)\},x\_\{t\+1\}^\{\(i\)\},x\_\{\\lceil t\+1\\rceil\}^\{\(i\)\}\)∼iid​qt\|t\+1​\(xt\|xt\+1\)​pt\+1\|⌈t\+1⌉​\(xt\+1\|x⌈t\+1⌉\)​p⌈t\+1⌉​\(x⌈t\+1⌉\)​V⌈t\+1⌉​\(x⌈t\+1⌉\)Z\\displaystyle\\overset\{\\text\{iid\}\}\{\\sim\}q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)p\_\{t\+1\|\\lceil t\+1\\rceil\}\(x\_\{t\+1\}\|x\_\{\\lceil t\+1\\rceil\}\)\\frac\{p\_\{\\lceil t\+1\\rceil\}\(x\_\{\\lceil t\+1\\rceil\}\)V\_\{\\lceil t\+1\\rceil\}\(x\_\{\\lceil t\+1\\rceil\}\)\}\{Z\}×pt\|t\+1​\(xt\|xt\+1\)qt\|t\+1​\(xt\|xt\+1\)​Vt​\(xt\)V⌈t\+1⌉​\(x⌈t\+1⌉\)\\displaystyle\\times\\frac\{p\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\}\{q\_\{t\|t\+1\}\(x\_\{t\}\|x\_\{t\+1\}\)\}\\frac\{V\_\{t\}\(x\_\{t\}\)\}\{V\_\{\\lceil t\+1\\rceil\}\(x\_\{\\lceil t\+1\\rceil\}\)\}\(42\)=1Z​pt,t\+1,⌈t\+1⌉​\(xt,xt\+1,x⌈t\+1⌉\)​Vt​\(xt\)\.\\displaystyle=\\frac\{1\}\{Z\}p\_\{t,t\+1,\\lceil t\+1\\rceil\}\(x\_\{t\},x\_\{t\+1\},x\_\{\\lceil t\+1\\rceil\}\)V\_\{t\}\(x\_\{t\}\)\.\(43\)By Lemma[B\.3](https://arxiv.org/html/2610.20669#A2.Thmtheorem3), we may conclude that

xt\(i\)​∼iid​πt∗​\(xt\)\.x\_\{t\}^\{\(i\)\}\\overset\{\\text\{iid\}\}\{\\sim\}\\pi\_\{t\}^\{\*\}\(x\_\{t\}\)\.\(44\)
The proof follows by induction for allt∈𝒢t\\in\\mathcal\{G\}\. ∎

### B\.3Probability path distance

We first record a simple inequality\.

###### Lemma B\.5\.

Random variableX∈ℝX\\in\\mathbb\{R\}satisfies0≤X≤M0\\leq X\\leq Mandμ=𝔼⁡\[X\]\\mu=\\mathbb\{E\}\[X\]\. Then,

𝔼⁡\[\|X−μ\|\]≤2​μ​\(1−μM\)\.\\mathbb\{E\}\[\|X\-\\mu\|\]\\leq 2\\mu\\left\(1\-\\frac\{\\mu\}\{M\}\\right\)\.\(45\)

###### Proof\.

Since

0=𝔼⁡\[X−μ\]=𝔼⁡\[\(X−μ\)​𝟏​\(X\>μ\)\+\(X−μ\)​𝟏​\(X≤μ\)\],\\displaystyle 0=\\mathbb\{E\}\[X\-\\mu\]=\\mathbb\{E\}\[\(X\-\\mu\)\\mathbf\{1\}\(X\>\\mu\)\+\(X\-\\mu\)\\mathbf\{1\}\(X\\leq\\mu\)\],we get the following identity:

𝔼⁡\[\(X−μ\)​𝟏​\(X\>μ\)\]=𝔼⁡\[\(μ−X\)​𝟏​\(X≤μ\)\]\.\\mathbb\{E\}\[\(X\-\\mu\)\\mathbf\{1\}\(X\>\\mu\)\]=\\mathbb\{E\}\[\(\\mu\-X\)\\mathbf\{1\}\(X\\leq\\mu\)\]\.\(46\)
Thus,

𝔼⁡\[\|X−μ\|\]\\displaystyle\\mathbb\{E\}\[\|X\-\\mu\|\]=𝔼⁡\[\(X−μ\)​𝟏​\(X\>μ\)\]\\displaystyle=\\mathbb\{E\}\[\(X\-\\mu\)\\mathbf\{1\}\(X\>\\mu\)\]\+𝔼⁡\[\(μ−X\)​𝟏​\(X≤μ\)\]\\displaystyle\+\\mathbb\{E\}\[\(\\mu\-X\)\\mathbf\{1\}\(X\\leq\\mu\)\]=2​𝔼​\[\(μ−X\)​𝟏​\(X≤μ\)\]\\displaystyle=2\\mathbb\{E\}\[\(\\mu\-X\)\\mathbf\{1\}\(X\\leq\\mu\)\]≤2​𝔼​\[μ⁡\(1−X/M\)\]=2​μ​\(1−μ/M\)\.\\displaystyle\\leq 2\\mathbb\{E\}\[\\mu\(1\-X/M\)\]=2\\mu\(1\-\\mu/M\)\.Applying the expectation to both sides:

𝔼⁡\[\|X−μ\|\]\\displaystyle\\mathbb\{E\}\[\|X\-\\mu\|\]≤2​𝔼​\[μ⁡\(1−XM\)\]\\displaystyle\\leq 2\\mathbb\{E\}\\left\[\\mu\\left\(1\-\\frac\{X\}\{M\}\\right\)\\right\]=2​μ​\(1−𝔼⁡\[X\]M\)=2​μ​\(1−μM\)\.\\displaystyle=2\\mu\\left\(1\-\\frac\{\\mathbb\{E\}\[X\]\}\{M\}\\right\)=2\\mu\\left\(1\-\\frac\{\\mu\}\{M\}\\right\)\.
![[Uncaptioned image]](https://arxiv.org/html/2610.20669v1/figures/Ymu.png)

Figure 5:Illustration of\(μ−X\)​𝟏​\(X≤μ\)≤μ⁡\(1−XM\)\(\\mu\-X\)\\mathbf\{1\}\(X\\leq\\mu\)\\leq\\mu\(1\-\\frac\{X\}\{M\}\)
∎

###### Proof of Theorem[4\.3](https://arxiv.org/html/2610.20669#S4.Thmtheorem3)\.

Ift∈𝒢t\\in\\mathcal\{G\}, thenπt∗=πt𝒢\\pi\_\{t\}^\{\*\}=\\pi\_\{t\}^\{\\mathcal\{G\}\}, so

𝐓𝐃⁡\(𝒢\)=1T​∑t∈𝒯−𝒢‖πt∗−πt𝒢‖1\.\\mathbf\{TD\}\(\\mathcal\{G\}\)=\\frac\{1\}\{T\}\\sum\_\{t\\in\\mathcal\{T\}\-\\mathcal\{G\}\}\\\|\\pi\_\{t\}^\{\*\}\-\\pi\_\{t\}^\{\\mathcal\{G\}\}\\\|\_\{1\}\.\(47\)
By Lemma[B\.1](https://arxiv.org/html/2610.20669#A2.Thmtheorem1),

V⌈t⌉\(x⌈t⌉\)=𝔼xt∼pt\|⌈t⌉\(⋅\|x⌈t⌉\)\[Vt\(xt\)\]\.V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)=\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\|\\lceil t\\rceil\}\(\\cdot\|x\_\{\\lceil t\\rceil\}\)\}\[V\_\{t\}\(x\_\{t\}\)\]\.\(48\)
Thus,

‖πt∗−πt𝒢‖1\\displaystyle\\\|\\pi\_\{t\}^\{\*\}\-\\pi\_\{t\}^\{\\mathcal\{G\}\}\\\|\_\{1\}=∑xt∈𝒳\|πt∗​\(xt\)−∑x⌈t⌉∈𝒳π⌈t⌉∗​\(x⌈t⌉\)​pt\|⌈t⌉​\(xt\|x⌈t⌉\)\|\\displaystyle=\\sum\_\{x\_\{t\}\\in\\mathcal\{X\}\}\|\\pi\_\{t\}^\{\*\}\(x\_\{t\}\)\-\\sum\_\{x\_\{\\lceil t\\rceil\}\\in\\mathcal\{X\}\}\\pi\_\{\\lceil t\\rceil\}^\{\*\}\(x\_\{\\lceil t\\rceil\}\)p\_\{t\|\\lceil t\\rceil\}\(x\_\{t\}\|x\_\{\\lceil t\\rceil\}\)\|\(49\)=1Z​∑xt∈𝒳\|∑x⌈t⌉∈𝒳p⌈t⌉​\(x⌈t⌉\)​pt\|⌈t⌉​\(xt\|x⌈t⌉\)​\(Vt​\(xt\)−V⌈t⌉​\(x⌈t⌉\)\)\|\\displaystyle=\\frac\{1\}\{Z\}\\sum\_\{x\_\{t\}\\in\\mathcal\{X\}\}\|\\sum\_\{x\_\{\\lceil t\\rceil\}\\in\\mathcal\{X\}\}p\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)p\_\{t\|\\lceil t\\rceil\}\(x\_\{t\}\|x\_\{\\lceil t\\rceil\}\)\(V\_\{t\}\(x\_\{t\}\)\-V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\)\|\(50\)≤1Z𝔼x⌈t⌉∼p⌈t⌉,xt∼pt\|⌈t⌉\(⋅\|x⌈t⌉\)\|Vt\(xt\)−V⌈t⌉\(x⌈t⌉\)\|\.\\displaystyle\\leq\\frac\{1\}\{Z\}\\mathbb\{E\}\_\{x\_\{\\lceil t\\rceil\}\\sim p\_\{\\lceil t\\rceil\},x\_\{t\}\\sim p\_\{t\|\\lceil t\\rceil\}\(\\cdot\|x\_\{\\lceil t\\rceil\}\)\}\|V\_\{t\}\(x\_\{t\}\)\-V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\|\.\(51\)
Applying Lemma[B\.5](https://arxiv.org/html/2610.20669#A2.Thmtheorem5)withX=Vt​\(xt\)X=V\_\{t\}\(x\_\{t\}\)andμ=V⌈t⌉​\(x⌈t⌉\)\\mu=V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)gives

𝔼xt∼pt\|⌈t⌉\(⋅\|x⌈t⌉\)\[\|Vt\(xt\)−V⌈t⌉\(x⌈t⌉\)\|\]≤2V⌈t⌉\(x⌈t⌉\)\(1−V⌈t⌉​\(x⌈t⌉\)Rmax\)\.\\mathbb\{E\}\_\{x\_\{t\}\\sim p\_\{t\|\\lceil t\\rceil\}\(\\cdot\|x\_\{\\lceil t\\rceil\}\)\}\[\|V\_\{t\}\(x\_\{t\}\)\-V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\|\]\\leq 2V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\(1\-\\frac\{V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\}\{R\_\{\\text\{max\}\}\}\)\.\(52\)
Therefore,

‖πt∗−πt𝒢‖1\\displaystyle\\\|\\pi\_\{t\}^\{\*\}\-\\pi\_\{t\}^\{\\mathcal\{G\}\}\\\|\_\{1\}≤2Z​𝔼x⌈t⌉∼p⌈t⌉​\[V⌈t⌉​\(x⌈t⌉\)​\(1−V⌈t⌉​\(x⌈t⌉\)Rmax\)\]\\displaystyle\\leq\\frac\{2\}\{Z\}\\mathbb\{E\}\_\{x\_\{\\lceil t\\rceil\}\\sim p\_\{\\lceil t\\rceil\}\}\[V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\(1\-\\frac\{V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\}\{R\_\{\\text\{max\}\}\}\)\]\(53\)=2​\(1−1Rmax​𝔼x⌈t⌉∼π⌈t⌉∗​\[V⌈t⌉​\(x⌈t⌉\)\]\)\.\\displaystyle=2\(1\-\\frac\{1\}\{R\_\{\\text\{max\}\}\}\\mathbb\{E\}\_\{x\_\{\\lceil t\\rceil\}\\sim\\pi^\{\*\}\_\{\\lceil t\\rceil\}\}\[V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\]\)\.\(54\)
∎

### B\.4DP algorithm

We aim to find𝒢\\mathcal\{G\}that maximizes

𝐒^​\(𝒢\):=∑t∈𝒯∖𝒢V^⌈t⌉\\hat\{\\mathbf\{S\}\}\(\\mathcal\{G\}\):=\\sum\_\{t\\in\\mathcal\{T\}\\setminus\\mathcal\{G\}\}\\hat\{V\}\_\{\\lceil t\\rceil\}\(55\)whereT′=\|𝒢\|T^\{\\prime\}=\|\\mathcal\{G\}\|is a given constant\.

Define

d​p​\[s\]​\[i\]:=max⁡∑t∈𝒯∖𝒢′;t\>s𝒢′⁡V^⌈t⌉dp\[s\]\[i\]:=\\max\_\{\\mathcal\{G\}^\{\\prime\}\}\\sum\_\{t\\in\\mathcal\{T\}\\setminus\\mathcal\{G\}^\{\\prime\};t\>s\}\\hat\{V\}\_\{\\lceil t\\rceil\}\(56\)where

𝒢′=\{s=ti<⋯<t1<T\}\.\\mathcal\{G\}^\{\\prime\}=\\\{s=t\_\{i\}<\\cdots<t\_\{1\}<T\\\}\.\(57\)
We can observe the recurrence relation

d​p​\[s\]​\[1\]\\displaystyle dp\[s\]\[1\]=\(T−s−1\)​V^T\\displaystyle=\(T\-s\-1\)\\hat\{V\}\_\{T\}\(58\)d​p​\[s\]​\[i\]\\displaystyle dp\[s\]\[i\]=maxs<s′≤T−i\+1⁡\[d​p​\[s′\]​\[i−1\]\+\(s′−s−1\)​V^s′\]\.\(i≥2\)\\displaystyle=\\max\_\{s<s^\{\\prime\}\\leq T\-i\+1\}\[dp\[s^\{\\prime\}\]\[i\-1\]\+\(s^\{\\prime\}\-s\-1\)\\hat\{V\}\_\{s^\{\\prime\}\}\]\.\\quad\(i\\geq 2\)\(59\)
Thus, Algorithm[6](https://arxiv.org/html/2610.20669#alg6)can compute the optimal𝒢\\mathcal\{G\}\.

Algorithm 6DP algorithm1:Initialize:

d​p​\[s\]​\[i\]←0dp\[s\]\[i\]\\leftarrow 0for all

s,is,i
2:for

s←T′−1,T′,⋯,T−1s\\leftarrow T^\{\\prime\}\-1,T^\{\\prime\},\\cdots,T\-1do

3:

d​p​\[s\]​\[1\]←\(T−s−1\)​V^Tdp\[s\]\[1\]\\leftarrow\(T\-s\-1\)\\hat\{V\}\_\{T\}
4:endfor

5:for

i←2,3,⋯,T′i\\leftarrow 2,3,\\cdots,T^\{\\prime\}do

6:for

s←T′−i,⋯,T−is\\leftarrow T^\{\\prime\}\-i,\\cdots,T\-ido

7:for

s′←s\+1s^\{\\prime\}\\leftarrow s\+1to

T−i\+1T\-i\+1do

8:

c←d​p​\[s′\]​\[i−1\]\+\(s′−s−1\)​V^s′c\\leftarrow dp\[s^\{\\prime\}\]\[i\-1\]\+\(s^\{\\prime\}\-s\-1\)\\hat\{V\}\_\{s^\{\\prime\}\}
9:if

c\>d​p​\[s\]​\[i\]c\>dp\[s\]\[i\]then

10:

d​p​\[s\]​\[i\]←cdp\[s\]\[i\]\\leftarrow c
11:

p​a​r​e​n​t​\[s\]​\[i\]←s′parent\[s\]\[i\]\\leftarrow s^\{\\prime\}
12:endif

13:endfor

14:endfor

15:endfor

16:

tT′←argmaxs​\(d​p​\[s\]​\[T′\]\+s​V^s\)t\_\{T^\{\\prime\}\}\\leftarrow\\text\{argmax\}\_\{s\}\(dp\[s\]\[T^\{\\prime\}\]\+s\\hat\{V\}\_\{s\}\)
17:for

j←T′−1,T′−2,⋯,1j\\leftarrow T^\{\\prime\}\-1,T^\{\\prime\}\-2,\\cdots,1do

18:

tj←p​a​r​e​n​t​\[tj\+1\]​\[j\+1\]t\_\{j\}\\leftarrow parent\[t\_\{j\+1\}\]\[j\+1\]
19:endfor

20:return

𝒢=\{t1,t2,⋯,tT′\}\\mathcal\{G\}=\\\{t\_\{1\},t\_\{2\},\\cdots,t\_\{T^\{\\prime\}\}\\\}

## Appendix CExperimental Details

### C\.1Experimental Setup

The Protein Fitness experiment was conducted on a single PRO 6000 GPU, while all other experiments were conducted on a single NVIDIA A6000 GPU\. Unlike SMC\-based evaluation protocols that generateNNcandidates but retain only the highest\-reward sample, we retain all generated samples to evaluate both alignment and sample diversity\.

For Binary MNIST, we use the masked diffusion model \(MDM\) from\[[29](https://arxiv.org/html/2610.20669#bib.bib3)\]and follow the implementation of\[[28](https://arxiv.org/html/2610.20669#bib.bib1)\]\. For Toxic Text Generation, we use the pretrained MDLM from\[[29](https://arxiv.org/html/2610.20669#bib.bib3)\]together with the reward model from\[[21](https://arxiv.org/html/2610.20669#bib.bib36)\], following the implementation of\[[28](https://arxiv.org/html/2610.20669#bib.bib1)\]\. For CIFAR\-10, we use the MDM and implementation described in the corresponding experimental setup\. For DNA generation, we follow the experimental setup of\[[28](https://arxiv.org/html/2610.20669#bib.bib1)\]\. For Protein Fitness, we use the pretrained MDLM from\[[29](https://arxiv.org/html/2610.20669#bib.bib3)\]together with a fitness oracle as the reward model\. The resampling configurations for each experiment are summarized in Table[5](https://arxiv.org/html/2610.20669#A3.T5)\.

Table 5:Resampling configurations for each experiment\. Adaptive resampling is triggered when the normalized Effective Sample Size \(ESS/N/N\) falls below the specified threshold\. Partial resampling denotes the number of particles replaced at each resampling step\.ExperimentESS thresholdPartial resamplingSystematic resamplingBinary MNIST0\.95⌊N/2⌋\\lfloor N/2\\rfloorYesToxic Text0\.98NNYesCIFAR\-100\.95⌊N/2⌋\\lfloor N/2\\rfloorYesDNA Design0\.98⌊N/2⌋\\lfloor N/2\\rfloorYesProtein Fitness0\.98⌊N/2⌋\\lfloor N/2\\rfloorYesContinuous MNIST0\.98⌊N/2⌋\\lfloor N/2\\rfloorYes
### C\.2Details on main experiment

#### C\.2\.1Details on Binary MNIST experiment

The warmup phase took 5\.78s for SMC\-base, and 24\.7s for SMC\-grad\. See Figure[6](https://arxiv.org/html/2610.20669#A3.F6),[7\(a\)](https://arxiv.org/html/2610.20669#A3.F7.sf1),[7\(b\)](https://arxiv.org/html/2610.20669#A3.F7.sf2)and Table[7](https://arxiv.org/html/2610.20669#A3.T7),[7](https://arxiv.org/html/2610.20669#A3.T7)\.

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/mnist_mdm_base_time.png)

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/mnist_mdm_base_unique_valid.png)

![Refer to caption](https://arxiv.org/html/2610.20669v1/figures/mnist_mdm_base_utd.png)

Figure 6:Binary MNIST \(SMC\-base\)\. Execution time, unique valid samples, and UTD as the guidance budgetT′T^\{\\prime\}varies\.Table 6:Binary MNIST \(SMC\-base\): execution time, unique valid, and UTD\.T′T^\{\\prime\}PolicyTime \[s\]↓\\downarrowUnique Valid↑\\uparrowUTD↓\\downarrow100Full1\.61±0\.011\.61\\pm 0\.0118\.66±1\.78\\mathbf\{18\.66\\pm 1\.78\}–6Interval 11\.46±0\.021\.46\\pm 0\.029\.36±3\.539\.36\\pm 3\.531\.0261\.026Interval 21\.46±0\.021\.46\\pm 0\.0211\.32±6\.1911\.32\\pm 6\.190\.4690\.469Interval 31\.46±0\.021\.46\\pm 0\.022\.84±2\.612\.84\\pm 2\.610\.9080\.908Interval 41\.46±0\.021\.46\\pm 0\.021\.74±1\.051\.74\\pm 1\.051\.3521\.352Interval 51\.46±0\.021\.46\\pm 0\.021\.68±1\.041\.68\\pm 1\.041\.7631\.763Top\-VV1\.46±0\.021\.46\\pm 0\.029\.24±5\.849\.24\\pm 5\.840\.5740\.574Top\-Δ​V\\Delta V1\.46±0\.021\.46\\pm 0\.0213\.34±4\.1213\.34\\pm 4\.120\.2170\.217Uniform1\.47±0\.091\.47\\pm 0\.0910\.20±3\.1210\.20\\pm 3\.120\.4160\.416VISTA1\.47±0\.021\.47\\pm 0\.0215\.58±2\.48\\mathbf\{15\.58\\pm 2\.48\}0\.116\\mathbf\{0\.116\}12Interval 11\.50±0\.031\.50\\pm 0\.0315\.76±3\.5415\.76\\pm 3\.540\.2270\.227Interval 21\.51±0\.111\.51\\pm 0\.1112\.30±6\.2112\.30\\pm 6\.210\.4280\.428Interval 31\.51±0\.101\.51\\pm 0\.103\.66±3\.473\.66\\pm 3\.470\.8610\.861Interval 41\.50±0\.041\.50\\pm 0\.041\.78±1\.091\.78\\pm 1\.091\.2841\.284Interval 51\.49±0\.021\.49\\pm 0\.021\.72±1\.051\.72\\pm 1\.051\.6511\.651Top\-VV1\.49±0\.021\.49\\pm 0\.029\.82±5\.949\.82\\pm 5\.940\.5570\.557Top\-Δ​V\\Delta V1\.51±0\.071\.51\\pm 0\.0716\.70±2\.9016\.70\\pm 2\.900\.1070\.107Uniform1\.52±0\.121\.52\\pm 0\.1215\.22±3\.9715\.22\\pm 3\.970\.2490\.249VISTA1\.50±0\.021\.50\\pm 0\.0217\.22±3\.45\\mathbf\{17\.22\\pm 3\.45\}3\.91×𝟏𝟎−𝟐\\mathbf\{3\.91\\times 10^\{\-2\}\}18Interval 11\.54±0\.021\.54\\pm 0\.0217\.26±3\.3317\.26\\pm 3\.330\.1310\.131Interval 21\.52±0\.021\.52\\pm 0\.0213\.50±5\.9513\.50\\pm 5\.950\.3870\.387Interval 31\.54±0\.111\.54\\pm 0\.114\.34±4\.034\.34\\pm 4\.030\.8100\.810Interval 41\.54±0\.081\.54\\pm 0\.081\.72±1\.091\.72\\pm 1\.091\.1961\.196Interval 51\.52±0\.021\.52\\pm 0\.021\.76±1\.081\.76\\pm 1\.081\.5381\.538Top\-VV1\.52±0\.021\.52\\pm 0\.0212\.14±6\.1412\.14\\pm 6\.140\.4510\.451Top\-Δ​V\\Delta V1\.53±0\.031\.53\\pm 0\.0316\.48±4\.0816\.48\\pm 4\.080\.1280\.128Uniform1\.56±0\.161\.56\\pm 0\.1615\.80±3\.6015\.80\\pm 3\.600\.2180\.218VISTA1\.53±0\.041\.53\\pm 0\.0417\.92±2\.47\\mathbf\{17\.92\\pm 2\.47\}1\.79×𝟏𝟎−𝟐\\mathbf\{1\.79\\times 10^\{\-2\}\}24Interval 11\.53±0\.021\.53\\pm 0\.0217\.82±2\.6017\.82\\pm 2\.602\.93×10−22\.93\\times 10^\{\-2\}Interval 21\.53±0\.031\.53\\pm 0\.0313\.28±5\.8213\.28\\pm 5\.820\.3730\.373Interval 31\.52±0\.021\.52\\pm 0\.025\.64±4\.355\.64\\pm 4\.350\.7430\.743Interval 41\.52±0\.021\.52\\pm 0\.021\.82±1\.161\.82\\pm 1\.161\.1031\.103Interval 51\.52±0\.031\.52\\pm 0\.031\.72±1\.051\.72\\pm 1\.051\.4261\.426Top\-VV1\.53±0\.101\.53\\pm 0\.1013\.96±5\.6113\.96\\pm 5\.610\.4050\.405Top\-Δ​V\\Delta V1\.53±0\.021\.53\\pm 0\.0217\.42±3\.0017\.42\\pm 3\.000\.1060\.106Uniform1\.55±0\.151\.55\\pm 0\.1516\.04±3\.6916\.04\\pm 3\.690\.1850\.185VISTA1\.55±0\.101\.55\\pm 0\.1017\.90±3\.05\\mathbf\{17\.90\\pm 3\.05\}1\.10×𝟏𝟎−𝟐\\mathbf\{1\.10\\times 10^\{\-2\}\}

Table 7:Binary MNIST \(SMC\-grad\): execution time, unique valid, and UTD\.T′T^\{\\prime\}PolicyTime \[s\]↓\\downarrowUnique Valid↑\\uparrowUTD↓\\downarrow100Full7\.13±0\.287\.13\\pm 0\.2818\.78±1\.73\\mathbf\{18\.78\\pm 1\.73\}–6Interval 12\.12±0\.482\.12\\pm 0\.4810\.14±3\.9310\.14\\pm 3\.930\.7950\.795Interval 22\.04±0\.362\.04\\pm 0\.3611\.68±6\.1111\.68\\pm 6\.110\.4610\.461Interval 32\.07±0\.382\.07\\pm 0\.382\.96±2\.572\.96\\pm 2\.570\.8840\.884Interval 42\.00±0\.332\.00\\pm 0\.331\.74±1\.031\.74\\pm 1\.031\.3141\.314Interval 51\.96±0\.301\.96\\pm 0\.301\.70±1\.071\.70\\pm 1\.071\.7631\.763Top\-VV2\.09±0\.442\.09\\pm 0\.445\.30±4\.125\.30\\pm 4\.120\.7140\.714Top\-Δ​V\\Delta V2\.06±0\.442\.06\\pm 0\.4414\.14±3\.7114\.14\\pm 3\.710\.1140\.114Uniform2\.03±0\.422\.03\\pm 0\.4210\.06±3\.7510\.06\\pm 3\.750\.3550\.355VISTA2\.10±0\.472\.10\\pm 0\.4715\.72±2\.85\\mathbf\{15\.72\\pm 2\.85\}6\.69×𝟏𝟎−𝟐\\mathbf\{6\.69\\times 10^\{\-2\}\}12Interval 12\.78±0\.492\.78\\pm 0\.4916\.22±2\.9716\.22\\pm 2\.975\.70×10−25\.70\\times 10^\{\-2\}Interval 22\.78±0\.482\.78\\pm 0\.4813\.10±6\.3613\.10\\pm 6\.360\.4180\.418Interval 32\.78±0\.482\.78\\pm 0\.483\.34±2\.943\.34\\pm 2\.940\.8270\.827Interval 42\.79±0\.472\.79\\pm 0\.471\.78±1\.071\.78\\pm 1\.071\.2391\.239Interval 52\.74±0\.482\.74\\pm 0\.481\.70±1\.021\.70\\pm 1\.021\.6511\.651Top\-VV2\.71±0\.522\.71\\pm 0\.525\.18±4\.415\.18\\pm 4\.410\.7140\.714Top\-Δ​V\\Delta V2\.70±0\.512\.70\\pm 0\.5116\.54±2\.8416\.54\\pm 2\.843\.19×10−23\.19\\times 10^\{\-2\}Uniform2\.72±0\.532\.72\\pm 0\.5315\.40±4\.4515\.40\\pm 4\.450\.1750\.175VISTA2\.68±0\.522\.68\\pm 0\.5216\.82±3\.33\\mathbf\{16\.82\\pm 3\.33\}9\.72×𝟏𝟎−𝟑\\mathbf\{9\.72\\times 10^\{\-3\}\}18Interval 13\.37±0\.153\.37\\pm 0\.1517\.68±2\.7917\.68\\pm 2\.791\.26×10−21\.26\\times 10^\{\-2\}Interval 23\.39±0\.133\.39\\pm 0\.1314\.04±5\.9914\.04\\pm 5\.990\.3790\.379Interval 33\.36±0\.213\.36\\pm 0\.214\.28±3\.614\.28\\pm 3\.610\.7700\.770Interval 43\.35±0\.233\.35\\pm 0\.231\.72±1\.031\.72\\pm 1\.031\.1641\.164Interval 53\.33±0\.273\.33\\pm 0\.271\.70±1\.021\.70\\pm 1\.021\.5381\.538Top\-VV3\.37±0\.263\.37\\pm 0\.265\.48±4\.555\.48\\pm 4\.550\.7140\.714Top\-Δ​V\\Delta V3\.31±0\.313\.31\\pm 0\.3118\.44±1\.67\\mathbf\{18\.44\\pm 1\.67\}6\.34×10−36\.34\\times 10^\{\-3\}Uniform3\.35±0\.423\.35\\pm 0\.4217\.44±2\.6517\.44\\pm 2\.650\.1260\.126VISTA3\.35±0\.243\.35\\pm 0\.2418\.32±1\.9618\.32\\pm 1\.962\.25×𝟏𝟎−𝟑\\mathbf\{2\.25\\times 10^\{\-3\}\}24Interval 13\.23±0\.313\.23\\pm 0\.3118\.44±2\.0418\.44\\pm 2\.041\.03×10−21\.03\\times 10^\{\-2\}Interval 23\.28±0\.273\.28\\pm 0\.2715\.02±5\.0615\.02\\pm 5\.060\.3580\.358Interval 33\.27±0\.283\.27\\pm 0\.285\.56±4\.475\.56\\pm 4\.470\.7140\.714Interval 43\.25±0\.303\.25\\pm 0\.301\.82±1\.351\.82\\pm 1\.351\.0701\.070Interval 53\.19±0\.303\.19\\pm 0\.301\.70±1\.021\.70\\pm 1\.021\.4261\.426Top\-VV3\.30±0\.303\.30\\pm 0\.305\.50±4\.225\.50\\pm 4\.220\.7140\.714Top\-Δ​V\\Delta V3\.24±0\.333\.24\\pm 0\.3318\.42±1\.6918\.42\\pm 1\.697\.43×10−37\.43\\times 10^\{\-3\}Uniform3\.24±0\.403\.24\\pm 0\.4017\.32±2\.3917\.32\\pm 2\.390\.1090\.109VISTA3\.23±0\.303\.23\\pm 0\.3018\.54±2\.15\\mathbf\{18\.54\\pm 2\.15\}1\.43×𝟏𝟎−𝟑\\mathbf\{1\.43\\times 10^\{\-3\}\}

\(a\)SMC\-base\(b\)SMC\-grad
Figure 7:VISTAschedules𝒢\\mathcal\{G\}for varyingT′T^\{\\prime\}on Binary MNIST
#### C\.2\.2Details on Toxic Text Generation

See Table[8](https://arxiv.org/html/2610.20669#A3.T8)and Figure[8](https://arxiv.org/html/2610.20669#A3.F8)\.

Table 8:Toxic text generation \(SMC\-grad\)\.T′T^\{\\prime\}PolicyTime \[s\]↓\\downarrowUnique Toxic Ratio↑\\uparrowS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)↑\\uparrow100Full204\.23±0\.65204\.23\\pm 0\.651\.00±0\.02\\mathbf\{1\.00\\pm 0\.02\}𝟎\\mathbf\{0\}5Interval 114\.10±0\.2014\.10\\pm 0\.200\.74±0\.340\.74\\pm 0\.3468\.9768\.97Interval 215\.08±0\.3215\.08\\pm 0\.320\.73±0\.280\.73\\pm 0\.2855\.6155\.61Interval 315\.01±0\.2515\.01\\pm 0\.250\.72±0\.350\.72\\pm 0\.3534\.8534\.85Interval 415\.15±0\.3515\.15\\pm 0\.350\.69±0\.420\.69\\pm 0\.4223\.6023\.60Interval 514\.34±0\.2514\.34\\pm 0\.250\.05±0\.170\.05\\pm 0\.175\.165\.16Top\-VV16\.01±0\.2716\.01\\pm 0\.270\.16±0\.300\.16\\pm 0\.3010\.2010\.20Top\-Δ​V\\Delta V15\.89±0\.2715\.89\\pm 0\.270\.80±0\.330\.80\\pm 0\.3373\.7373\.73Uniform16\.97±0\.2716\.97\\pm 0\.270\.77±0\.340\.77\\pm 0\.3469\.2669\.26VISTA17\.58±0\.2517\.58\\pm 0\.250\.81±0\.35\\mathbf\{0\.81\\pm 0\.35\}78\.73\\mathbf\{78\.73\}10Interval 124\.09±0\.2424\.09\\pm 0\.240\.78±0\.300\.78\\pm 0\.3064\.8764\.87Interval 225\.07±0\.2825\.07\\pm 0\.280\.78±0\.310\.78\\pm 0\.3153\.6153\.61Interval 325\.03±0\.2425\.03\\pm 0\.240\.74±0\.280\.74\\pm 0\.2832\.9332\.93Interval 425\.01±0\.2825\.01\\pm 0\.280\.73±0\.370\.73\\pm 0\.3722\.0822\.08Interval 524\.31±0\.2724\.31\\pm 0\.270\.35±0\.410\.35\\pm 0\.414\.894\.89Top\-VV25\.09±0\.2825\.09\\pm 0\.280\.27±0\.340\.27\\pm 0\.345\.735\.73Top\-Δ​V\\Delta V29\.93±0\.2429\.93\\pm 0\.240\.90±0\.210\.90\\pm 0\.2171\.1271\.12Uniform30\.94±0\.2130\.94\\pm 0\.210\.94±0\.090\.94\\pm 0\.0970\.0270\.02VISTA28\.28±0\.2228\.28\\pm 0\.220\.95±0\.11\\mathbf\{0\.95\\pm 0\.11\}74\.92\\mathbf\{74\.92\}15Interval 133\.98±0\.2333\.98\\pm 0\.230\.82±0\.200\.82\\pm 0\.2063\.3063\.30Interval 234\.82±0\.2034\.82\\pm 0\.200\.87±0\.180\.87\\pm 0\.1850\.4750\.47Interval 334\.76±0\.1834\.76\\pm 0\.180\.89±0\.180\.89\\pm 0\.1832\.6132\.61Interval 434\.76±0\.2434\.76\\pm 0\.240\.78±0\.340\.78\\pm 0\.3420\.1420\.14Interval 534\.09±0\.2234\.09\\pm 0\.220\.58±0\.420\.58\\pm 0\.424\.624\.62Top\-VV35\.02±0\.2735\.02\\pm 0\.270\.50±0\.420\.50\\pm 0\.425\.435\.43Top\-Δ​V\\Delta V43\.07±0\.2143\.07\\pm 0\.210\.96±0\.110\.96\\pm 0\.1167\.8467\.84Uniform44\.91±0\.1944\.91\\pm 0\.190\.94±0\.190\.94\\pm 0\.1964\.9664\.96VISTA38\.16±0\.2438\.16\\pm 0\.240\.99±0\.03\\mathbf\{0\.99\\pm 0\.03\}70\.86\\mathbf\{70\.86\}

Figure 8:VISTAschedules𝒢\\mathcal\{G\}for varyingT′T^\{\\prime\}\(Toxic Text, SMC\-grad\)
#### C\.2\.3Details on CIFAR\-10 Image Generation

See Table[9](https://arxiv.org/html/2610.20669#A3.T9)and Figure[9](https://arxiv.org/html/2610.20669#A3.F9)\.

Table 9:CIFAR\-10 class\-conditional generation: execution time, unique success count, andS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)\.T′T^\{\\prime\}PolicyTime \[s\]↓\\downarrowUnique Valid↑\\uparrowS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)↑\\uparrow1000Full43\.51±0\.0743\.51\\pm 0\.072\.18±1\.02\\mathbf\{2\.18\\pm 1\.02\}−⁣−\{\-\-\}5Interval 120\.03±0\.0420\.03\\pm 0\.040\.88±0\.460\.88\\pm 0\.4636\.7736\.77Interval 219\.99±0\.0319\.99\\pm 0\.031\.02±0\.221\.02\\pm 0\.22329\.40329\.40Interval 319\.98±0\.0119\.98\\pm 0\.010\.72±0\.260\.72\\pm 0\.26276\.90276\.90Interval 419\.98±0\.0219\.98\\pm 0\.020\.64±0\.280\.64\\pm 0\.28155\.30155\.30Interval 519\.98±0\.0219\.98\\pm 0\.020\.60±0\.300\.60\\pm 0\.3021\.2121\.21Top\-VV19\.98±0\.0119\.98\\pm 0\.010\.74±0\.320\.74\\pm 0\.32234\.60234\.60Top\-Δ​V\\Delta V19\.98±0\.0119\.98\\pm 0\.012\.42±0\.822\.42\\pm 0\.82395\.30395\.30Uniform19\.98±0\.0119\.98\\pm 0\.011\.26±0\.341\.26\\pm 0\.34384\.60384\.60VISTA19\.98±0\.0119\.98\\pm 0\.012\.72±0\.76\\mathbf\{2\.72\\pm 0\.76\}486\.00\\mathbf\{486\.00\}10Interval 120\.17±0\.0320\.17\\pm 0\.032\.06±0\.742\.06\\pm 0\.7467\.6367\.63Interval 220\.09±0\.0220\.09\\pm 0\.021\.04±0\.281\.04\\pm 0\.28328\.90328\.90Interval 320\.09±0\.0220\.09\\pm 0\.020\.78±0\.240\.78\\pm 0\.24278\.40278\.40Interval 420\.09±0\.0320\.09\\pm 0\.030\.66±0\.300\.66\\pm 0\.30154\.40154\.40Interval 520\.09±0\.0220\.09\\pm 0\.020\.56±0\.280\.56\\pm 0\.2821\.1021\.10Top\-VV20\.10±0\.0120\.10\\pm 0\.010\.76±0\.380\.76\\pm 0\.38233\.50233\.50Top\-Δ​V\\Delta V20\.10±0\.0120\.10\\pm 0\.013\.02±1\.383\.02\\pm 1\.38441\.30441\.30Uniform20\.09±0\.0220\.09\\pm 0\.022\.02±0\.462\.02\\pm 0\.46439\.30439\.30VISTA20\.09±0\.0220\.09\\pm 0\.023\.68±1\.02\\mathbf\{3\.68\\pm 1\.02\}494\.20\\mathbf\{494\.20\}15Interval 120\.29±0\.0220\.29\\pm 0\.022\.34±0\.942\.34\\pm 0\.94102\.70102\.70Interval 220\.23±0\.0220\.23\\pm 0\.020\.98±0\.280\.98\\pm 0\.28329\.50329\.50Interval 320\.22±0\.0120\.22\\pm 0\.010\.86±0\.360\.86\\pm 0\.36278\.10278\.10Interval 420\.22±0\.0320\.22\\pm 0\.030\.70±0\.320\.70\\pm 0\.32153\.00153\.00Interval 520\.22±0\.0320\.22\\pm 0\.030\.60±0\.260\.60\\pm 0\.2620\.9920\.99Top\-VV20\.22±0\.0120\.22\\pm 0\.010\.80±0\.360\.80\\pm 0\.36232\.00232\.00Top\-Δ​V\\Delta V20\.21±0\.0120\.21\\pm 0\.013\.86±1\.223\.86\\pm 1\.22446\.10446\.10Uniform20\.21±0\.0320\.21\\pm 0\.033\.08±0\.643\.08\\pm 0\.64452\.50452\.50VISTA20\.21±0\.0220\.21\\pm 0\.024\.28±1\.36\\mathbf\{4\.28\\pm 1\.36\}495\.20\\mathbf\{495\.20\}20Interval 120\.41±0\.0320\.41\\pm 0\.032\.32±1\.002\.32\\pm 1\.00134\.50134\.50Interval 220\.34±0\.0220\.34\\pm 0\.021\.00±0\.481\.00\\pm 0\.48336\.80336\.80Interval 320\.34±0\.0320\.34\\pm 0\.030\.80±0\.280\.80\\pm 0\.28277\.50277\.50Interval 420\.34±0\.0320\.34\\pm 0\.030\.64±0\.300\.64\\pm 0\.30152\.40152\.40Interval 520\.34±0\.0220\.34\\pm 0\.020\.58±0\.300\.58\\pm 0\.3020\.8920\.89Top\-VV20\.34±0\.0220\.34\\pm 0\.020\.82±0\.400\.82\\pm 0\.40254\.30254\.30Top\-Δ​V\\Delta V20\.34±0\.0220\.34\\pm 0\.023\.96±1\.303\.96\\pm 1\.30457\.30457\.30Uniform20\.34±0\.0220\.34\\pm 0\.023\.42±0\.723\.42\\pm 0\.72454\.90454\.90VISTA20\.32±0\.0220\.32\\pm 0\.024\.10±1\.12\\mathbf\{4\.10\\pm 1\.12\}495\.20\\mathbf\{495\.20\}25Interval 120\.46±0\.0220\.46\\pm 0\.022\.60±1\.142\.60\\pm 1\.14174\.10174\.10Interval 220\.45±0\.0220\.45\\pm 0\.021\.12±0\.321\.12\\pm 0\.32339\.20339\.20Interval 320\.45±0\.0120\.45\\pm 0\.010\.88±0\.260\.88\\pm 0\.26272\.80272\.80Interval 420\.45±0\.0220\.45\\pm 0\.020\.70±0\.300\.70\\pm 0\.30151\.50151\.50Interval 520\.44±0\.0220\.44\\pm 0\.020\.60±0\.320\.60\\pm 0\.3220\.7820\.78Top\-VV20\.46±0\.0220\.46\\pm 0\.020\.80±0\.380\.80\\pm 0\.38254\.20254\.20Top\-Δ​V\\Delta V20\.45±0\.0220\.45\\pm 0\.024\.34±0\.824\.34\\pm 0\.82457\.90457\.90Uniform20\.45±0\.0220\.45\\pm 0\.023\.90±0\.563\.90\\pm 0\.56457\.50457\.50VISTA20\.43±0\.0220\.43\\pm 0\.024\.64±1\.34\\mathbf\{4\.64\\pm 1\.34\}494\.90\\mathbf\{494\.90\}

Figure 9:VISTAschedules𝒢\\mathcal\{G\}for varyingT′T^\{\\prime\}\(CIFAR\-10, SMC\-base\)
#### C\.2\.4Details on Biology Design

See Figure[10](https://arxiv.org/html/2610.20669#A3.F10),[11](https://arxiv.org/html/2610.20669#A3.F11)\.

Figure 10:VISTAschedules𝒢\\mathcal\{G\}for varyingT′T^\{\\prime\}\(DNA, SMC\-grad\)Figure 11:VISTAschedules𝒢\\mathcal\{G\}for varyingT′T^\{\\prime\}\(Protein Fitness, MDM, SMC\-base\)

### C\.3Experiments beyond masked diffusion models

The weighted version of𝐓𝐃⁡\(𝒢\)\\mathbf\{TD\}\(\\mathcal\{G\}\)is defined as

𝐓𝐃⁡\(𝒢\):=∑t=0T−1λ⁡\(t\)​‖πt∗−πt𝒢‖1,\\mathbf\{TD\}\(\\mathcal\{G\}\):=\\sum\_\{t=0\}^\{T\-1\}\\lambda\(t\)\\\|\\pi\_\{t\}^\{\*\}\-\\pi\_\{t\}^\{\\mathcal\{G\}\}\\\|\_\{1\},\(60\)whereλ⁡\(t\)\\lambda\(t\)is nonnegative weight\. We used uniform weight in masked diffusion models\. However, in uniform discrete diffusion or continuous diffusion models, if the state takes the wrong path at the beginning of sampling, it could be corrected later\. Thus, we should give bigger weight whenttis close to 0\.

For the weighted version, by the proof of Theorem[4\.3](https://arxiv.org/html/2610.20669#S4.Thmtheorem3), the following inequality holds:

∑t=0T−1λ⁡\(t\)​‖πt∗−πt𝒢‖1\\displaystyle\\sum\_\{t=0\}^\{T\-1\}\\lambda\(t\)\\\|\\pi\_\{t\}^\{\*\}\-\\pi\_\{t\}^\{\\mathcal\{G\}\}\\\|\_\{1\}≤2​∑t∈𝒯∖𝒢λ⁡\(t\)​\[1−1Rm​a​x​𝔼x⌈t⌉∼π⌈t⌉∗​\[V⌈t⌉​\(x⌈t⌉\)\]\]\\displaystyle\\leq 2\\sum\_\{t\\in\\mathcal\{T\}\\setminus\\mathcal\{G\}\}\\lambda\(t\)\[1\-\\frac\{1\}\{R\_\{max\}\}\\mathbb\{E\}\_\{x\_\{\\lceil t\\rceil\}\\sim\\pi\_\{\\lceil t\\rceil\}^\{\*\}\}\[V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\]\]\(61\)≤2​∑t=0T−1λ⁡\(t\)−2Rm​a​x​∑t∈𝒯∖𝒢λ⁡\(t\)​𝔼x⌈t⌉∼π⌈t⌉∗​\[V⌈t⌉​\(x⌈t⌉\)\]\.\\displaystyle\\leq 2\\sum\_\{t=0\}^\{T\-1\}\\lambda\(t\)\-\\frac\{2\}\{R\_\{max\}\}\\sum\_\{t\\in\\mathcal\{T\}\\setminus\\mathcal\{G\}\}\\lambda\(t\)\\mathbb\{E\}\_\{x\_\{\\lceil t\\rceil\}\\sim\\pi\_\{\\lceil t\\rceil\}^\{\*\}\}\[V\_\{\\lceil t\\rceil\}\(x\_\{\\lceil t\\rceil\}\)\]\.\(62\)
Thus, we aim to maximize the surrogate score

S^​\(𝒢\):=∑t∈𝒯∖𝒢λ⁡\(t\)​V^⌈t⌉,\\hat\{S\}\(\\mathcal\{G\}\):=\\sum\_\{t\\in\\mathcal\{T\}\\setminus\\mathcal\{G\}\}\\lambda\(t\)\\hat\{V\}\_\{\\lceil t\\rceil\},\(63\)in weighted VISTA, and the corresponding DP algorithm is modified into Alg[7](https://arxiv.org/html/2610.20669#alg7)\. Further investigation on finding the optimal weightλ⁡\(t\)\\lambda\(t\)will be left to future work\.

Algorithm 7DP algorithm1:Initialize:

d​p​\[s\]​\[i\]←0dp\[s\]\[i\]\\leftarrow 0for all

s,is,i
2:for

s←T′−1,T′,⋯,T−1s\\leftarrow T^\{\\prime\}\-1,T^\{\\prime\},\\cdots,T\-1do

3:

d​p​\[s\]​\[1\]←\(∑u\>sλ⁡\(u\)\)​V^Tdp\[s\]\[1\]\\leftarrow\(\\sum\_\{u\>s\}\\lambda\(u\)\)\\hat\{V\}\_\{T\}
4:endfor

5:for

i←2,3,⋯,T′i\\leftarrow 2,3,\\cdots,T^\{\\prime\}do

6:for

s←T′−i,⋯,T−is\\leftarrow T^\{\\prime\}\-i,\\cdots,T\-ido

7:for

s′←s\+1s^\{\\prime\}\\leftarrow s\+1to

T−i\+1T\-i\+1do

8:

c←d​p​\[s′\]​\[i−1\]\+\(∑s<u<s′λ⁡\(u\)\)​V^s′c\\leftarrow dp\[s^\{\\prime\}\]\[i\-1\]\+\(\\sum\_\{s<u<s^\{\\prime\}\}\\lambda\(u\)\)\\hat\{V\}\_\{s^\{\\prime\}\}
9:if

c\>d​p​\[s\]​\[i\]c\>dp\[s\]\[i\]then

10:

d​p​\[s\]​\[i\]←cdp\[s\]\[i\]\\leftarrow c
11:

p​a​r​e​n​t​\[s\]​\[i\]←s′parent\[s\]\[i\]\\leftarrow s^\{\\prime\}
12:endif

13:endfor

14:endfor

15:endfor

16:

tT′←argmaxs​\(d​p​\[s\]​\[T′\]\+\(∑u<sλ⁡\(u\)\)​V^s\)t\_\{T^\{\\prime\}\}\\leftarrow\\text\{argmax\}\_\{s\}\(dp\[s\]\[T^\{\\prime\}\]\+\(\\sum\_\{u<s\}\\lambda\(u\)\)\\hat\{V\}\_\{s\}\)
17:for

j←T′−1,T′−2,⋯,1j\\leftarrow T^\{\\prime\}\-1,T^\{\\prime\}\-2,\\cdots,1do

18:

tj←p​a​r​e​n​t​\[tj\+1\]​\[j\+1\]t\_\{j\}\\leftarrow parent\[t\_\{j\+1\}\]\[j\+1\]
19:endfor

20:return

𝒢=\{t1,t2,⋯,tT′\}\\mathcal\{G\}=\\\{t\_\{1\},t\_\{2\},\\cdots,t\_\{T^\{\\prime\}\}\\\}

#### C\.3\.1Protein Fitness \(Uniform discrete diffusion\)

We useλ⁡\(t\)=1−t/T\\lambda\(t\)=1\-t/T\. We use reward modelr⁡\(x0\)r\(x\_\{0\}\)as oracle that predicts fitness\. We setT=128,N=32,J=3,M=3,α=0\.2T=128,N=32,J=3,M=3,\\alpha=0\.2\. A samplex0x\_\{0\}is considered “valid” ifr⁡\(x0\)≥0\.5r\(x\_\{0\}\)\\geq 0\.5\. Two sequences with Hamming distance≤1\\leq 1are treated as identical\. Results are averaged over 20 independent runs, yielding 640 generated samples per policy\. The warmup phase took 18\.7 s\. See Table[10](https://arxiv.org/html/2610.20669#A3.T10)\.

Table 10:Protein fitness generation, Uniform Discrete Diffusion \(SMC\-base, 32 particles×\\times20 runs\): execution time, unique valid count, andS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)\.T′T^\{\\prime\}PolicyTime \[s\]↓\\downarrowUnique Valid↑\\uparrowS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)↑\\uparrow128Full3\.263\.263\.15±2\.37\\mathbf\{3\.15\\pm 2\.37\}–16Interval 11\.421\.424\.45±3\.194\.45\\pm 3\.192\.71×1032\.71\\times 10^\{3\}Interval 21\.411\.417\.50±3\.727\.50\\pm 3\.726\.05×1036\.05\\times 10^\{3\}Interval 31\.411\.416\.10±3\.936\.10\\pm 3\.937\.86×1037\.86\\times 10^\{3\}Interval 41\.411\.415\.20±2\.845\.20\\pm 2\.843\.53×1033\.53\\times 10^\{3\}Interval 51\.411\.412\.20±1\.202\.20\\pm 1\.20602\.24602\.24Top\-VV1\.421\.425\.95±3\.595\.95\\pm 3\.591\.19×1041\.19\\times 10^\{4\}Top\-Δ​V\\Delta V1\.411\.416\.40±3\.916\.40\\pm 3\.911\.35×1041\.35\\times 10^\{4\}Uniform1\.411\.416\.20±3\.906\.20\\pm 3\.901\.10×1041\.10\\times 10^\{4\}VISTA1\.421\.428\.75±4\.76\\mathbf\{8\.75\\pm 4\.76\}2\.06×𝟏𝟎𝟒\\mathbf\{2\.06\\times 10^\{4\}\}32Interval 11\.851\.858\.90±5\.318\.90\\pm 5\.311\.07×1041\.07\\times 10^\{4\}Interval 21\.861\.866\.80±4\.646\.80\\pm 4\.646\.02×1036\.02\\times 10^\{3\}Interval 31\.851\.856\.65±4\.326\.65\\pm 4\.326\.08×1036\.08\\times 10^\{3\}Interval 41\.851\.854\.70±3\.344\.70\\pm 3\.344\.35×1034\.35\\times 10^\{3\}Interval 51\.851\.852\.95±2\.502\.95\\pm 2\.50443\.11443\.11Top\-VV1\.851\.855\.60±2\.955\.60\\pm 2\.951\.06×1041\.06\\times 10^\{4\}Top\-Δ​V\\Delta V1\.851\.856\.05±3\.526\.05\\pm 3\.521\.08×1041\.08\\times 10^\{4\}Uniform1\.841\.845\.45±2\.145\.45\\pm 2\.148\.88×1038\.88\\times 10^\{3\}VISTA1\.851\.859\.95±5\.30\\mathbf\{9\.95\\pm 5\.30\}1\.97×𝟏𝟎𝟒\\mathbf\{1\.97\\times 10^\{4\}\}

#### C\.3\.2Continuous MNIST \(Continuous diffusion\)

We useλ⁡\(t\)=\(1−t/T\)20\\lambda\(t\)=\(1\-t/T\)^\{20\}\. We use reward modelr⁡\(x0\)=log⁡p⁡\(c\|x0\)r\(x\_\{0\}\)=\\log p\(c\|x\_\{0\}\)whereccindicates the condition ‘image is 4’\. We setT=100,N=20,J=1,M=3,α=2\.0T=100,N=20,J=1,M=3,\\alpha=2\.0\. A samplex0x\_\{0\}is considered “valid” ifr⁡\(x0\)\>−0\.1r\(x\_\{0\}\)\>\-0\.1\. If the binarized images have Jaccard similarity≥0\.85\\geq 0\.85, then we regard them as duplicates\. Results are averaged over 30 independent runs, yielding 600 generated samples per policy\. The warmup phase took 41\.0 s\. See Table[11](https://arxiv.org/html/2610.20669#A3.T11)\.

Table 11:MNIST \(TDS, reproduce full/sparse\): execution time, unique valid count, andS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)\.T′T^\{\\prime\}PolicyTime \[s\]↓\\downarrowUnique Valid↑\\uparrowS^​\(𝒢\)\\hat\{S\}\(\\mathcal\{G\}\)↑\\uparrow100Full15\.47±0\.0815\.47\\pm 0\.0810\.33±3\.45\\mathbf\{10\.33\\pm 3\.45\}–6Interval 14\.29±0\.204\.29\\pm 0\.201\.93±1\.441\.93\\pm 1\.440\.060\.06Interval 24\.24±0\.034\.24\\pm 0\.031\.60±1\.251\.60\\pm 1\.250\.360\.36Interval 34\.24±0\.024\.24\\pm 0\.027\.77±4\.647\.77\\pm 4\.643\.403\.40Interval 44\.25±0\.034\.25\\pm 0\.0312\.87±6\.7712\.87\\pm 6\.775\.085\.08Interval 54\.15±0\.034\.15\\pm 0\.031\.73±1\.281\.73\\pm 1\.280\.110\.11Top\-VV4\.26±0\.034\.26\\pm 0\.0311\.73±5\.9111\.73\\pm 5\.914\.924\.92Top\-Δ​V\\Delta V4\.26±0\.034\.26\\pm 0\.0314\.13±4\.9114\.13\\pm 4\.914\.344\.34Uniform4\.16±0\.024\.16\\pm 0\.0214\.10±6\.2114\.10\\pm 6\.214\.124\.12VISTA4\.26±0\.034\.26\\pm 0\.0316\.83±5\.02\\mathbf\{16\.83\\pm 5\.02\}5\.11\\mathbf\{5\.11\}12Interval 14\.95±0\.374\.95\\pm 0\.371\.93±1\.011\.93\\pm 1\.010\.090\.09Interval 24\.86±0\.054\.86\\pm 0\.051\.30±1\.091\.30\\pm 1\.091\.211\.21Interval 34\.85±0\.044\.85\\pm 0\.0413\.43±4\.9013\.43\\pm 4\.904\.424\.42Interval 44\.86±0\.054\.86\\pm 0\.0514\.97±5\.5714\.97\\pm 5\.575\.065\.06Interval 54\.77±0\.054\.77\\pm 0\.051\.87±1\.381\.87\\pm 1\.380\.030\.03Top\-VV4\.86±0\.374\.86\\pm 0\.378\.20±4\.218\.20\\pm 4\.213\.073\.07Top\-Δ​V\\Delta V4\.90±0\.424\.90\\pm 0\.4214\.03±5\.4014\.03\\pm 5\.404\.674\.67Uniform4\.75±0\.024\.75\\pm 0\.0215\.73±5\.7215\.73\\pm 5\.723\.963\.96VISTA5\.31±0\.975\.31\\pm 0\.9717\.13±3\.80\\mathbf\{17\.13\\pm 3\.80\}5\.11\\mathbf\{5\.11\}

### C\.4Sensitivity Analysis

Throughout the paper, we useMMwarmup simulations to estimate the value function for schedule optimization\. Although this is a relatively small number of simulations, we find that it is sufficient to recover the high\-level structure of the optimized guidance schedule\. To verify this, we varyMMand visualize the resulting schedules in Figures[12](https://arxiv.org/html/2610.20669#A3.F12),[13](https://arxiv.org/html/2610.20669#A3.F13),[14](https://arxiv.org/html/2610.20669#A3.F14),[15](https://arxiv.org/html/2610.20669#A3.F15),[16](https://arxiv.org/html/2610.20669#A3.F16),[17](https://arxiv.org/html/2610.20669#A3.F17),[18](https://arxiv.org/html/2610.20669#A3.F18), and[19](https://arxiv.org/html/2610.20669#A3.F19)\. Across all tasks, the value\-critical guidance regions remain largely unchanged onceM=3M=3\(M=5M=5in CIFAR\-10\) is reached, indicating that only a small number of warmup simulations is required in practice\.

Figure 12:Optimized guidance schedules𝒢\\mathcal\{G\}for varyingMM\(Binary MNIST, SMC\-base\)Figure 13:Optimized guidance schedules𝒢\\mathcal\{G\}for varyingMM\(Binary MNIST, SMC\-grad\)Figure 14:Optimized guidance schedules𝒢\\mathcal\{G\}for varyingMM\(Toxic Text, SMC\-grad\)Figure 15:Optimized guidance schedules𝒢\\mathcal\{G\}for varyingMM\(CIFAR\-10, SMC\-base\)Figure 16:Optimized guidance schedules𝒢\\mathcal\{G\}for varyingMM\(DNA, SMC\-grad\)Figure 17:Optimized guidance schedules𝒢\\mathcal\{G\}for varyingMM\(Protein Fitness, SMC\-base, MDM\)Figure 18:Optimized guidance schedules𝒢\\mathcal\{G\}for varyingMM\(Protein Fitness, SMC\-base, UDM\)Figure 19:Optimized guidance schedules𝒢\\mathcal\{G\}for varyingMM\(MNIST, TDS\)

## Appendix DSocietal Impacts

##### Positive impacts\.

VISTAreduces the cost of SMC\-based reward guidance\. This can make inference\-time alignment more accessible to researchers with limited hardware and can reduce the energy cost of repeated reward\-guided generation\.

##### Potential negative impacts\.

More efficient control of generative models could be misused to generate harmful or manipulative content at lower cost\. The toxic\-text experiment is included only as a controlled benchmark for testing reward steering; it is not an endorsement of generating toxic content\.

##### Mitigation\.

The main risk depends on the reward model and application\. We recommend usingVISTAwith ethically aligned reward functions, safety filters, monitoring, and application\-specific release controls\. The scheduler itself is neutral: it decides when a given reward should be applied, but it does not define the reward\.