Accelerating Diffusion Sampling via Speculative Draft Trees

# Accelerating Diffusion Sampling viaSpeculative Draft Trees
Source: [https://arxiv.org/html/2609.17691](https://arxiv.org/html/2609.17691)
## Accelerating Diffusion Sampling via Speculative Draft Trees

Marcello BulloImperial College Londonm\.bullo21@imperial\.ac\.ukYanxiao LiuImperial College Londony\.liu2@imperial\.ac\.ukÖykü Sıla GünerImperial College Londono\.guner25@imperial\.ac\.uk Arpan MukherjeeImperial College Londona\.mukherjee@imperial\.ac\.ukDeniz GündüzImperial College Londond\.gunduz@imperial\.ac\.uk

###### Abstract

Speculative sampling accelerates diffusion model generation by drafting inexpensive candidate states and correcting them under a coupling that preserves the target distributionexactly, reducing the number of expensive target evaluations\. Existing diffusion samplers, notably those based on reflection maximal coupling, are topologically constrained: their lookahead drafts form a chain graph, a single linear sequence, which inherently limits the acceptance rate per target evaluation\. We connect speculative sampling in diffusion models to relative entropy coding \(REC\)\. This perspective shows the lookahead need not be linear and motivates our central contribution,draft trees, which enrich the candidates considered per round and lower the target function evaluations\. We further adopt greedy rejection sampling, an REC algorithm, as the draft–target coupling, improving acceptance while guaranteeing exact target samples\. Experiments across diverse target and draft models demonstrate up to 8\.3% acceleration over the reflection coupling baseline in practical settings\.

## 1 Introduction

Denoising diffusion probabilistic models \(DDPMs\)\([Sohl\-Dickstein et al\., 2015](https://arxiv.org/html/2609.17691#bib.bib5);[Ho et al\., 2020](https://arxiv.org/html/2609.17691#bib.bib4);[Song et al\., 2021b](https://arxiv.org/html/2609.17691#bib.bib6)\)have become the dominant paradigm for high\-fidelity image generation\. By learning to reverse a gradual noising process, diffusion models synthesize images through a sequence of denoising steps that progressively transform Gaussian noise into a sample from the target data distribution\. Their remarkable generation quality has established diffusion models as the state of the art across a wide range of generative tasks\([Yang et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib41)\)\.

Figure 1:Efficiency frontier of speculative sampling\.The plot compares the maximal speedup attainable over a standard sampler at a given proposal budgetBBfor the setting described in Section[5\.1](https://arxiv.org/html/2609.17691#S5.SS1)\. By allocating budget strictly to sequential depth, RMC rapidly saturates near a2\.00×2\.00\\timesspeedup\. In contrast, D\-GRS allocates budget to multiple proposals per node, effectively raising the per\-step acceptance probability\. This branching strategy allows D\-GRS to monotonically convert larger budgets into measurable acceleration, up to a14%14\\%reduction in target model evaluations relative to RMC\.A major drawback of diffusion models, however, is their high inference cost\. Image generation requires repeatedly evaluating a large neural network over tens to hundreds of denoising iterations, making sampling extremely slow\. This computational bottleneck has motivated a broad range of acceleration techniques\. Training\-free approaches include improved solvers to reduce the number of sampling steps\([Zhang and Chen, 2023](https://arxiv.org/html/2609.17691#bib.bib53);[Lu et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib43)\), while training\-based methods learn few\-step generators, either by preserving the sampling trajectory, as in consistency models\([Song et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib44)\)and their two\-time flow\-map generalizations\([Geng et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib45);[Zhou et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib46);[Sabour et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib47)\), recently extended to learn the solution map of the underlying SDE\([McCallum et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib28)\), or by directly matching the output distribution\([Yin et al\., 2024a](https://arxiv.org/html/2609.17691#bib.bib48);[Sauer et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib49)\)\. While effective in reducing latency, these approaches typically incur either additional training cost or degradation in sample quality\([Dieleman, 2026](https://arxiv.org/html/2609.17691#bib.bib50);[Lai et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib40)\)\. Parallel simulation provides an orthogonal acceleration strategy, trading additional computation and memory for reduced latency\([Shih et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib23);[Tang et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib51);[Chen et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib24);[Zhou and Sugiyama, 2025](https://arxiv.org/html/2609.17691#bib.bib52)\)\. Despite reducing the sequential depth of diffusion sampling, it typically requires multiple iterative refinement rounds and substantial parallel resources\.

Recently,*speculative diffusion sampling*\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\)has emerged as a complementary paradigm for accelerating diffusion inference\. Inspired by speculative decoding for large language models\([Leviathan et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib16);[Chen et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib17)\), these methods first generate candidate denoising states using an inexpensive source \(referred to as aproposal\) before verifying them using the target diffusion model\. Whenever multiple draft states are accepted, several denoising iterations are effectively skipped, leading directly to lower inference latency\. Crucially, unlike approximate acceleration techniques, speculative diffusion sampling is*exact*: it preserves the sampling distribution of the target model, and therefore incurs no loss in image quality\.

Despite these advantages, existing speculative diffusion samplers are fundamentally*topology constrained*\. They generate drafts sequentially along the diffusion trajectory, resulting in a linear chain of proposal states\. From a broader perspective, speculative sampling is a*budget allocation*problem: given a fixed amount of parallel computation, how should proposal evaluations be allocated to maximize the probability of accepting draft states? Restricting proposals to a chain represents only one possible allocation strategy and can leave substantial parallelism underutilized, particularly in the low\-acceptance regime that commonly arises in practice\.

In this paper, we take a first step toward lifting this restriction\. Our contributions are two\-fold\. First, we generalize speculative diffusion sampling from linear draft chains to*draft trees*, enabling substantially richer allocations of parallel computation across candidate denoising trajectories\. Second, we establish a novel connection between speculative sampling and*relative entropy coding*\(REC\)\([Li, 2024](https://arxiv.org/html/2609.17691#bib.bib26);[Li and El Gamal, 2018](https://arxiv.org/html/2609.17691#bib.bib7);[Flamich and Theis, 2023](https://arxiv.org/html/2609.17691#bib.bib10);[Flamich, 2023](https://arxiv.org/html/2609.17691#bib.bib11)\)\. This connection leads naturally to a greedy rejection sampling \(GRS\)\([Harsha et al\., 2010](https://arxiv.org/html/2609.17691#bib.bib9)\)strategy for budget allocation, providing an efficient and principled mechanism for constructing draft trees\. Extensive experiments demonstrate that our approach consistently improves speculative diffusion sampling, achieving speedups of up to 7\.4% over existing methods for practical batch sizes≤256\\leq 256, while preserving exact sampling \(see Figure[1](https://arxiv.org/html/2609.17691#S1.F1)\)\.

## 2 Problem Setting

##### DDPM\.

DDPMs define a fixed forward diffusion process that gradually perturbs data samples with Gaussian noise\. LetX0∼pdataX\_\{0\}\\sim p\_\{\\rm data\}denote a data sample\. The forward process\(Xt\)t∈\[0,1\]\(X\_\{t\}\)\_\{t\\in\[0,1\]\}evolves according to the stochastic differential equation \(SDE\)

⁡d​Xt=ft​\(Xt\)​⁡d​t\+gt​⁡d​Bt,\\displaystyle\\mathop\{\}\\\!\\mathrm\{d\}X\_\{t\}=f\_\{t\}\(X\_\{t\}\)\\,\\mathop\{\}\\\!\\mathrm\{d\}t\+g\_\{t\}\\,\\mathop\{\}\\\!\\mathrm\{d\}B\_\{t\}\\ ,\(1\)where\(Bt\)t∈\[0,1\]\(B\_\{t\}\)\_\{t\\in\[0,1\]\}is add\-dimensional Brownian motion\. Image generation is performed by simulating the associated reverse\-time process\(Yt\)t∈\[0,1\]\(Y\_\{t\}\)\_\{t\\in\[0,1\]\}, which satisfies

⁡d​Yt\\displaystyle\\mathop\{\}\\\!\\mathrm\{d\}Y\_\{t\}=btq​\(Yt\)​⁡d​t\+ε​g1−t​⁡d​Wt\\displaystyle=b\_\{t\}^\{q\}\(Y\_\{t\}\)\\,\\mathop\{\}\\\!\\mathrm\{d\}t\+\\varepsilon g\_\{1\-t\}\\,\\mathop\{\}\\\!\\mathrm\{d\}W\_\{t\}\\,\(2\)btq​\(x\)\\displaystyle b\_\{t\}^\{q\}\(x\)=−f1−t​\(x\)\+1\+ε22​g1−t2​s1−t​\(x\),\\displaystyle=\-f\_\{1\-t\}\(x\)\+\\frac\{1\+\\varepsilon^\{2\}\}\{2\}g\_\{1\-t\}^\{2\}s\_\{1\-t\}\(x\)\\ ,\(3\)where\(Wt\)t∈\[0,1\]\(W\_\{t\}\)\_\{t\\in\[0,1\]\}is add\-dimensional Brownian motion,st​\(x\)s\_\{t\}\(x\)is the Stein score function, andε∈\[0,1\]\\varepsilon\\in\[0,1\]is the*churn parameter*controlling the stochasticity of the reverse process\. In particular,ε=1\\varepsilon=1recovers the standard reverse\-time SDE, whereasε=0\\varepsilon=0yields the deterministic probability\-flow ordinary differential equation \(ODE\)\. To simulate this reverse process, we introduce a uniform time grid

tn=nγ,n=0,…,N,γ=1N,t\_\{n\}=n\\gamma\\ ,\\qquad n=0,\\ldots,N\\ ,\\qquad\\gamma=\\frac\{1\}\{N\}\\ ,and writeYn:=YtnY\_\{n\}:=Y\_\{t\_\{n\}\}\. Applying the Euler–Maruyama discretization withξn∼𝒩⁡\(0,𝕀d\)\\xi\_\{n\}\\sim\\mathcal\{N\}\(0,\\mathbb\{I\}\_\{d\}\)gives

Yn\+1=Yn\+γ​btnq​\(Yn\)\+γ​ε​g1−tn​ξn\.\\displaystyle Y\_\{n\+1\}=Y\_\{n\}\+\\gamma b\_\{t\_\{n\}\}^\{q\}\(Y\_\{n\}\)\+\\sqrt\{\\gamma\}\\,\\varepsilon g\_\{1\-t\_\{n\}\}\\,\\xi\_\{n\}\\ \.\(4\)Hence, conditional onYn=ynY\_\{n\}=y\_\{n\}, the next state followsYn\+1\|Yn=yn∼𝒩⁡\(mnq​\(yn\),σn2​𝕀d\)Y\_\{n\+1\}\\,\\big\|\\,Y\_\{n\}=y\_\{n\}\\sim\\mathcal\{N\}\\\!\\left\(m\_\{n\}^\{q\}\(y\_\{n\}\),\\sigma\_\{n\}^\{2\}\\mathbb\{I\}\_\{d\}\\right\), where

mnq​\(y\):=y\+γ​btnq​\(y\)andσn:=γ​ε​g1−tn\.\\displaystyle m\_\{n\}^\{q\}\(y\):=y\+\\gamma b\_\{t\_\{n\}\}^\{q\}\(y\)\\quad\\text\{and\}\\quad\\sigma\_\{n\}:=\\sqrt\{\\gamma\}\\,\\varepsilon g\_\{1\-t\_\{n\}\}\\ \.\(5\)Equivalently, denoting the target transition kernel byℚn\(⋅∣yn\)\\mathbb\{Q\}\_\{n\}\(\\cdot\\mid y\_\{n\}\), we haveℚn\(⋅∣yn\)=𝒩\(mnq\(yn\),σn2𝕀d\),\\mathbb\{Q\}\_\{n\}\(\\cdot\\mid y\_\{n\}\)=\\mathcal\{N\}\\\!\\left\(m\_\{n\}^\{q\}\(y\_\{n\}\),\\sigma\_\{n\}^\{2\}\\mathbb\{I\}\_\{d\}\\right\),and the discretized reverse process forms a Markov chain with joint density

q⁡\(y0,…,yN\)=q0​\(y0\)​∏n=0N−1qn​\(yn\+1∣yn\),\\displaystyle q\(y\_\{0\},\\ldots,y\_\{N\}\)=q\_\{0\}\(y\_\{0\}\)\\displaystyle\\prod\_\{n=0\}^\{N\-1\}q\_\{n\}\(y\_\{n\+1\}\\mid y\_\{n\}\)\\ ,\(6\)whereq0q\_\{0\}denotes the initial Gaussian density of the reverse process\.

##### Speculative diffusion sampling\.

The principal computational bottleneck in diffusion inference is the repeated evaluation of the reverse driftbtnqb\_\{t\_\{n\}\}^\{q\}, which is parameterized by a large neural network\. Since Euler–Maruyama simulation proceeds sequentially, each stateYn\+1Y\_\{n\+1\}depends on the reverse drift evaluated at the preceding stateYnY\_\{n\}, limiting the utilization of modern parallel hardware\. Speculative diffusion sampling addresses this bottleneck by exploiting parallel computation\. Rather than simulating one denoising step at a time, an inexpensive*proposal*model first generates multiple draft states\. The key assumption underlying speculative sampling is that*drafting is significantly cheaper than verification*\. The drafted states are subsequently verified using the target reverse drift in parallel\. Consequently, given a parallel compute budget ofBB, one batched target\-model evaluation, corresponding to one neural function evaluation \(NFE\), can verify all proposed drafts\.

The effectiveness of speculative sampling depends critically on the quality of the proposal model\. Better proposals lead to higher acceptance rates, allowing more denoising steps to be advanced per NFE and thereby increasing the overall speedup while preserving exact sampling\. Existing speculative diffusion methods\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\)consider two proposal mechanisms\. The first trains a lightweight diffusion model to approximate the target reverse drift\. The second, referred to as the*delayed reverse drift*, avoids additional training by replacing the current reverse drift with one evaluated at an earlier state\. Specifically, the target transition meanmnq​\(yn\)=yn\+γ​btnq​\(yn\)m\_\{n\}^\{q\}\(y\_\{n\}\)=y\_\{n\}\+\\gamma\{\\color\[rgb\]\{1,0,0\}b\_\{t\_\{n\}\}^\{q\}\(y\_\{n\}\)\}is approximated by the proposal mean

mnp​\(yn\)=yn\+γ​btkq​\(yk\),k<n,\\displaystyle m\_\{n\}^\{p\}\(y\_\{n\}\)=y\_\{n\}\+\\gamma\{\\color\[rgb\]\{0,0,1\}b\_\{t\_\{k\}\}^\{q\}\(y\_\{k\}\)\},\\qquad k<n\\ ,\(7\)while retaining the same transition varianceσn2=γ​ε2​g1−tn2\\sigma\_\{n\}^\{2\}=\\gamma\\varepsilon^\{2\}g\_\{1\-t\_\{n\}\}^\{2\}\. Empirically, delayed reverse drift has been shown to produce higher\-quality proposals than lightweight draft models\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\)\. Accordingly, we adopt it as the proposal mechanism throughout this paper\.

## 3 Speculative Algorithms

In this section, we first state the general template that both algorithms instantiate, then review the reflection maximal coupling \(RMC\) algorithm of\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\), and finally introduce our diffusion\-GRS \(D\-GRS\) algorithm\. At a high level, every speculative sampling algorithm consists of three phases: a*drafting*phase, a*verification*phase, and an*acceptance*phase\. Given a parallel compute budget, the drafting phase sequentially generates candidate states according to a prescribed draft topology, the verification phase evaluates all drafted states in parallel using the target model, and the acceptance phase sequentially examines the verified states to determine whether a proposal can be accepted as the next target state\. Consequently, speculative sampling algorithms differ primarily in the choice ofdraft topologyand theacceptance criterionused after verification\. Both RMC and D\-GRS rely on a common one\-dimensional reduction that exploits the translation and rotational invariance of isotropic Gaussian distributions, described next\. Throughout, we useY^\\widehat\{Y\}\(orS^\\widehat\{S\}\) to denote a proposed state andYY\(orSS\) to denote the accepted target state\.

### 3\.1 A General Template For Speculative Diffusion

We begin from the general procedure, stated so that the two algorithms of this section differ only in the two components the paper varies: the*draft topology*and the*verification rule*\. The template is agnostic to how proposals are produced and it is shown in Algorithm[1](https://arxiv.org/html/2609.17691#alg1)\.

Algorithm 1Speculative diffusion sampling for an arbitrary draft tree𝒯\\mathcal\{T\}\.0:draft and target mean functions

mpm^\{p\},

mqm^\{q\}, scales

\{σn\}\\\{\\sigma\_\{n\}\\\}, horizon

NN, draft tree

𝒯=\(𝒱,pa\)\\mathcal\{T\}=\(\\mathcal\{V\},\\mathrm\{pa\}\)of depth

LL, ruleVerify

0:trajectory

Y1:NY\_\{1:N\}with the law \(6\) of the target chain

1:

n←0n\\leftarrow 0;

Y0∼q0Y\_\{0\}\\sim q\_\{0\}
2:while

n<Nn<Ndo

3:

Ln←min⁡\{L,N−n\}L\_\{n\}\\leftarrow\\min\\\{L,N\-n\\\},

Y^r←Yn\\widehat\{Y\}\_\{\\mathrm\{r\}\}\\leftarrow Y\_\{n\},

𝒯n←𝒯\|Ln\\mathcal\{T\}\_\{n\}\\leftarrow\\mathcal\{T\}\|\_\{L\_\{n\}\},

ℐn←ℐ⁡\(𝒯n\)\\mathcal\{I\}\_\{n\}\\leftarrow\\mathcal\{I\}\(\\mathcal\{T\}\_\{n\}\)
4:// Phase 1 – Drafting

5:for

ℓ=1\\ell=1to

LnL\_\{n\}do

6:for all

u∈𝒱ℓ−1u\\in\\mathcal\{V\}\_\{\\ell\-1\}in paralleldo

7:

\(Y^v\)v∈𝒞⁡\(u\)​∼i\.i\.d\.​𝒩​\(mn\+ℓ−1p​\(Y^u\),σn\+ℓ−12​Id\)\\big\(\\widehat\{Y\}\_\{v\}\\big\)\_\{v\\in\\mathcal\{C\}\(u\)\}\\overset\{\\text\{i\.i\.d\.\}\}\{\\sim\}\\mathcal\{N\}\\big\(m^\{p\}\_\{n\+\\ell\-1\}\(\\widehat\{Y\}\_\{u\}\),\\,\\sigma^\{2\}\_\{n\+\\ell\-1\}I\_\{d\}\\big\)
8:endfor

9:endfor

10:// Phase 2 – Verification

11:for all

u∈ℐnu\\in\\mathcal\{I\}\_\{n\}in paralleldo

12:evaluate

mn\+\|u\|q​\(Y^u\)m^\{q\}\_\{n\+\|u\|\}\\big\(\\widehat\{Y\}\_\{u\}\\big\)
13:endfor

14:// Phase 3 – Acceptance

15:

u←ru\\leftarrow\\mathrm\{r\};

Nα←0N\_\{\\alpha\}\\leftarrow 0
16:for

ℓ=1\\ell=1to

LnL\_\{n\}do

17:

\(Yn\+ℓ,accepted,v⋆\)←Verify​\(mn\+ℓ−1p​\(Y^u\),mn\+ℓ−1q​\(Y^u\),σn\+ℓ−1,\{Y^v\}v∈𝒞⁡\(u\)\)\\big\(Y\_\{n\+\\ell\},\\,\\texttt\{accepted\},\\,v^\{\\star\}\\big\)\\leftarrow\\textsc\{Verify\}\\big\(m^\{p\}\_\{n\+\\ell\-1\}\(\\widehat\{Y\}\_\{u\}\),\\,m^\{q\}\_\{n\+\\ell\-1\}\(\\widehat\{Y\}\_\{u\}\),\\,\\sigma\_\{n\+\\ell\-1\};\\,\\\{\\widehat\{Y\}\_\{v\}\\\}\_\{v\\in\\mathcal\{C\}\(u\)\}\\big\)
18:ifnotacceptedthen

19:

Nα←Nα\+1N\_\{\\alpha\}\\leftarrow N\_\{\\alpha\}\+1
20:break

21:endif

22:

Nα←Nα\+1N\_\{\\alpha\}\\leftarrow N\_\{\\alpha\}\+1;

u←v⋆u\\leftarrow v^\{\\star\}
23:endfor

24:

n←n\+min⁡\{Nα,Ln\}n\\leftarrow n\+\\min\\\{N\_\{\\alpha\},\\,L\_\{n\}\\\}
25:endwhile

26:return

Y1:NY\_\{1:N\}

#### 3\.1 Setting

Fix a round beginning at stepnn\. The template requires only that the draft and target transitions be isotropic Gaussians sharing the variance schedule and differing in their means,

ℙn\(⋅∣y\)\\displaystyle\\mathbb\{P\}\_\{n\}\(\\cdot\\mid y\)=𝒩⁡\(mnp​\(y\),σn2​Id\),\\displaystyle=\\mathcal\{N\}\\big\(m^\{p\}\_\{n\}\(y\),\\sigma\_\{n\}^\{2\}I\_\{d\}\\big\),\(8\)ℚn\(⋅∣y\)\\displaystyle\\mathbb\{Q\}\_\{n\}\(\\cdot\\mid y\)=𝒩⁡\(mnq​\(y\),σn2​Id\),\\displaystyle=\\mathcal\{N\}\\big\(m^\{q\}\_\{n\}\(y\),\\sigma\_\{n\}^\{2\}I\_\{d\}\\big\),withmnqm^\{q\}\_\{n\}as in \([5](https://arxiv.org/html/2609.17691#S2.E5)\) andmnpm^\{p\}\_\{n\}arbitrary\. This is exactly the structure the rank\-11reduction in[3\.1\.4](https://arxiv.org/html/2609.17691#S3.SS1.SSS4)exploits, so both Algorithm[2](https://arxiv.org/html/2609.17691#alg2)and Algorithm[3](https://arxiv.org/html/2609.17691#alg3)below apply unchanged at every node\.

#### 3\.1 Draft Topology

A*node*is one candidate realisation of the process state at one step\. Its depth is the step it realises and its parent is the state it was conditioned on\. Because the reverse chain is Markov, each node is drawn conditionally on exactly one parent, so a draft set always carries the structure of a rooted tree, and we take such a tree as the general description of a draft topology\.

A*draft tree*is a finite rooted tree𝒯=\(𝒱,pa\)\\mathcal\{T\}=\(\\mathcal\{V\},\\mathrm\{pa\}\)with vertex set𝒱\\mathcal\{V\}, rootr\\mathrm\{r\}, and parent mappa:𝒱∖\{r\}→𝒱\\mathrm\{pa\}:\\mathcal\{V\}\\setminus\\\{\\mathrm\{r\}\\\}\\to\\mathcal\{V\}\. We write𝒞​\(u\):=pa−1​\(u\)\\mathcal\{C\}\(u\):=\\mathrm\{pa\}^\{\-1\}\(u\)for the children ofuu,\|u\|\|u\|for its depth, and

𝒱0=\{r\},\\displaystyle\\mathcal\{V\}\_\{0\}=\\\{\\mathrm\{r\}\\\},\(9\)𝒱ℓ:=\{u∈𝒱:\|u\|=ℓ\},\\displaystyle\\mathcal\{V\}\_\{\\ell\}:=\\\{u\\in\\mathcal\{V\}:\\ \|u\|=\\ell\\\},ℐ⁡\(𝒯\):=\{u∈𝒱:𝒞⁡\(u\)≠∅\},\\displaystyle\\mathcal\{I\}\(\\mathcal\{T\}\):=\\\{u\\in\\mathcal\{V\}:\\ \\mathcal\{C\}\(u\)\\neq\\emptyset\\\},for the layers and the internal nodes\. The root holds the last accepted stateY^r=Yn\\widehat\{Y\}\_\{\\mathrm\{r\}\}=Y\_\{n\}and nodeu≠ru\\neq\\mathrm\{r\}holds the drafted stateY^u\\widehat\{Y\}\_\{u\}, a candidate realisation of stepn\+\|u\|n\+\|u\|\. The number of drafted states is\|𝒱\|−1\|\\mathcal\{V\}\|\-1, and the target model can be evaluated at all the tree nodesu∈𝒱u\\in\\mathcal\{V\}, or only at the internal onesu∈ℐ⁡\(𝒯\)u\\in\\mathcal\{I\}\(\\mathcal\{T\}\)\.

The children of a node are i\.i\.d\. candidates for the*same*step, of which verification retains at most one\. Because the accepted candidate node is determined exclusively during the verification phase, the proposal mechanism must exhaustively expand all sibling branches\. The uniform\(K,L\)\(K,L\)family used in the paper is the case\|𝒞⁡\(u\)\|=K\|\\mathcal\{C\}\(u\)\|=Kfor everyuuof depth at mostL−1L\-1, and𝒱ℓ=∅\\mathcal\{V\}\_\{\\ell\}=\\emptysetforℓ\>L\\ell\>L\. Therefore,\|𝒱ℓ\|=Kℓ\|\\mathcal\{V\}\_\{\\ell\}\|=K^\{\\ell\}and

\|ℐ⁡\(𝒯\)\|=∑ℓ=0L−1Kℓ,\|𝒱\|=∑ℓ=0L\|𝒱ℓ\|=∑ℓ=0LKℓ\.\|\\mathcal\{I\}\(\\mathcal\{T\}\)\|=\\sum\_\{\\ell=0\}^\{L\-1\}K^\{\\ell\},\\qquad\|\\mathcal\{V\}\|=\\sum\_\{\\ell=0\}^\{L\}\|\\mathcal\{V\}\_\{\\ell\}\|=\\sum\_\{\\ell=0\}^\{L\}K^\{\\ell\}\.\(10\)During the verification phase, when the target model is evaluated atu∈𝒱u\\in\\mathcal\{V\}, a round costsB=\|𝒱\|B=\|\\mathcal\{V\}\|target evaluations, while it costsB=\|ℐ⁡\(𝒯\)\|B=\|\\mathcal\{I\}\(\\mathcal\{T\}\)\|when the target model is evaluated atu∈ℐ⁡\(𝒯\)u\\in\\mathcal\{I\}\(\\mathcal\{T\}\)\. The chain of RMC corresponds to the special case whereK=1K=1, while a single\-node proposal list toL=1L=1\. Depth\-dependent widthsKℓK\_\{\\ell\}, or irregular trees \(e\.g\., pruned\), are covered by𝒯\\mathcal\{T\}directly and require no change to what follows\.

Let𝒱≤m:=⋃ℓ=0m𝒱ℓ\\mathcal\{V\}\_\{\\leq m\}:=\\bigcup\_\{\\ell=0\}^\{m\}\\mathcal\{V\}\_\{\\ell\}denote the set of nodes of depth at mostmm, and let

𝒯\|m:=𝒯⁡\[𝒱≤m\]=\(𝒱≤m,pa\|𝒱≤m∖\{r\}\)\\mathcal\{T\}\|\_\{m\}:=\\mathcal\{T\}\\big\[\\mathcal\{V\}\_\{\\leq m\}\\big\]=\\big\(\\mathcal\{V\}\_\{\\leq m\},\\ \\mathrm\{pa\}\|\_\{\\mathcal\{V\}\_\{\\leq m\}\\setminus\\\{\\mathrm\{r\}\\\}\}\\big\)\(11\)denote the subtree of𝒯\\mathcal\{T\}induced by𝒱≤m\\mathcal\{V\}\_\{\\leq m\}\.

A round starting at stepnntherefore operates on𝒯\|Ln\\mathcal\{T\}\|\_\{L\_\{n\}\}withLn:=min⁡\{L,N−n\}L\_\{n\}:=\\min\\\{L,N\-n\\\}, which equals𝒯\\mathcal\{T\}in every round but the last few\.

#### 3\.1 Round Structure

Algorithm[1](https://arxiv.org/html/2609.17691#alg1)describes the speculative diffusion sampling procedure for an arbitrary draft tree𝒯\\mathcal\{T\}\. Each iteration of the algorithm involves three distinct phases\.

1. 1\.Thedrafting phaseexecutes a layer\-wise expansion of the tree: this expansion is strictly sequential with respect to depth, as child nodes cannot be sampled prior to their respective parents\. However, all nodes residing at the same depth level are generated concurrently\.
2. 2\.Theverification phasecomputes the target means for all internal nodes of the current tree via a unified batched operation, representing the sole evaluation of the target model within a given iteration\.
3. 3\.Finally, theacceptance phasetraverses downward from the root node, iteratively applying the verification criterion at each depth level and terminating immediately upon encountering the first rejection\.

The verification rule, denoted asVerify, serves as a modular component within this framework and is governed by the following formal contract\. Given a parent nodeuu, its associated proposal and target means, the corresponding step scale, and its set of drafted children\{Y^v\}v∈𝒞⁡\(u\)\\\{\\widehat\{Y\}\_\{v\}\\\}\_\{v\\in\\mathcal\{C\}\(u\)\}, the rule outputs a tuple\(Y,accepted,v⋆\)\(Y,\\texttt\{accepted\},v^\{\\star\}\)\. The primary component,YY, constitutes an exact sample drawn from the target transition distribution,Y∼ℚ\(⋅∣Y^u\)Y\\sim\\mathbb\{Q\}\\big\(\\cdot\\mid\\widehat\{Y\}\_\{u\}\\big\)\. This sample is obtained through one of two mutually exclusive mechanisms: either a valid child node is identified, wherein the boolean flagacceptedevaluates to true andY=Y^v⋆Y=\\widehat\{Y\}\_\{v^\{\\star\}\}for a specificv⋆∈𝒞⁡\(u\)v^\{\\star\}\\in\\mathcal\{C\}\(u\); alternatively, if no child satisfies the criterion,acceptedevaluates to false, andYYis instead sampled from the normalized residual distribution formulated in \([17](https://arxiv.org/html/2609.17691#S3.E17)\) below\. Algorithm[3](https://arxiv.org/html/2609.17691#alg3)fulfills this contract for any arbitrary branching factorK≥1K\\geq 1, whereas Algorithm[2](https://arxiv.org/html/2609.17691#alg2)is specialized forK=1K=1\.

The template leaves exactly two components free: the draft topology𝒯\\mathcal\{T\}and the verification ruleVerify\. The remainder of this section instantiates them\. RMC takes𝒯\\mathcal\{T\}to be a chain \(K=1K=1\) andVerifyto be the reflection maximal coupling by\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\); D\-GRS takes𝒯\\mathcal\{T\}to be aKK\-ary tree andVerifyto be our proposed D\-GRS algorithm\. Both rely on a common one dimensional reduction that exploits the translation and rotational invariance of isotropic Gaussian distributions, described next\.

#### 3\.1 Rank\-1 Projection

Since the proposalℙ:=𝒩⁡\(μp,σ2​𝕀d\)\\mathbb\{P\}:=\\mathcal\{N\}\(\\mu\_\{p\},\\sigma^\{2\}\\mathbb\{I\}\_\{d\}\)and targetℚ:=𝒩⁡\(μq,σ2​𝕀d\)\\mathbb\{Q\}:=\\mathcal\{N\}\(\\mu\_\{q\},\\sigma^\{2\}\\mathbb\{I\}\_\{d\}\)are isotropic Gaussians differing only in their means, we can reduce thedd\-dimensional coupling to a single dimension\. Let

Δ:=\(μq−μp\)/σ,δ:=‖Δ‖,e:=Δ/δ,\\displaystyle\\Delta:=\(\\mu\_\{q\}\-\\mu\_\{p\}\)/\\sigma,\\quad\\delta:=\\\|\\Delta\\\|,\\quad e:=\\Delta/\\delta,\(12\)be the normalized mean displacement, its Euclidean norm, and the corresponding unit vector, respectively\. Standardizing a stateYYasZ:=\(Y−μp\)/σZ:=\(Y\-\\mu\_\{p\}\)/\\sigmayieldsZ∼𝒩⁡\(0,𝕀d\)Z\\sim\\mathcal\{N\}\(0,\\mathbb\{I\}\_\{d\}\)underℙ\\mathbb\{P\}andZ∼𝒩⁡\(δ​e,𝕀d\)Z\\sim\\mathcal\{N\}\(\\delta e,\\mathbb\{I\}\_\{d\}\)underℚ\\mathbb\{Q\}\. Note that any realizationz∈ℝdz\\in\\mathbb\{R\}^\{d\}admits the orthogonal decomposition

z=e​e⊤​z⏟s\+\(𝕀d−e​e⊤\)​z⏟z⟂\.z=e\\underbrace\{e^\{\\top\}z\}\_\{s\}\+\\underbrace\{\(\\mathbb\{I\}\_\{d\}\-ee^\{\\top\}\)z\}\_\{z\_\{\\perp\}\}\\ \.\(13\)Under either distribution, the random variablesS:=e⊤​ZS:=e^\{\\top\}ZandZ⟂Z\_\{\\perp\}are independent, and the orthogonal residual isZ⟂∼𝒩⁡\(0,𝕀d−e​e⊤\)Z\_\{\\perp\}\\sim\\mathcal\{N\}\(0,\\mathbb\{I\}\_\{d\}\-ee^\{\\top\}\)\.Only the scalar projection differs:S∼𝒩⁡\(0,1\)S\\sim\\mathcal\{N\}\(0,1\)underℙ\\mathbb\{P\}, andS∼𝒩⁡\(δ,1\)S\\sim\\mathcal\{N\}\(\\delta,1\)underℚ\\mathbb\{Q\}\. Consequently, given a proposal sampleY^∼ℙ\\widehat\{Y\}\\sim\\mathbb\{P\}, exact sampling fromℚ\\mathbb\{Q\}requires transforming only the scalar projectionS^\\widehat\{S\}from𝒩⁡\(0,1\)\\mathcal\{N\}\(0,1\)to𝒩⁡\(δ,1\)\\mathcal\{N\}\(\\delta,1\), while retaining the proposal’s orthogonal componentZ⟂Z\_\{\\perp\}unchanged:

S^:=e⊤​\(Y^−μp\)/σ,Z⟂=\(𝕀d−e​e⊤\)​\(Y^−μp\)/σ\.\\widehat\{S\}:=e^\{\\top\}\(\\widehat\{Y\}\-\\mu\_\{p\}\)/\\sigma,\\quad Z\_\{\\perp\}=\(\\mathbb\{I\}\_\{d\}\-ee^\{\\top\}\)\(\\widehat\{Y\}\-\\mu\_\{p\}\)/\\sigma\\ \.\(14\)Once a valid target scalarS∼𝒩⁡\(δ,1\)S\\sim\\mathcal\{N\}\(\\delta,1\)is obtained, the final state is reconstructed as

Y=μp\+σ⁡\(S​e\+Z⟂\)\.Y=\\mu\_\{p\}\+\\sigma\(Se\+Z\_\{\\perp\}\)\\ \.\(15\)One of the major distinctions between RMC and GRS lies in how they transformS^∼𝒩⁡\(0,1\)\\widehat\{S\}\\sim\\mathcal\{N\}\(0,1\)intoS∼𝒩⁡\(δ,1\)S\\sim\\mathcal\{N\}\(\\delta,1\)\.

### 3\.2 Reflection Maximal Coupling \(RMC\)

RMC\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\)instantiates the speculative sampling framework using achain topologyandreflection maximal couplingfor acceptance\. Specifically, the drafting phase sequentially generates a chain ofLLproposal states, whereLLis referred to as the*lookahead*\. During the acceptance phase, each proposal is maximally coupled with the corresponding target transition, thereby minimizing the probability of rejecting the drafted state\.

Algorithm 2Reflection maximal coupling \(RMC\)0:proposal sample

Y^∼ℙ=𝒩⁡\(μp,σ2​𝕀d\)\\widehat\{Y\}\\sim\\mathbb\{P\}=\\mathcal\{N\}\(\\mu\_\{p\},\\sigma^\{2\}\\mathbb\{I\}\_\{d\}\); target mean

μq\\mu\_\{q\}; shared scale

σ\\sigma
0:target sample

Y∼ℚ=𝒩⁡\(μq,σ2​𝕀d\)Y\\sim\\mathbb\{Q\}=\\mathcal\{N\}\(\\mu\_\{q\},\\sigma^\{2\}\\mathbb\{I\}\_\{d\}\); acceptance indicator

accepted∈\{true,false\}\\texttt\{accepted\}\\in\\\{\\texttt\{true\},\\texttt\{false\}\\\}
1:Obtain

\(S^,Z⟂,δ,e\)\(\\widehat\{S\},Z\_\{\\perp\},\\delta,e\)from \([12](https://arxiv.org/html/2609.17691#S3.E12)\) and \([14](https://arxiv.org/html/2609.17691#S3.E14)\)// rank\-1 projection

2:

R←ϕ⁡\(S^−δ\)ϕ⁡\(S^\)R\\leftarrow\\dfrac\{\\phi\(\\widehat\{S\}\-\\delta\)\}\{\\phi\(\\widehat\{S\}\)\}// target\-to\-proposal likelihood ratio

3:

U∼Uniform⁡\[0,1\]U\\sim\\mathrm\{Uniform\}\[0,1\]
4:

accepted←𝟙\{U≤1∧R\}\\texttt\{accepted\}\\leftarrow\\mathds\{1\}\\\{U\\leq 1\\wedge R\\\}
5:ifnotacceptedthen

6:

S^←δ−S^\\widehat\{S\}\\leftarrow\\delta\-\\widehat\{S\}// reflectS^\\widehat\{S\}aboutδ/2\\delta/2

7:endif

8:

Y←μp\+σ⁡\(S^​e\+Z⟂\)Y\\leftarrow\\mu\_\{p\}\+\\sigma\(\\widehat\{S\}e\+Z\_\{\\perp\}\)//Y∼ℚY\\sim\\mathbb\{Q\}

9:return

Y,accepted,1Y,\\;\\texttt\{accepted\},\\;1

Given a proposalY^\\widehat\{Y\}, RMC first decomposes it into a scalar proposalS^\\widehat\{S\}along the displacement direction and an orthogonal componentZ⟂Z\_\{\\perp\}using \([12](https://arxiv.org/html/2609.17691#S3.E12)\) and \([14](https://arxiv.org/html/2609.17691#S3.E14)\)\. The scalar proposal is accepted with probability equal to the target\-to\-proposal likelihood ratio, capped at one\. If accepted, RMC retainsS^\\widehat\{S\}and proceeds to the next proposal in the chain\. Otherwise, it draws an exact sample from the residual distribution and terminates the current speculative trajectory, after which a new lookahead chain is drafted\. For Gaussian proposal and target distributions with equal covariance, residual sampling admits a simple geometric implementation: the rejected scalar proposal is reflected about the midpoint between the proposal and target means\. The resulting scalar is then combined with the unchanged orthogonal component to reconstruct an exact sample from the target distribution\. The complete procedure is summarized in Algorithm[2](https://arxiv.org/html/2609.17691#alg2)\. Here,ϕ\\phidenotes the standard Gaussian density function\. In line88, the residual sample is obtained by reflectingS^\\widehat\{S\}about the midpointδ/2\\delta/2, yieldingS=δ−S^S=\\delta\-\\widehat\{S\}\.

Drawbacks of RMC:Despite its simplicity and exactness, RMC is fundamentally constrained by its chain topology\. Given a fixed parallel compute budget ofBBdraft evaluations, RMC allocates the entire budget to a single lookahead trajectory of lengthBB\(settingL=BL=B\)\. Consequently, if a proposal is rejected early in the acceptance phase, all subsequent drafted states become unusable and are discarded, resulting in underutilization of the available parallel compute budget\. This inefficiency becomes increasingly pronounced as the compute budget grows\. To better exploit parallelism, we instead propose a tree\-based draft topology that allocates the same budget across multiple speculative trajectories, as described next\.

### 3\.3 Diffusion Greedy Rejection Sampling \(D\-GRS\)

GRS departs fundamentally from RMC in both its drafting topology and, consequently, its acceptance criterion\.

Figure 2:Allocation of a fixed compute budget under chain\- and tree\-structured drafting forB=\|𝒱\|B=\|\\mathcal\{V\}\|\. Given a compute budget ofW=7W=7states, RMC constructs a single proposal chain\. If an early proposal is rejected, all downstream proposals become unusable and are discarded\. In contrast, D\-GRS constructs aKK\-ary draft tree, here withK=2K=2andL=2L=2\. At each level, D\-GRS may select among multiple candidate states, increasing the probability that at least one proposal can be accepted\.Topology:The chain employed by RMC represents only one possible allocation of a fixed parallel compute budget\. To exploit this budget more effectively, we introduce a more flexible topology based on*draft trees*, depicted in Figure[2](https://arxiv.org/html/2609.17691#S3.F2)\. AKK\-ary draft tree expands every non\-leaf node intoKKchildren, each representing a candidate realization of the next target state\. Suppose drafting begins at time stepnn, and letYnY\_\{n\}denote the current accepted target state, obtained either from the preceding draft tree or through residual sampling\. We treatYnY\_\{n\}as the root of the tree and drawKKconditionally independent and identically distributed \(i\.i\.d\.\) proposals from the proposal transition kernel at timen\+1n\+1\. Each resulting proposal is then recursively expanded intoKKchildren, yielding a tree withLLlevels, excluding the root\. DefiningWWas the number of target evaluations that fit in a single parallel batch, the pair\(K,L\)\(K,L\)must satisfy

B≤W,\\displaystyle B\\leq W,\(16\)withB=\|𝒱\|B=\|\\mathcal\{V\}\|orB=\|ℐ⁡\(𝒯\)\|B=\|\\mathcal\{I\}\(\\mathcal\{T\}\)\|, based on the verification mode\.

During drafting, every node at a given level generatesKKchildren using a proposal transition constructed from the target reverse drift already evaluated at the tree rootYnY\_\{n\}\. Since the target model has not yet been evaluated at the intermediate parent states, the same root drift is reused throughout the draft tree\. Conditional on a parent state, itsKKchildren are sampled independently from this delayed\-drift proposal kernel\. Once the full tree has been constructed, all drafted states are passed to the target model in parallel, yielding the parent\-specific reverse drifts required for verification\.

Algorithm 3Diffusion Greedy Rejection Sampling \(D\-GRS\)0:proposals

Y^1:K∼i\.i\.d\.ℙ=𝒩\(μp,σ2𝕀d\)\\widehat\{Y\}\_\{1:K\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathbb\{P\}=\\mathcal\{N\}\(\\mu\_\{p\},\\sigma^\{2\}\\mathbb\{I\}\_\{d\}\); target mean

μq\\mu\_\{q\}; shared scale

σ\\sigma
0:target sample

Y∼ℚ=𝒩⁡\(μq,σ2​𝕀d\)Y\\sim\\mathbb\{Q\}=\\mathcal\{N\}\(\\mu\_\{q\},\\sigma^\{2\}\\mathbb\{I\}\_\{d\}\); acceptance indicator

accepted∈\{true,false\}\\texttt\{accepted\}\\in\\\{\\texttt\{true\},\\texttt\{false\}\\\}
1:for

k=1k=1to

KKdo

2:Obtain

\(S^k,Z⟂,δ,e\)\(\\widehat\{S\}\_\{k\},Z\_\{\\perp\},\\delta,e\)from \([12](https://arxiv.org/html/2609.17691#S3.E12)\) and \([14](https://arxiv.org/html/2609.17691#S3.E14)\)// rank\-1 projection

3:endfor

4:

λ0←0\\lambda\_\{0\}\\leftarrow 0,

S1←1S\_\{1\}\\leftarrow 1
5:for

k=1k=1to

KKdo

6:

ρk←ϕ⁡\(S^k−δ\)ϕ⁡\(S^k\),βk←1∧\(ρk−λk−1\)\+Sk\\rho\_\{k\}\\leftarrow\\dfrac\{\\phi\(\\widehat\{S\}\_\{k\}\-\\delta\)\}\{\\phi\(\\widehat\{S\}\_\{k\}\)\},\\hskip 18\.49988pt\\beta\_\{k\}\\leftarrow 1\\wedge\\dfrac\{\(\\rho\_\{k\}\-\\lambda\_\{k\-1\}\)\_\{\+\}\}\{S\_\{k\}\}
7:

Uk∼Uniform⁡\[0,1\]U\_\{k\}\\sim\\mathrm\{Uniform\}\[0,1\]
8:if

Uk≤βkU\_\{k\}\\leq\\beta\_\{k\}then

9:

Y←μp\+σ⁡\(S^k​e\+Z⟂,k\)Y\\leftarrow\\mu\_\{p\}\+\\sigma\(\\widehat\{S\}\_\{k\}e\+Z\_\{\\perp,k\}\)
10:

accepted←true\\texttt\{accepted\}\\leftarrow\\texttt\{true\}
11:return

Y,accepted,kY,\\;\\texttt\{accepted\},\\;k
12:endif

13:

λk←λk−1\+Gk\\lambda\_\{k\}\\leftarrow\\lambda\_\{k\-1\}\+G\_\{k\}
14:

ℋk←\{s∈ℝ:ρ⁡\(s\)≥λk\}\\mathcal\{H\}\_\{k\}\\leftarrow\\\{s\\in\\mathbb\{R\}:\\rho\(s\)\\geq\\lambda\_\{k\}\\\}
15:

Gk\+1←ℚ⁡\(ℋk\)−λk​ℙ​\(ℋk\)G\_\{k\+1\}\\leftarrow\\mathbb\{Q\}\(\\mathcal\{H\}\_\{k\}\)\-\\lambda\_\{k\}\\mathbb\{P\}\(\\mathcal\{H\}\_\{k\}\)// remaining target mass

16:endfor

17:

S∼rK\+1S\\sim r\_\{K\+1\}defined in \([17](https://arxiv.org/html/2609.17691#S3.E17)\)

18:

Y←μp\+σ⁡\(S​e\+Z⟂,1\)Y\\leftarrow\\mu\_\{p\}\+\\sigma\(Se\+Z\_\{\\perp,1\}\)
19:

accepted←false\\texttt\{accepted\}\\leftarrow\\texttt\{false\}
20:return

Y,accepted,K\+1Y,\\;\\texttt\{accepted\},\\;K\+1

Acceptance:The introduction of a tree topology fundamentally changes the acceptance problem\. In contrast to RMC, where each proposed state is individually coupled with the target through a maximal coupling, tree\-based speculative sampling requires selecting one state from a*list of proposals*and coupling the selected state with the target\. This naturally gives rise to a*list\-coupling*problem\. More precisely, maximal coupling minimizes the Hamming cost𝟙\{Y^≠Y\}\\mathds\{1\}\\\{\\widehat\{Y\}\\neq Y\\\}between a single proposalY^∼ℙ\\widehat\{Y\}\\sim\\mathbb\{P\}and a target sampleY∼ℚY\\sim\\mathbb\{Q\}\. At each node of our draft tree, however, we are givenKKconditionally i\.i\.d\. proposalsY^1,…,Y^K∼ℙ\\widehat\{Y\}\_\{1\},\\ldots,\\widehat\{Y\}\_\{K\}\\sim\\mathbb\{P\}and seek a coupling that minimizes the membership cost𝟙\{Y∉\{Y^1,…,Y^K\}\}\\mathds\{1\}\\bigl\\\{Y\\notin\\\{\\widehat\{Y\}\_\{1\},\\ldots,\\widehat\{Y\}\_\{K\}\\\}\\bigr\\\}\. The optimal transport formulation of[Sun et al\. \(2023\)](https://arxiv.org/html/2609.17691#bib.bib3)characterizes the coupling that minimizes this cost\. However, computing the optimal transport plan requires solving a linear program over the target domain, whose complexity grows exponentially with the number of proposalsKK\. This is already costly for finite discrete state spaces and does not admit a tractable direct extension to continuous diffusion transitions\. Computationally efficient approaches to list coupling have been studied in the context of language models; we provide a survey in the appendix\. These methods, however, do not remain computationally tractable for continuous diffusion models\.

As a workaround, we draw inspiration from relative entropy coding \(REC\)\([Li, 2024](https://arxiv.org/html/2609.17691#bib.bib26);[Flamich and Gündüz, 2026](https://arxiv.org/html/2609.17691#bib.bib34)\), which is closely connected to speculative sampling, as elaborated below\. REC is a class of algorithms that tackles a*remote sampling*problem: a sender, which has access to a target distributionℚ\\mathbb\{Q\}and a proposal distributionℙ\\mathbb\{P\}, communicates an index to a receiver, which sharesℙ\\mathbb\{P\}, allowing the receiver to reproduce an*exact sample*fromℚ\\mathbb\{Q\}\. The fundamental communication cost of the index is governed, up to lower\-order terms, by the relative entropyDKL\(ℚ∥ℙ\)D\_\{\\mathrm\{KL\}\}\(\\mathbb\{Q\}\\\|\\mathbb\{P\}\), hence the name REC\. Correspondingly, REC implementations, which can be based on Poisson processes\([Li and El Gamal, 2018](https://arxiv.org/html/2609.17691#bib.bib7);[Liu et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib8)\), dithered quantization\([Zamir and Feder, 1992](https://arxiv.org/html/2609.17691#bib.bib35)\), or rejection sampling\([Harsha et al\., 2010](https://arxiv.org/html/2609.17691#bib.bib9);[Flamich and Theis, 2023](https://arxiv.org/html/2609.17691#bib.bib10);[Flamich, 2023](https://arxiv.org/html/2609.17691#bib.bib11)\), need to examineΩ~\(exp\(DKL\(ℚ∥ℙ\)\)\)\\tilde\{\\Omega\}\(\\exp\(D\_\{\\mathrm\{KL\}\}\(\\mathbb\{Q\}\\\|\\mathbb\{P\}\)\)\)shared proposals before identifying the communicated sample\. This viewpoint parallels speculative sampling: both problems seek to couple proposal samples with an exact target sample while maximizing the probability that the target can be recovered from the available proposals\. While RMC optimizes the single\-proposal overlap, measured by the total variation distance, REC algorithms construct exact couplings by processing anordered sequenceof proposal samples\. Motivated by this connection, we replace the unrestricted list\-coupling problem with an order\-preserving*sequence\-coupling*problem\. In list coupling, theKKproposals may be jointly matched to the target without regard to their ordering; in sequence coupling, proposals are examined in their sampled order, and the first proposal satisfying the acceptance rule is selected\.

We then adapt the GRS procedure underlying the communication protocol of[Harsha et al\. \(2010\)](https://arxiv.org/html/2609.17691#bib.bib9)to continuous diffusion transitions and embed it within our draft\-tree topology\. At each node, GRS sequentially examines theKKverified children, greedily assigns as much remaining target mass as possible to each proposal, and accepts the first proposal selected by the resulting coupling\. If none of theKKchildren is accepted, the algorithm samples from the remaining target residual and terminates the current speculative path\. More specifically, GRS maintains a sequence of scalar levels\{λk\}k=0K\\\{\\lambda\_\{k\}\\\}\_\{k=0\}^\{K\}, together with residual masses\{Gk\}k=1K\+1\\\{G\_\{k\}\\\}\_\{k=1\}^\{K\+1\}\. The levelλk\\lambda\_\{k\}records how much of the target\-to\-proposal likelihood ratio has already been assigned to the firstkkproposals, whileGk\+1G\_\{k\+1\}denotes the target mass that remains unassigned after roundkk\. Initially, no target mass has been assigned, soλ0=0\\lambda\_\{0\}=0andG1=1G\_\{1\}=1\. Letρ​\(s\):=d​ℚd​ℙ​\(s\)=ϕ⁡\(s−δ\)ϕ⁡\(s\)\\rho\(s\):=\\frac\{\\mathrm\{d\}\\mathbb\{Q\}\}\{\\mathrm\{d\}\\mathbb\{P\}\}\(s\)=\\frac\{\\phi\(s\-\\delta\)\}\{\\phi\(s\)\}denote the target\-to\-proposal likelihood ratio in the one\-dimensional coordinate\. At roundkk, GRS considers only the portion ofρ⁡\(S^k\)\\rho\(\\widehat\{S\}\_\{k\}\)that remains above the previously assigned levelλk−1\\lambda\_\{k\-1\}\. The proposal is therefore accepted with probability

βk:=1∧\(ρ⁡\(S^k\)−λk−1\)\+Sk\.\\beta\_\{k\}:=1\\wedge\\frac\{\\bigl\(\\rho\(\\widehat\{S\}\_\{k\}\)\-\\lambda\_\{k\-1\}\\bigr\)\_\{\+\}\}\{S\_\{k\}\}\\ \.This greedily assigns the largest admissible slice of the remaining target mass to the current proposal while ensuring that the acceptance probability does not exceed one\. If the proposal is rejected, the assigned level is increased toλk=λk−1\+Gk\\lambda\_\{k\}=\\lambda\_\{k\-1\}\+G\_\{k\}, which induces the super\-level set

ℋk=\{s∈ℝ:ρ⁡\(s\)≥λk\}\.\\mathcal\{H\}\_\{k\}=\\\{s\\in\\mathbb\{R\}:\\rho\(s\)\\geq\\lambda\_\{k\}\\\}\\ \.The remaining target mass is thenGk\+1=ℚ⁡\(ℋk\)−λk​ℙ​\(ℋk\)G\_\{k\+1\}=\\mathbb\{Q\}\(\\mathcal\{H\}\_\{k\}\)\-\\lambda\_\{k\}\\mathbb\{P\}\(\\mathcal\{H\}\_\{k\}\)\. AfterKKunsuccessful rounds, GRS samples from the normalized residual measure with density

rK\+1​\(s\)=\(q⁡\(s\)−λK​p​\(s\)\)\+GK\+1,\\displaystyle r\_\{K\+1\}\(s\)=\\frac\{\\bigl\(q\(s\)\-\\lambda\_\{K\}p\(s\)\\bigr\)\_\{\+\}\}\{G\_\{K\+1\}\}\\ ,\(17\)thereby preserving exactness\. The D\-GRS procedure is summarized in Algorithm[3](https://arxiv.org/html/2609.17691#alg3)\.

## 4 Performance Analysis

In this section, we provide performance analyses of the proposed D\-GRS algorithm, and compare it to the RMC baseline\. Specifically, we show theexactnessof D\-GRS and compute itsacceptance probability\. The complete proofs are deferred to Appendix[B](https://arxiv.org/html/2609.17691#A2)\.

###### Theorem 1\(Exactness\)\.

The outputYYof Algorithm[3](https://arxiv.org/html/2609.17691#alg3)with target distributionℚ=𝒩⁡\(μq,σ2​𝕀d\)\\mathbb\{Q\}=\\mathcal\{N\}\(\\mu\_\{q\};\\sigma^\{2\}\\mathbb\{I\}\_\{d\}\)satisfiesY∼ℚY\\sim\\mathbb\{Q\}\.

The proof follows from the residual\-measure characterization of GRS\([Flamich and Theis, 2023](https://arxiv.org/html/2609.17691#bib.bib10), Lemmas III\.1\-2\): one can explicitly calculate the target mass generated by the firstKKproposal branches, and the amount contributed by a sample from its normalized residual distribution\. The proposal and residual branches together recover the target distributionℚ\\mathbb\{Q\}exactly\. Next, we explicitly characterize the probability that Algorithm[3](https://arxiv.org/html/2609.17691#alg3)accepts at least one drafted child\.

###### Theorem 2\(Acceptance Probability\)\.

Denoteδ:=‖mq−mp‖2σ\\delta:=\\frac\{\\left\\\|m^\{q\}\-m^\{p\}\\right\\\|\_\{2\}\}\{\\sigma\}\. For Algorithm[3](https://arxiv.org/html/2609.17691#alg3), the probability of accepting at least one of theKKproposals is given by

ℙ\(∃k∈\{1,⋯,K\}:Y=Y^k\)=1−GK\+1\.\\mathbb\{P\}\(\\exists k\\in\\\{1,\\cdots,K\\\}:Y=\\widehat\{Y\}\_\{k\}\)=1\-G\_\{K\+1\}\\ \.\(18\)Furthermore, for anyδ∈ℝ\+\\delta\\in\\mathbb\{R\}\_\{\+\}, we have

GK\+1=Φ¯​\(1δ​ln⁡λK−δ2\)−λK​Φ¯​\(1δ​ln⁡λK\+δ2\),\\displaystyle G\_\{K\+1\}=\\overline\{\\Phi\}\\left\(\\frac\{1\}\{\\delta\}\\ln\\lambda\_\{K\}\-\\frac\{\\delta\}\{2\}\\right\)\-\\lambda\_\{K\}\\overline\{\\Phi\}\\left\(\\frac\{1\}\{\\delta\}\\ln\\lambda\_\{K\}\+\\frac\{\\delta\}\{2\}\\right\),\(19\)whereΦ¯​\(x\)\\overline\{\\Phi\}\(x\)is the standard Gaussian survival function\.

Note that settingK=1K=1andδ∈ℝ\+\\delta\\in\\mathbb\{R\}\_\{\+\}, we obtain

ℙ\(∃k∈\{1,⋯,K\}:Y=Y^k\)=2Φ¯\(δ2\),\\mathbb\{P\}\(\\exists k\\in\\\{1,\\cdots,K\\\}:Y=\\widehat\{Y\}\_\{k\}\)=2\\overline\{\\Phi\}\\left\(\\frac\{\\delta\}\{2\}\\right\)\\ ,\(20\)theoptimalacceptance probability for maximal coupling which is achieved by RMC\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1), Proposition3\.13\.1\)\. The per\-step acceptance probabilities for different value ofKKare plotted in Figure[6](https://arxiv.org/html/2609.17691#A2.F6)in Appendix[B\.2](https://arxiv.org/html/2609.17691#A2.SS2)\.

## 5 Experiments

We evaluate the performance of D\-GRS relative to RMC across three generative tasks: \(i\) diffusion sampling from a Gaussian mixture target utilizing analytically tractable scores; \(ii\) pixel\-space generation, including conditional sampling on CIFAR\-10 \(32×3232\\times 32\)\([Krizhevsky and Hinton, 2009](https://arxiv.org/html/2609.17691#bib.bib36)\)and unconditional sampling on FFHQ \(64×6464\\times 64\)\([Karras et al\., 2019](https://arxiv.org/html/2609.17691#bib.bib37)\)datasets; and \(iii\) text\-to\-image latent space sampling via Stable Diffusion 3 \(SD3\)\([Esser et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib38)\)\.

##### Protocol\.

All methods are evaluated under the*self\-speculative*regime of\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\)forB=\|𝒱\|B=\|\\mathcal\{V\}\|\. Specifically, the draft is constructed directly from the target denoiser by freezing the velocity field computed at the root of the speculation tree \(or the window head for RMC\) and reusing it throughout the draft trajectory\. Sampling follows the Euler\-Maruyama discretization of the linear probability pathxσ=\(1−σ\)​x0\+σ​ξx\_\{\\sigma\}=\(1\-\\sigma\)x\_\{0\}\+\\sigma\\,\\xi,ξ∼𝒩⁡\(0,𝕀d\)\\xi\\sim\\mathcal\{N\}\(0,\\mathbb\{I\}\_\{d\}\), with stochasticity level \(churn\)ε∈\(0,1\]\\varepsilon\\in\(0,1\]111ε=0\\varepsilon=0recovers the deterministic flow sampler, rendering speculation vacuous as the acceptance probability drops to zero\.\.

##### Cost metric and matched budgets\.

Following\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\)we measure the computational cost in NFEs\. To compare D\-GRS and RMC at equal cost per target call, we evaluate every D\-GRS configuration\(K,L\)\(K,L\)against an RMC chain of lengthBB\. Both methods therefore process an identical verification batch of sizeBBper round, differing only in structural allocation\. Consequently, the proposal budgetBBserves as the naturalxx\-axis throughout our comparisons\. All costs are reported as expected NFE per trajectory/image\.

### 5\.1 Gaussian Mixture Target

##### Setting\.

We reproduce the low\-dimensional setting of\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1), App\. I\.2\)\. The target is a mixture of55isotropic Gaussians inℝd,d=500\\mathbb\{R\}^\{d\},d=500, withμi∼𝒰⁡\(\[−2,2\]d\)\\mu\_\{i\}\\sim\\mathcal\{U\}\\\!\\left\(\[\-2,2\]^\{d\}\\right\)andσi∼𝒰⁡\(\[0\.10,0\.25\]\)\\sigma\_\{i\}\\sim\\mathcal\{U\}\\\!\\left\(\[0\.10,\\,0\.25\]\\right\)drawn once from a fixed seed and then held fixed across all runs\. We use the*exact*velocity field which, under the linear probability path, is available in closed form\. We useT=30T=30generation steps with stochasticityε=0\.1\\varepsilon=0\.1\. We sweep the full gridL∈\{2,…,7\}×K∈\{1,…,7\}L\\in\\\{2,\\dots,7\\\}\\times K\\in\\\{1,\\dots,7\\\}, and run100100independent trajectories per entry\. Speedups are reported relative to the reference cost of the standard sampler,\(K,L\)=\(1,1\)\(K,L\)=\(1,1\), which spends3030target calls per trajectory\.

##### Results\.

Figure[3](https://arxiv.org/html/2609.17691#S5.F3)shows the expected speedups across the\(K,L\)\(K,L\)grid and corresponding budgetsBB\. RMC speedup plateaus \(≈1\.96×\\approx 1\.96\\times–2\.00×2\.00\\times\) across six orders of magnitude inBB, bottlenecked by the geometric decay of its linear sequential look\-ahead\. Conversely, D\-GRS allocates budget toKKproposals per node to boost per\-step acceptance, avoiding the rapid saturation of look\-ahead scaling\. ForL≥3L\\geq 3, D\-GRS strictly dominates, yielding a2\.33×2\.33\\timesspeedup at\(6,6\)\(6,6\)compared to RMC’s1\.99×1\.99\\times\. Overall, D\-GRS reduces target calls by 8–9% \(2\.14×2\.14\\timesvs\.1\.96×1\.96\\timesatB=155B=155\) without compromising exactness\.

![Refer to caption](https://arxiv.org/html/2609.17691v1/gm_heatmap_calls.png)Figure 3:Expected speedup in the Gaussian mixture setting over100100independent trajectories per\(K,L\)\(K,L\)pairs\. Performance for RMC \(left\) remains uniform across the grid, whereas D\-GRS \(right\) exhibits steady improvement asBBis increasingly allocated to widthKK\.

### 5\.2 Pixel\-Space Diffusion

For both conditional and unconditional diffusion sampling over the pixel\-space, we adopt pretrained denoisers provided by[Karras et al\. \(2022\)](https://arxiv.org/html/2609.17691#bib.bib21)\. At time steptt, the modelDθ​\(xt,ς\)=𝔼⁡\[x0∣xt=x0\+ς​ξ\]D\_\{\\theta\}\(x\_\{t\};\\varsigma\)=\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}=x\_\{0\}\+\\varsigma\\xi\]estimates the uncorrupted datax0x\_\{0\}from a noisy observationxtx\_\{t\}, perturbed by standard Gaussian noiseξ\\xiat a noise scaleς\\varsigma\. To integrate this denoiser into our setting, we recast it as a velocity fieldvθ​\(xt,t\)v\_\{\\theta\}\(x\_\{t\},t\)along a linear interpolant\. By applyingς=t/\(1−t\)\\varsigma=t/\(1\-t\), the velocity field evaluates tovθ​\(xt,t\)=1t​xt−Dθ​\(xt1−t,t1−t\)v\_\{\\theta\}\(x\_\{t\},t\)=\\frac\{1\}\{t\}x\_\{t\}\-D\_\{\\theta\}\\left\(\\frac\{x\_\{t\}\}\{1\-t\};\\frac\{t\}\{1\-t\}\\right\)\. Finally, to ensure numerical stability during generation, the evaluated noise scale is clipped toς∈\[10−3,80\]\\varsigma\\in\[10^\{\-3\},80\], which remains strictly within the original model’s native sampling range\.

Figure 4:NFE speed\-up relative to proposal budgetBBfor CIFAR\-10 \(top\) and FFHQ \(bottom\) at varying stochasticityϵ∈\{0\.1,0\.3,0\.6\}\\epsilon\\in\\\{0\.1,0\.3,0\.6\\\}\.##### Experimental Setup\.

Each sampler generates100100images over100100time steps\. For CIFAR\-10, the generation is conditioned on labels sampled uniformly at random from the ten available classes\. We evaluate four configuration pairs\(K,L\)∈\{\(2,3\),\(2,5\),\(3,4\),\(4,4\)\}\(K,L\)\\in\\\{\(2,3\),\\allowbreak\(2,5\),\\allowbreak\(3,4\),\\allowbreak\(4,4\)\\\}, which correspond to computational budgetsB∈\{14,62,120,340\}B\\in\\\{14,62,120,340\\\}, with RMC executed on the matched windows\(1,B\)\(1,B\)\. This procedure is repeated across three stochasticity levels,ε∈\{0\.1,0\.3,0\.6\}\\varepsilon\\in\\\{0\.1,0\.3,0\.6\\\}\. Figure[4](https://arxiv.org/html/2609.17691#S5.F4)illustrates the resulting mean and standard deviation of the speedups relative to the baseline standard sampler, which incurs a fixed cost of100100NFEs per image\.

##### Results\.

In Figure[4](https://arxiv.org/html/2609.17691#S5.F4), we report the NFE speedups per image against the computational budgetBBfor two datasets with distinct spatial resolutions: CIFAR\-10 \(32×3232\\times 32, top row\) and FFHQ \(64×6464\\times 64, bottom row\)\. Across both tasks, increasing the churn level degrades the sampling efficiency of both methods\. Specifically, higher stochasticity increases the divergence between the draft and target transition kernels, widening the gap between their respective means, thereby attenuating the sampling acceptance rates across both coupling mechanisms\.

On CIFAR\-10, RMC exhibits a marginal empirical advantage at the minimal budget \(B=14B=14\), where its linear sequential topology achieves greater effective depth compared to the shallow branching of a tree\. However, forB≥62B\\geq 62the efficiency trend reverses: RMC suffers from plateaued scaling, yielding a fixed speedup irrespective ofBB, whereas D\-GRS efficiently leverages its speculative tree topology to maintain monotonically increasing speedups\.

On the higher\-dimensional FFHQ benchmark, absolute speedups are inherently reduced \(≈2\.0\\approx 2\.0–2\.2×2\.2\\times, down from2\.22\.2–2\.6×2\.6\\timeson CIFAR\-10\), as the mean discrepancy between the draft and target kernels scales withd\\sqrt\{d\}\. Crucially, the relative advantage of D\-GRS becomes absolute in this higher dimensional regime, strictly dominating RMC at*every*tested budget and churn level\. To empirically verify exactness, we evaluate FID on 50,000 samples generated withK=2K=2,L=3L=3, andε=0\.3\\varepsilon=0\.3\. On CIFAR\-10, D\-GRS and RMC achieve FID scores of 1\.824 and 1\.822, respectively, closely tracking the 1\.79 baseline from[Karras et al\. \(2022\)](https://arxiv.org/html/2609.17691#bib.bib21)\. On FFHQ, D\-GRS and RMC yield respective scores of 2\.86 and 2\.88, performing comparably to, though slightly above, the 2\.39 baseline\([Karras et al\., 2022](https://arxiv.org/html/2609.17691#bib.bib21)\)\.

Figure 5:Expected NFE speed\-up over the vanilla target sampler as a function ofKK\(512×512512\\times 512resolution,ε=0\.8\\varepsilon=0\.8,L=2L=2\)\.Table 1:CLIP scores \(mean±\\pmstandard error of the mean overn=100n=100prompts\) forL=2L=2, comparing D\-GRS and RMC across differentKKvalues against the target baseline\.

### 5\.3 High\-dimensional Latent Space Diffusion

Stable Diffusion 3\.5 \(SD3\)222The model is publicly available at Huggingface asstabilityai/stable\-diffusion\-3\.5\-medium\.is natively trained via rectified flow\. It therefore relies on ODE integration, with transitions being deterministic point masses\. Consequently, draft and target proposals exhibit a total variation distance of one, rendering speculation vacuous\. We resolve this by sampling from a marginal\-preserving SDE family along the identical linear probability path, requiring no retraining\. The score is computed directly from the velocity assθ=−\(xσ\+\(1−σ\)vθ\)/σs\_\{\\theta\}=\-\(x\_\{\\sigma\}\+\(1\-\\sigma\)v\_\{\\theta\}\)/\\sigma\. Adding the drift−12​ε2​g2​sθ\-\\frac\{1\}\{2\}\\varepsilon^\{2\}g^\{2\}s\_\{\\theta\}\(whereg2​\(σ\)=2​σ/\(1−σ\)g^\{2\}\(\\sigma\)=2\\sigma/\(1\-\\sigma\)\) and noise of scaleε​g\\varepsilon gensures every marginalpσp\_\{\\sigma\}remains invariant\. Under an Euler–Maruyama discretization, each step yields a Gaussian kernel𝒩⁡\(m,ς2​I\)\\mathcal\{N\}\(m,\\varsigma^\{2\}I\)\. Since the standard deviationς=ε​g​−γ\\varsigma=\\varepsilon g\\sqrt\{\-\\gamma\}is purely a function of the time schedule, draft and target kernels diverge*only by their means*—the exact mathematical structure required by our coupling framework\. A detailed derivation is deferred to Appendix[C\.3](https://arxiv.org/html/2609.17691#A3.SS3)\.

##### Experimental Setup\.

We evaluate high\-resolution generation at512×512512\\times 512, utilizingT=50T=50generation steps, a free\-guidance scale of7\.07\.0, and a stochasticity level ofε=0\.8\\varepsilon=0\.8\. To adapt the native resolution to ours, we adopt a timestep shift of3\.03\.0\. Throughout the simulations, we fix the lookahead atL=2L=2and sweeping the branching factorK∈\{2,3,4,5,6,7\}K\\in\\\{2,3,4,5,6,7\\\}, yielding budgetsB∈\{6,12,20,30,42,56\}B\\in\\\{6,12,20,30,42,56\\\}\. Generative sampling is executed within the VAE latent space of dimensiond=16×64×64=65,536d=16\\times 64\\times 64=65\{,\}536\. All configurations generate100100images conditioned on an identical, pre\-sampled set of100100prompts drawn uniformly without replacement from the COCO 2014 validation set\([Lin et al\., 2014](https://arxiv.org/html/2609.17691#bib.bib39)\)\.

##### Results\.

Figure[5](https://arxiv.org/html/2609.17691#S5.F5)illustrates the empirical speed\-up over the standard target sampler as a function ofKKand a constrained lookahead depth ofL=2L=2\. As the number of proposals per node \(KK\) scales from22to77, the corresponding computational budgetBBexpands from66to5656\. Consistent with prior topological analyses, the proposed method effectively translates this expanded budget into measurable acceleration\. By allocating compute towardKK, D\-GRS yields a monotonically increasing speedup that peaks near1\.18×1\.18\\timesatK=6K=6\(B=42B=42\), with an increment of6\.7%6\.7\\%relative to RMC\. In contrast, RMC fails to meaningfully exploit the additional budget\. Bottlenecked by the geometric decay of its sequential acceptance, the reflection sampler exhibits severe saturation, plateauing between1\.10×1\.10\\timesand1\.12×1\.12\\timesacross the entire parameter sweep\. This widening performance gap clearly demonstrates that scaling multiple proposals per node \(KK\) successfully improves per\-step acceptance rates, allowing the branching topology to strictly dominate the linear chain in high\-dimensional text\-to\-image synthesis\. To empirically show the exactness of both sampling procedures, we report in Table[1](https://arxiv.org/html/2609.17691#S5.T1)the CLIP scores\([Radford et al\., 2021](https://arxiv.org/html/2609.17691#bib.bib42)\)for each budget configuration, computed usingopenai/clip\-vit\-large\-patch14\.

## 6 Conclusions and Future Work

We identify a key topological limitation of reflection maximal coupling \(RMC\), which allocates its speculative budget to a single draft chain\. To better utilize parallel computation, we introduce draft trees and propose D\-GRS, which adapts greedy rejection sampling from the relative entropy coding literature to the resulting list\-coupling problem\. Across our experiments, D\-GRS improves NFE speed\-up over RMC by up to8\.3%8\.3\\%in practical settings\.

This work is a first step toward more efficient speculative diffusion sampling\. Important open directions include overcoming the rapid degradation of exact coupling efficiency with generation dimension and adapting the draft topology online to the proposal quality and available compute budget\.

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## Appendix ALiterature Review

### A\.1 Diffusion Models

Diffusion models implicitly learn data distributions from empirical samples\. The generative process is initialized with a sample from a tractable prior, typically a Gaussian distribution, and iteratively denoises it to recover a sample from the target data distribution\([Lai et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib40)\)\. This mechanism is formalized by defining a fixed forward process that progressively corrupts the data with noise, coupled with a parameterized neural network trained to reverse this corruption\. Several theoretical frameworks formulate this concept, including the Denoising Diffusion Probabilistic Model \(DDPM\)\([Sohl\-Dickstein et al\., 2015](https://arxiv.org/html/2609.17691#bib.bib5);[Ho et al\., 2020](https://arxiv.org/html/2609.17691#bib.bib4)\)and noise\-conditioned score networks\([Song and Ermon, 2019](https://arxiv.org/html/2609.17691#bib.bib68)\)\. These were subsequently unified as discretizations of a single continuous\-time stochastic differential equation \(SDE\)\([Song et al\., 2021b](https://arxiv.org/html/2609.17691#bib.bib6)\), alongside the more recent flow matching paradigm\([Lipman et al\., 2022](https://arxiv.org/html/2609.17691#bib.bib69);[Liu et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib70)\)\. Fundamentally, these formulations are mathematically equivalent up to a transformation of the model’s output parameterization and noise schedule\([Kingma et al\., 2021](https://arxiv.org/html/2609.17691#bib.bib71);[Lai et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib40)\)\. A critical practical consequence of this equivalence is that the training objective does not uniquely dictate the sampling algorithm: models trained via denoising can be sampled deterministically using a probability flow ordinary differential equation \(ODE\)\([Song et al\., 2021b](https://arxiv.org/html/2609.17691#bib.bib6)\), while models trained under a flow matching objective can employ stochastic sampling schemes\([Liu et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib70);[Hu et al\., 2025a](https://arxiv.org/html/2609.17691#bib.bib72)\)\. These generative frameworks have established state\-of\-the\-art performance across a diverse array of modalities, encompassing image synthesis\([Dhariwal and Nichol, 2021](https://arxiv.org/html/2609.17691#bib.bib73);[Rombach et al\., 2022](https://arxiv.org/html/2609.17691#bib.bib74);[Karras et al\., 2022](https://arxiv.org/html/2609.17691#bib.bib21)\), video generation\([Ho et al\., 2022](https://arxiv.org/html/2609.17691#bib.bib75);[Bar\-Tal et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib84)\), protein design\([Watson et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib76)\), prediction of biomolecular structure\([Abramson et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib77)\), and numerous other applications\([Yang et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib41)\)\.

Despite their empirical success, a primary limitation of diffusion models is the iterative nature of the sampling process\. Efforts to accelerate inference by reducing the number of integration steps have been extensively investigated and generally branch into training\-free and training\-based methodologies\([Dieleman, 2024](https://arxiv.org/html/2609.17691#bib.bib25);[Dieleman, 2026](https://arxiv.org/html/2609.17691#bib.bib50);[Lai et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib40)\)\. Training\-free approaches leverage the fixed, pre\-trained model and achieve acceleration through the deployment of advanced ODE and SDE solvers that allow for larger step sizes without compromising accuracy\([Song et al\., 2021a](https://arxiv.org/html/2609.17691#bib.bib78);[Zhang and Chen, 2023](https://arxiv.org/html/2609.17691#bib.bib53);[Lu et al\., 2022](https://arxiv.org/html/2609.17691#bib.bib22);[Lu et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib43)\)\. Conversely, training\-based methods explicitly optimize a model to execute large synthesis steps\. These approaches vary along two primary axes\. The first is the source of supervision: they may distill knowledge from a pretrained teacher model\([Luhman and Luhman, 2021](https://arxiv.org/html/2609.17691#bib.bib79);[Salimans and Ho, 2022](https://arxiv.org/html/2609.17691#bib.bib80);[Song et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib44);[Sabour et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib47)\)or train a few\-step model entirely from scratch\([Song et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib44);[Frans et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib81);[Geng et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib29);[Boffi et al\., 2025a](https://arxiv.org/html/2609.17691#bib.bib27)\)\. The second axis concerns the matching objective: certain techniques enforce the preservation of trajectories connecting noise to data\([Salimans and Ho, 2022](https://arxiv.org/html/2609.17691#bib.bib80);[Song et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib44);[Boffi et al\., 2025b](https://arxiv.org/html/2609.17691#bib.bib82)\), whereas others constrain the final model output to match the target marginal distribution\([Yin et al\., 2024b](https://arxiv.org/html/2609.17691#bib.bib83);[Yin et al\., 2024a](https://arxiv.org/html/2609.17691#bib.bib48);[Sauer et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib49)\)\. While initially developed for ODE trajectories, these acceleration techniques have recently been generalized to SDEs\. This includes methods that learn transition kernels to replicate the process’s marginal distributions\([Potaptchik et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib30);[Holderrieth et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib31)\), as well as those that learn the solution map pathwise, conditioned on specific realizations of the driving noise\([McCallum et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib28)\)\.

### A\.2 Speculative Sampling for Language Models

Standard autoregressive decoding approaches, such as top\-kksampling\([Radford et al\., 2019](https://arxiv.org/html/2609.17691#bib.bib12);[Fan et al\., 2018](https://arxiv.org/html/2609.17691#bib.bib13)\), nucleus sampling\([Holtzman et al\., 2020](https://arxiv.org/html/2609.17691#bib.bib15)\), and the permute\-and\-flip decoder\([Zhao et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib14)\), generate tokens one at a time and therefore incur the inherent latency of one target\-model call for each token\. Speculative sampling\([Chen et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib17);[Leviathan et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib16)\)alleviates this bottleneck by employing a smaller draft model to propose multiple future tokens, while the larger target model verifies these proposed continuations in parallel, with a rejection\-correction step ensuring that the exact target\-model distribution is preserved\. The resulting efficiency gain depends on how closely the draft distribution approximates the target distribution: more accurate proposals yield higher acceptance rates and longer verified blocks\. Recent works have further advanced this paradigm through optimal\-transport\-based token selection\([Sun et al\., 2023](https://arxiv.org/html/2609.17691#bib.bib3)\), tree\-based inference and verification\([Miao et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib18)\), and multi\-draft architectures\([Khisti et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib19);[Hu et al\., 2025b](https://arxiv.org/html/2609.17691#bib.bib32);[Lin et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib33)\)\. Multi\-draft speculative decoding can be viewed as an application of list\-coupling schemes\([Rowan et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib20)\), which strictly extend ordinary coupling for speculative decoding\([Daliri et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib54)\)\. Recently, speculative sampling has been used to accelerating the diffusion modesl\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1);[Soen et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib2)\)\.

### A\.3 Relative Entropy Coding

The sequential coupling methodology introduced here for speculative diffusion is theoretically grounded in greedy rejection sampling\([Harsha et al\., 2010](https://arxiv.org/html/2609.17691#bib.bib9)\)\. This technique falls within a broader family of remote coupling methods identified by various terms in the literature, including channel simulation\([Li, 2024](https://arxiv.org/html/2609.17691#bib.bib26)\), channel synthesis\([Cuff, 2013](https://arxiv.org/html/2609.17691#bib.bib55)\), and reverse channel coding\([Bennett et al\., 2002](https://arxiv.org/html/2609.17691#bib.bib56)\)\. Hereafter, we refer to this class of schemes as relative entropy coding \(REC\)\([Flamich and Gündüz, 2026](https://arxiv.org/html/2609.17691#bib.bib34)\)\. Fundamentally, REC is formulated as a lossy compression problem designed to establish the minimal communication overhead required to simulate a target stochastic channel over a noiseless link\. The nomenclature derives from the theoretical result that, in the regime of exact simulation, the fundamental lower bound on this communication cost is governed by a relative entropy measure\.

In this work, we employ greedy rejection sampling\([Harsha et al\., 2010](https://arxiv.org/html/2609.17691#bib.bib9)\), a methodology whose capabilities have been significantly expanded in recent literature\([Flamich and Theis, 2023](https://arxiv.org/html/2609.17691#bib.bib10);[Flamich, 2023](https://arxiv.org/html/2609.17691#bib.bib11);[Hill et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib57)\)\. A parallel and prominent class of REC frameworks relies on Poisson processes\. Rooted in the Poisson functional representation\([Li and El Gamal, 2018](https://arxiv.org/html/2609.17691#bib.bib7)\)\-a concept intimately connected toA∗\\mathrm\{A\}^\{\*\}sampling\([Maddison et al\., 2014](https://arxiv.org/html/2609.17691#bib.bib58);[Maddison, 2016](https://arxiv.org/html/2609.17691#bib.bib59)\)\-these schemes have found broad application across diverse domains\. Notable examples include hypothesis testing\([Guo et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib62)\), network information theory\([Li and Anantharam, 2021](https://arxiv.org/html/2609.17691#bib.bib64);[Liu and Li, 2025](https://arxiv.org/html/2609.17691#bib.bib60);[Liu et al\., 2026](https://arxiv.org/html/2609.17691#bib.bib61)\), and machine learning\([He et al\., 2024](https://arxiv.org/html/2609.17691#bib.bib63)\)\. Related methodologies for approximate channel simulation have been proposed by[Block and Polyanskiy \(2023\)](https://arxiv.org/html/2609.17691#bib.bib66);[Kobus et al\. \(2024\)](https://arxiv.org/html/2609.17691#bib.bib67);[Flamich and Wells \(2024\)](https://arxiv.org/html/2609.17691#bib.bib65)\. Nevertheless, since these approaches yield inexact simulations and therefore fall short of the theoretical relative entropy limit, they are precluded from the standard definition of REC\.

## Appendix BOn Performance Analysis

In this section, we provide proofs of the theoretical guarantees discussed in Section[4](https://arxiv.org/html/2609.17691#S4)\. For completeness, we restate the theorems before presenting their proofs\.

Throughout this sectionℙ=𝒩⁡\(μp,σ2​Id\)\\mathbb\{P\}=\\mathcal\{N\}\(\\mu\_\{p\},\\sigma^\{2\}I\_\{d\}\)andℚ=𝒩⁡\(μq,σ2​Id\)\\mathbb\{Q\}=\\mathcal\{N\}\(\\mu\_\{q\},\\sigma^\{2\}I\_\{d\}\)denote the proposal and target distributions onΩ=ℝd\\Omega=\\mathbb\{R\}^\{d\}\. Furthermore, letJJdenote the index of the proposal accepted by Algorithm[3](https://arxiv.org/html/2609.17691#alg3), so thatY=Y^JY=\\widehat\{Y\}\_\{J\}\. Let also\(λj\)j≥0\(\\lambda\_\{j\}\)\_\{j\\geq 0\}be an increasing sequence of levels that define superlevel sets of the density ratioρ=⁡d​ℚ⁡d​ℙ\\rho=\\frac\{\\mathop\{\}\\\!\\mathrm\{d\}\\mathbb\{Q\}\}\{\\mathop\{\}\\\!\\mathrm\{d\}\\mathbb\{P\}\}, i\.e\.,

ℋj:=\{y∈Ω:ρ⁡\(y\)≥λj\}\.\\mathcal\{H\}\_\{j\}:=\\left\\\{y\\in\\Omega:\\rho\(y\)\\geq\\lambda\_\{j\}\\right\\\}\.Finally,Φ¯​\(x\):=1−Φ​\(x\)\\overline\{\\Phi\}\(x\):=1\-\\Phi\(x\)is the standard Gaussian survival function, and\(x\)\+:=max⁡\{x,0\}\(x\)\_\{\+\}:=\\max\\\{x,0\\\}\.

### B\.1 Proof of Exactness

###### Theorem 3\(Exactness\)\.

The outputYYof Algorithm[3](https://arxiv.org/html/2609.17691#alg3)with target distributionℚ=𝒩⁡\(μq,σ2​𝕀d\)\\mathbb\{Q\}=\\mathcal\{N\}\(\\mu\_\{q\};\\sigma^\{2\}\\mathbb\{I\}\_\{d\}\)satisfiesY∼ℚY\\sim\\mathbb\{Q\}\.

###### Proof\.

LetY^1:K∼i\.i\.d\.ℙ\\widehat\{Y\}\_\{1:K\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathbb\{P\}beKKproposals\. By the residual\-measure characterization of greedy rejection sampling\([Flamich and Theis, 2023](https://arxiv.org/html/2609.17691#bib.bib10), Lemma III\.1\-2\), for every measurable setA⊆ℝdA\\subseteq\\mathbb\{R\}^\{d\}, the target mass that remains after the firstjjproposals is

Δj​\(A\)\\displaystyle\\Delta\_\{j\}\(A\)=ℚ⁡\(A\)−ℙ⁡\(J≤j,Y^J∈A\)\\displaystyle=\\mathbb\{Q\}\(A\)\-\\mathbb\{P\}\\left\(J\\leq j,\\,\\widehat\{Y\}\_\{J\}\\in A\\right\)=∫A\(ρ⁡\(y\)−λj\)\+​ℙ​\(𝑑y\)\.\\displaystyle=\\int\_\{A\}\\left\(\\rho\(y\)\-\\lambda\_\{j\}\\right\)\_\{\+\}\\mathbb\{P\}\(\\mathrm\{d\}y\)\.\(21\)Note that if we takeA=ΩA=\\Omega, we have

Δj​\(Ω\)=1−ℙ⁡\(J≤j\)=P⁡\(J\>j\):=Gj\+1\.\\displaystyle\\Delta\_\{j\}\(\\Omega\)=1\-\\mathbb\{P\}\(J\\leq j\)=P\(J\>j\):=G\_\{j\+1\}\.\(22\)Hence whenGj\+1\>0G\_\{j\+1\}\>0, the normalized residual distribution is

rj​\(A\)=Δj​\(A\)Gj\+1\.r\_\{j\}\(A\)=\\frac\{\\Delta\_\{j\}\(A\)\}\{G\_\{j\+1\}\}\.\(23\)Therefore, by combining \([21](https://arxiv.org/html/2609.17691#A2.E21)\), \([22](https://arxiv.org/html/2609.17691#A2.E22)\), and \([23](https://arxiv.org/html/2609.17691#A2.E23)\), for every measurableA⊆ℝdA\\subseteq\\mathbb\{R\}^\{d\}, we have

ℙ⁡\(Y∈A\)\\displaystyle\\mathbb\{P\}\(Y\\in A\)=ℙ⁡\(J≤j,Y^J∈A\)\+ℙ⁡\(J\>j\)​rj​\(A\)\\displaystyle=\\mathbb\{P\}\\left\(J\\leq j,\\widehat\{Y\}\_\{J\}\\in A\\right\)\+\\mathbb\{P\}\(J\>j\)\\ r\_\{j\}\(A\)=ℚ⁡\(A\)−Δj​\(A\)\+Gj\+1​Δj​\(A\)Gj\+1\\displaystyle=\\mathbb\{Q\}\(A\)\-\\Delta\_\{j\}\(A\)\+G\_\{j\+1\}\\frac\{\\Delta\_\{j\}\(A\)\}\{G\_\{j\+1\}\}=ℚ⁡\(A\)\.\\displaystyle=\\mathbb\{Q\}\(A\)\.ThereforeY∼ℚY\\sim\\mathbb\{Q\}exactly\. ∎

### B\.2 Proof of Acceptance Probability

###### Theorem 4\(Acceptance Probability\)\.

Denote

Δ:=\(μq−μp\)/σ,δ:=‖Δ‖,e:=Δ/δ\.\\displaystyle\\Delta:=\(\\mu\_\{q\}\-\\mu\_\{p\}\)/\\sigma,\\quad\\delta:=\\\|\\Delta\\\|,\\quad e:=\\Delta/\\delta\.For Algorithm[3](https://arxiv.org/html/2609.17691#alg3), the probability of accepting at least one of theKKproposals is given by

ℙ\(∃k∈\{1,⋯,K\}:Y=Y^k\)=1−GK\+1\.\\mathbb\{P\}\(\\exists k\\in\\\{1,\\cdots,K\\\}:Y=\\widehat\{Y\}\_\{k\}\)=1\-G\_\{K\+1\}\\ \.\(24\)Furthermore, for anyδ∈ℝ\+\\delta\\in\\mathbb\{R\}\_\{\+\}, we have

GK\+1=Φ¯​\(1δ​ln⁡λK−δ2\)−λK​Φ¯​\(1δ​ln⁡λK\+δ2\),\\displaystyle G\_\{K\+1\}=\\overline\{\\Phi\}\\left\(\\frac\{1\}\{\\delta\}\\ln\\lambda\_\{K\}\-\\frac\{\\delta\}\{2\}\\right\)\-\\lambda\_\{K\}\\overline\{\\Phi\}\\left\(\\frac\{1\}\{\\delta\}\\ln\\lambda\_\{K\}\+\\frac\{\\delta\}\{2\}\\right\),\(25\)whereΦ¯​\(x\)=1−Φ​\(x\)\\overline\{\\Phi\}\(x\)=1\-\\Phi\(x\)is the standard Gaussian survival function\.

###### Proof\.

By Lemma III\.1 of[Flamich and Theis \(2023\)](https://arxiv.org/html/2609.17691#bib.bib10),GjG\_\{j\}is the probability that greedy rejection sampling reaches proposaljj, i\.e\.Gj=Pr⁡\(J≥j\)G\_\{j\}=\\Pr\(J\\geq j\)\. Consequently the probability that the firstjjproposals are all rejected is

Pr⁡\(J\>j\)=Pr⁡\(J≥j\+1\)=Gj\+1,\\Pr\(J\>j\)=\\Pr\(J\\geq j\+1\)=G\_\{j\+1\},soPr⁡\(J≤j\)=1−Gj\+1\\Pr\(J\\leq j\)=1\-G\_\{j\+1\}, and therefore

Pr\(∃k∈\{1,…,K\}:Y=Y^k\)\\displaystyle\\Pr\\big\(\\exists k\\in\\\{1,\\dots,K\\\}:\\,Y=\\widehat\{Y\}\_\{k\}\\big\)=Pr⁡\(J≤K\)\\displaystyle=\\Pr\(J\\leq K\)=1−GK\+1\.\\displaystyle=1\-G\_\{K\+1\}\.The remainder of the proof computesGK\+1G\_\{K\+1\}\. Combining \([21](https://arxiv.org/html/2609.17691#A2.E21)\) and \([22](https://arxiv.org/html/2609.17691#A2.E22)\), we obtain

Gj\+1=Δj​\(Ω\)=∫Ω\(ρ⁡\(y\)−λj\)\+​ℙ​\(⁡𝑑y\)\.\\displaystyle G\_\{j\+1\}=\\Delta\_\{j\}\(\\Omega\)=\\int\_\{\\Omega\}\\left\(\\rho\(y\)\-\\lambda\_\{j\}\\right\)\_\{\+\}\\,\\mathbb\{P\}\(\\mathop\{\}\\\!\\mathrm\{d\}y\)\.\(26\)Since the integrand in \([26](https://arxiv.org/html/2609.17691#A2.E26)\) vanishes outside the superlevel set

ℋj:=\{y∈Ω:ρ⁡\(y\)≥λj\},\\mathcal\{H\}\_\{j\}:=\\left\\\{y\\in\\Omega:\\ \\rho\(y\)\\geq\\lambda\_\{j\}\\right\\\},we have

Gj\+1=∫ℋj\(ρ⁡\(y\)​λj\)​ℙ​\(⁡𝑑y\)=ℚ⁡\(ℋj\)−λj​ℙ​\(ℋj\)\.G\_\{j\+1\}=\\int\_\{\\mathcal\{H\}\_\{j\}\}\\left\(\\rho\(y\)\\lambda\_\{j\}\\right\)\\mathbb\{P\}\(\\mathop\{\}\\\!\\mathrm\{d\}y\)=\\mathbb\{Q\}\(\\mathcal\{H\}\_\{j\}\)\-\\lambda\_\{j\}\\,\\mathbb\{P\}\(\\mathcal\{H\}\_\{j\}\)\.\(27\)ComputingGj\+1G\_\{j\+1\}has thus been reduced to computing theℙ\\mathbb\{P\}\- andℚ\\mathbb\{Q\}\-masses of a superlevel set of the density ratio\. For Gaussianℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}with a common covarianceσ2​Id\\sigma^\{2\}I\_\{d\}, the log target\-to\-proposal ratio becomes:

ln⁡ρ⁡\(y\)\\displaystyle\\ln\\rho\(y\)=‖y−μp‖2−‖y−μq‖22​σ2\\displaystyle=\\frac\{\\\|y\-\\mu\_\{p\}\\\|^\{2\}\-\\\|y\-\\mu\_\{q\}\\\|^\{2\}\}\{2\\sigma^\{2\}\}=2​\(μq−μp\)⊤​y−\(‖μq‖2−‖μp‖2\)2​σ2\\displaystyle=\\frac\{2\(\\mu\_\{q\}\-\\mu\_\{p\}\)^\{\\top\}y\-\\big\(\\\|\\mu\_\{q\}\\\|^\{2\}\-\\\|\\mu\_\{p\}\\\|^\{2\}\\big\)\}\{2\\sigma^\{2\}\}=\(μq−μp\)⊤​\(y−μ¯\)σ2=δ​e⊤​\(y−μ¯\)σ,\\displaystyle=\\frac\{\(\\mu\_\{q\}\-\\mu\_\{p\}\)^\{\\top\}\\big\(y\-\\bar\{\\mu\}\\big\)\}\{\\sigma^\{2\}\}=\\frac\{\\delta\\,e^\{\\top\}\(y\-\\bar\{\\mu\}\)\}\{\\sigma\},withμ¯:=12​\(μp\+μq\)\\bar\{\\mu\}:=\\tfrac\{1\}\{2\}\(\\mu\_\{p\}\+\\mu\_\{q\}\)\. Let

s⁡\(y\):=1σ​e⊤​\(y−μ¯\),s\(y\):=\\frac\{1\}\{\\sigma\}\\,e^\{\\top\}\\\!\\left\(y\-\\bar\{\\mu\}\\right\),then the likelihood ratio becomes a monotone function of the scalar statisticss,

ρ⁡\(y\)=exp⁡\(δ​s​\(y\)\)\.\\rho\(y\)=\\exp\\left\(\\delta\\,s\(y\)\\right\)\.
Becauseδ\>0\\delta\>0, the mapt↦eδ​tt\\mapsto e^\{\\delta t\}is strictly increasing, and the superlevel set is the half\-space

ℋj=\{y:ρ⁡\(y\)≥λj\}=\{y:s⁡\(y\)≥τj\},τj=ln⁡λjδ\.\\displaystyle\\mathcal\{H\}\_\{j\}=\\left\\\{y:\\ \\rho\(y\)\\geq\\lambda\_\{j\}\\right\\\}=\\left\\\{y:\\ s\(y\)\\geq\\tau\_\{j\}\\right\\\},\\quad\\tau\_\{j\}=\\frac\{\\ln\\lambda\_\{j\}\}\{\\delta\}\.Now, to characterize the massesℚ⁡\(ℋj\)\\mathbb\{Q\}\(\\mathcal\{H\}\_\{j\}\)andℙ⁡\(ℋj\)\\mathbb\{P\}\(\\mathcal\{H\}\_\{j\}\), we are left with the characterization of the the laws ofsswhenX∼ℙX\\sim\\mathbb\{P\}andY∼ℚY\\sim\\mathbb\{Q\}\.ssis an affine functional of a Gaussian vector, hence Gaussian with means

𝔼⁡\[s⁡\(X\)\]\\displaystyle\\mathbb\{E\}\[s\(X\)\]=e⊤​\(μp−μ¯\)σ=e⊤​\(μp−μq\)2​σ=−δ2,\\displaystyle=\\frac\{e^\{\\top\}\(\\mu\_\{p\}\-\\bar\{\\mu\}\)\}\{\\sigma\}=\\frac\{e^\{\\top\}\(\\mu\_\{p\}\-\\mu\_\{q\}\)\}\{2\\sigma\}=\-\\frac\{\\delta\}\{2\},𝔼⁡\[s⁡\(Y\)\]\\displaystyle\\mathbb\{E\}\[s\(Y\)\]=e⊤​\(μq−μ¯\)σ=e⊤​\(μq−μp\)2​σ=\+δ2,\\displaystyle=\\frac\{e^\{\\top\}\(\\mu\_\{q\}\-\\bar\{\\mu\}\)\}\{\\sigma\}=\\frac\{e^\{\\top\}\(\\mu\_\{q\}\-\\mu\_\{p\}\)\}\{2\\sigma\}=\+\\frac\{\\delta\}\{2\},and same unit variance

Var⁡\[s⁡\(X\)\]=Var⁡\[s⁡\(Y\)\]=e⊤​\(σ2​Id\)​eσ2=‖e‖2=1\.\\displaystyle\\operatorname\{Var\}\[s\(X\)\]=\\operatorname\{Var\}\[s\(Y\)\]=\\frac\{e^\{\\top\}\(\\sigma^\{2\}I\_\{d\}\)e\}\{\\sigma^\{2\}\}=\\\|e\\\|^\{2\}=1\.Therefore,

s⁡\(X\)∼𝒩⁡\(−δ2,1\),s⁡\(Y\)∼𝒩⁡\(δ2,1\)\.s\(X\)\\sim\\mathcal\{N\}\\left\(\-\\frac\{\\delta\}\{2\},1\\right\),\\qquad s\(Y\)\\sim\\mathcal\{N\}\\left\(\\frac\{\\delta\}\{2\},1\\right\)\.Standardising the two laws ofssthen gives

ℚ⁡\(ℋj\)\\displaystyle\\mathbb\{Q\}\(\\mathcal\{H\}\_\{j\}\)=Φ¯​\(τj−δ2\),ℙ⁡\(ℋj\)=Φ¯​\(τj\+δ2\),\\displaystyle=\\overline\{\\Phi\}\\left\(\\tau\_\{j\}\-\\frac\{\\delta\}\{2\}\\right\),\\qquad\\mathbb\{P\}\(\\mathcal\{H\}\_\{j\}\)=\\overline\{\\Phi\}\\left\(\\tau\_\{j\}\+\\frac\{\\delta\}\{2\}\\right\),which, substituted into \([27](https://arxiv.org/html/2609.17691#A2.E27)\), becomes

Gj\+1\\displaystyle G\_\{j\+1\}=ℚ⁡\(ℋj\)−λj​ℙ​\(ℋj\)\\displaystyle=\\mathbb\{Q\}\(\\mathcal\{H\}\_\{j\}\)\-\\lambda\_\{j\}\\,\\mathbb\{P\}\(\\mathcal\{H\}\_\{j\}\)=Φ¯​\(τj−δ2\)−λj​Φ¯​\(τj\+δ2\)\.\\displaystyle=\\overline\{\\Phi\}\\left\(\\tau\_\{j\}\-\\frac\{\\delta\}\{2\}\\right\)\-\\lambda\_\{j\}\\overline\{\\Phi\}\\left\(\\tau\_\{j\}\+\\frac\{\\delta\}\{2\}\\right\)\.Takingj=Kj=Kcompletes the proof\. ∎

000\.50\.5111\.51\.5222\.52\.5333\.53\.5444\.54\.5000\.20\.20\.40\.40\.60\.60\.80\.811Normalized Mean Mismatchδ=‖μq−μp‖2/σ\\delta=\\\|\\mu\_\{q\}\-\\mu\_\{p\}\\\|\_\{2\}/\\sigmaPer\-step Acceptance ProbabilityD\-GRS Per\-step Acceptance ProbabilityK=1K=1K=2K=2K=4K=4K=8K=8Figure 6:Per\-step acceptance probability of D\-GRS for different numberKKof draft proposals\.

## Appendix CExtended Experimental Results

In this section, we provide more details about our experimental settings\.

### C\.1 Implementation Details

##### The delayed\-drift proposal\.

In our experiments, we instantiate the template with \([7](https://arxiv.org/html/2609.17691#S2.E7)\), taking the target drift evaluated at the root:m~n′p​\(y\)=y\+γ​btnq​\(Yn\)\\tilde\{m\}^\{p\}\_\{n^\{\\prime\}\}\(y\)=y\+\\gamma\\,b^\{q\}\_\{t\_\{n\}\}\(Y\_\{n\}\)for every stepn′≤Ln^\{\\prime\}\\leq Lof the round\. The proposal mean is therefore the same map at every depth, so the drafting phase in Algorithm[1](https://arxiv.org/html/2609.17691#alg1)specializes to

Y^v∼𝒩⁡\(m~np​\(Y^u\),σn\+ℓ−12​Id\),v∈𝒞⁡\(u\),\|u\|=ℓ−1,\\widehat\{Y\}\_\{v\}\\sim\\mathcal\{N\}\\big\(\\tilde\{m\}^\{p\}\_\{n\}\(\\widehat\{Y\}\_\{u\}\),\\,\\sigma^\{2\}\_\{n\+\\ell\-1\}I\_\{d\}\\big\),\\ v\\in\\mathcal\{C\}\(u\),\\ \|u\|=\\ell\-1,withm~np​\(Y^u\)=Y^u\+γ​btnq​\(Yn\)\\tilde\{m\}^\{p\}\_\{n\}\(\\widehat\{Y\}\_\{u\}\)=\\widehat\{Y\}\_\{u\}\+\\gamma b^\{q\}\_\{t\_\{n\}\}\(Y\_\{n\}\),Yn=YrY\_\{n\}=Y\_\{\\mathrm\{r\}\}\. Note that this would require 1 extra target call per round \(other than the one needed for verification\) to compute the target drift at the root nodebtnq​\(Yn\)b^\{q\}\_\{t\_\{n\}\}\(Y\_\{n\}\)\.

##### Root\-drift Prefetching\.

When the delayed\-drift is used as proposal, the drafting phase requires evaluatingbtnq​\(Yn\)b^\{q\}\_\{t\_\{n\}\}\(Y\_\{n\}\)at the root of every round, that is, one target evaluation in addition to the one used for verification\. Following[De Bortoli et al\. \(2025\)](https://arxiv.org/html/2609.17691#bib.bib1), we avoid this by reusing a drift already computed during the preceding round, a device we refer to as*root\-drift prefetching*:btnq​\(Yn\)b^\{q\}\_\{t\_\{n\}\}\(Y\_\{n\}\)is replaced bybtn′q​\(Y~\)b^\{q\}\_\{t\_\{n^\{\\prime\}\}\}\(\\tilde\{Y\}\)for some pair\(n′,Y~\)\(n^\{\\prime\},\\tilde\{Y\}\)whose drift is already available\.

### C\.2 The Value ofKKandLL

![Refer to caption](https://arxiv.org/html/2609.17691v1/speedup_vs_K.png)Figure 7:NFE speedup over standard sampling on the Gaussian\-mixture target as a function of tree widthKK\. Points indicate the mean per\-replicate speedup, averaged over 100 independent trials\. Coloured lines depict D\-GRS across lookahead depthsL∈\{2,…,7\}L\\in\\\{2,\\dots,7\\\}\(see color bar\), with shaded bands showing±1\\pm 1standard error\.We next isolate the two mechanisms by which a proposal budget may be allocated: the branching factorKK, i\.e\. the number of candidate children expanded per node, and the lookahead depthLL\. Figure[7](https://arxiv.org/html/2609.17691#A3.F7)reports the speedup over standard sampling as a function ofKK, stratified byLL, with each point averaged over100100independent replicates\.

RMC admits no branching, as it drafts a single linear chain, and consequently saturates at≈2×\\approx 2\\timesirrespective of the chain length\. This ceiling admits a closed\-form characterization\. Under the assumption that the per\-step acceptance probability of the couplingα\\alphais fixed for each step, the expected length of the accepted prefix of a chain ofnndrafted states is

∑i=1nαi=α⁡\(1−αn\)1−α→n→∞α1−α,\\sum\_\{i=1\}^\{n\}\\alpha^\{i\}\\;=\\;\\frac\{\\alpha\\left\(1\-\\alpha^\{n\}\\right\)\}\{1\-\\alpha\}\\;\\xrightarrow\[n\\to\\infty\]\{\}\\;\\frac\{\\alpha\}\{1\-\\alpha\},so that, since each round commits the accepted prefix together with one resampled state at the cost of a single target evaluation, the number of committed steps per target network function evaluation is bounded above by1/\(1−α\)1/\(1\-\\alpha\)\. The measured acceptance probabilityα≈0\.52\\alpha\\approx 0\.52yields a predicted ceiling of2\.09×2\.09\\times, in close agreement with the observed2×2\\times\. Extending the chain beyond this regime therefore incurs additional draft computation without any attendant reduction in target evaluations: the geometric decay of the acceptance probability along the chain, rather than the draft budget, constitutes the binding constraint\.

The relative value of depthLLand widthKKis governed by the per\-step acceptance probability of the coupling, which is controlled by the ratioδ\\deltadefined in \([12](https://arxiv.org/html/2609.17691#S3.E12)\)\. Consequently,K,LK,Lare complementary rather than interchangeable, and their relative priority is separated by a problem\-dependent crossover\. Below it, depth is the binding constraint: a shallow tree exhausts its lookahead before branching can be exploited, and noKKrecovers the deficit\. AtL=2L=2the best configuration attains only1\.876×1\.876\\times, short of the RMC ceiling, whereas atK=1K=1the tree degenerates to a chain and D\-GRS reproduces that ceiling\. D\-GRS surpasses RMC only once the lookahead is deep enough to sustain an accepted run, here from\(K,L\)=\(2,3\)\(K,L\)=\(2,3\)onward\. Above the crossover, the two parameters compete for the same budget: the marginal benefit of depth is exhausted byL≈5L\\approx 5, beyond which theL∈\{5,6,7\}L\\in\\\{5,6,7\\\}curves lie within one standard error of one another, while widening fromK=5K=5toK=7K=7atL=6L=6still yields0\.09×0\.09\\times, up to a best of2\.34×2\.34\\timesat\(K,L\)=\(7,6\)\(K,L\)=\(7,6\)\. Once the lookahead is sufficient, additional budget is better spent on width than on depth\.

### C\.3 SD3 Conversion

Unlike DDPMs, SD3 is trained as a rectified flow \(flow matching\) model, i\.e\. it directly parameterizes a probability\-flow ODE\. We recall here how this ODE is recast as the SDE in \([2](https://arxiv.org/html/2609.17691#S2.E2)\), which requires identifying the driftftf\_\{t\}, the diffusion coefficientgtg\_\{t\}and the scorests\_\{t\}implied by the SD3 interpolant\.

Unlike DDPMs, SD3 is trained as a rectified flow model: it parameterises a probability\-flow ODE, whose transitions are deterministic point masses, yielding a total variation distance of one\. Speculation is therefore vacuous for the native SD3 sampler, and applying our framework requires transitions that are genuinely Gaussian\.

We obtain them without retraining, by exhibiting SD3 as a member of the family of reverse\-time SDEs \([2](https://arxiv.org/html/2609.17691#S2.E2)\) that share its marginals\. Concretely, the conversion identifies the driftftf\_\{t\}, the diffusion coefficientgtg\_\{t\}and the scorests\_\{t\}implied by the SD3 interpolant, so that the reverse drift \([2](https://arxiv.org/html/2609.17691#S2.E2)\) and its Euler–Maruyama discretisation \([4](https://arxiv.org/html/2609.17691#S2.E4)\)–\([5](https://arxiv.org/html/2609.17691#S2.E5)\) apply verbatim\. The outcome, recorded in \([36](https://arxiv.org/html/2609.17691#A3.E36)\)–\([37](https://arxiv.org/html/2609.17691#A3.E37)\) below, is a Gaussian transition kernel whose variance is independent of the state, which is precisely the structure the rank\-11reduction \([12](https://arxiv.org/html/2609.17691#S3.E12)\)–\([15](https://arxiv.org/html/2609.17691#S3.E15)\) requires\.

##### Interpolant and velocity field\.

LetX0∼pdataX\_\{0\}\\sim p\_\{\\rm data\}andξ∼𝒩⁡\(0,𝕀d\)\\xi\\sim\\mathcal\{N\}\(0,\\mathbb\{I\}\_\{d\}\)be independent\. Fort∈\[0,1\]t\\in\[0,1\], SD3 perturbs data along the linear interpolant

Xt=αt​X0\+βt​ξ,αt=1−t,βt=t,\\displaystyle X\_\{t\}=\\alpha\_\{t\}X\_\{0\}\+\\beta\_\{t\}\\,\\xi,\\quad\\alpha\_\{t\}=1\-t,\\quad\\beta\_\{t\}=t,\(28\)so thatX0∼pdataX\_\{0\}\\sim p\_\{\\rm data\}andX1∼𝒩⁡\(0,𝕀d\)X\_\{1\}\\sim\\mathcal\{N\}\(0,\\mathbb\{I\}\_\{d\}\)\. Along \([28](https://arxiv.org/html/2609.17691#A3.E28)\) the velocity is constant,⁡d​Xt⁡d​t=ξ−X0\\tfrac\{\\mathop\{\}\\\!\\mathrm\{d\}X\_\{t\}\}\{\\mathop\{\}\\\!\\mathrm\{d\}t\}=\\xi\-X\_\{0\}, and the networkvθv\_\{\\theta\}is trained by conditional flow matching to regress it from the corrupted state alone\. Since the pair\(X0,ξ\)\(X\_\{0\},\\xi\)is not identifiable fromXtX\_\{t\}, the minimiser of the training objective is the conditional expectation, that is the*marginal*velocity field

vt​\(x\)\\displaystyle v\_\{t\}\(x\):=𝔼⁡\[⁡d​Xt⁡d​t\|Xt=x\]\\displaystyle:=\\mathbb\{E\}\\\!\\left\[\\frac\{\\mathop\{\}\\\!\\mathrm\{d\}X\_\{t\}\}\{\\mathop\{\}\\\!\\mathrm\{d\}t\}\\,\\bigg\|\\,X\_\{t\}=x\\right\]\(29\)=𝔼⁡\[ξ∣Xt=x\]−𝔼⁡\[X0∣Xt=x\]\.\\displaystyle=\\mathbb\{E\}\[\\xi\\mid X\_\{t\}=x\]\-\\mathbb\{E\}\[X\_\{0\}\\mid X\_\{t\}=x\]\.\(30\)

##### Drift and diffusion coefficient\.

Any affine interpolant of the form \([28](https://arxiv.org/html/2609.17691#A3.E28)\) is the marginal law of a linear SDE

⁡d​Xt=ft​Xt​⁡d​t\+gt​⁡d​Bt,\\mathop\{\}\\\!\\mathrm\{d\}X\_\{t\}=f\_\{t\}X\_\{t\}\\,\\mathop\{\}\\\!\\mathrm\{d\}t\+g\_\{t\}\\,\\mathop\{\}\\\!\\mathrm\{d\}B\_\{t\},since for such an SDE the conditional mean and variance ofXt\|X0X\_\{t\}\\mid X\_\{0\}obey⁡d​αt⁡d​t=ft​αt\\tfrac\{\\mathop\{\}\\\!\\mathrm\{d\}\\alpha\_\{t\}\}\{\\mathop\{\}\\\!\\mathrm\{d\}t\}=f\_\{t\}\\alpha\_\{t\}and⁡d​βt2⁡d​t=2​ft​βt2\+gt2\\tfrac\{\\mathop\{\}\\\!\\mathrm\{d\}\\beta\_\{t\}^\{2\}\}\{\\mathop\{\}\\\!\\mathrm\{d\}t\}=2f\_\{t\}\\beta\_\{t\}^\{2\}\+g\_\{t\}^\{2\}\. Inverting these relations,

ft=1αt​⁡d​αt⁡d​t,gt2=2​βt​\(⁡d​βt⁡d​t−ft​βt\),\\displaystyle f\_\{t\}=\\frac\{1\}\{\\alpha\_\{t\}\}\\frac\{\\mathop\{\}\\\!\\mathrm\{d\}\\alpha\_\{t\}\}\{\\mathop\{\}\\\!\\mathrm\{d\}t\},\\qquad g\_\{t\}^\{2\}=2\\beta\_\{t\}\\\!\\left\(\\frac\{\\mathop\{\}\\\!\\mathrm\{d\}\\beta\_\{t\}\}\{\\mathop\{\}\\\!\\mathrm\{d\}t\}\-f\_\{t\}\\beta\_\{t\}\\right\),\(31\)and substitutingαt=1−t\\alpha\_\{t\}=1\-t,βt=t\\beta\_\{t\}=tyields the SD3 coefficients

ft​\(x\)=−x1−t,gt2=2​t1−t\.\\displaystyle f\_\{t\}\(x\)=\-\\frac\{x\}\{1\-t\},\\qquad g\_\{t\}^\{2\}=\\frac\{2t\}\{1\-t\}\.\(32\)

##### Score\.

The reverse drift \([3](https://arxiv.org/html/2609.17691#S2.E3)\) is expressed through the Stein score, which the flow matching parameterization does not supply directly\. It is, however, an affine function of the velocity\. Taking conditional expectations in \([28](https://arxiv.org/html/2609.17691#A3.E28)\) gives

x=\(1−t\)​𝔼​\[X0∣Xt=x\]\+t​𝔼​\[ξ∣Xt=x\]\.x=\(1\-t\)\\,\\mathbb\{E\}\[X\_\{0\}\\mid X\_\{t\}=x\]\+t\\,\\mathbb\{E\}\[\\xi\\mid X\_\{t\}=x\]\.Eliminating𝔼⁡\[X0∣Xt=x\]\\mathbb\{E\}\[X\_\{0\}\\mid X\_\{t\}=x\]from \([30](https://arxiv.org/html/2609.17691#A3.E30)\) then leaves

vt​\(x\)=𝔼⁡\[ξ∣Xt=x\]−x1−t\.\\displaystyle v\_\{t\}\(x\)=\\frac\{\\mathbb\{E\}\[\\xi\\mid X\_\{t\}=x\]\-x\}\{1\-t\}\.\(33\)Combining \([33](https://arxiv.org/html/2609.17691#A3.E33)\) withst\(x\)=−𝔼\[ξ∣Xt=x\]/βts\_\{t\}\(x\)=\-\\mathbb\{E\}\[\\xi\\mid X\_\{t\}=x\]/\\beta\_\{t\}[Karras et al\. \(2022\)](https://arxiv.org/html/2609.17691#bib.bib21), we have

st​\(x\)=−x\+\(1−t\)​vt​\(x\)t\.\\displaystyle s\_\{t\}\(x\)=\-\\frac\{x\+\(1\-t\)\\,v\_\{t\}\(x\)\}\{t\}\.\(34\)

##### Reverse process\.

Substitutingf1−t\(x\)=−x/tf\_\{1\-t\}\(x\)=\-x/t,g1−t2=2​\(1−t\)/tg^\{2\}\_\{1\-t\}=2\(1\-t\)/tand \([34](https://arxiv.org/html/2609.17691#A3.E34)\), all evaluated at time1−t1\-t, into the reverse drift \([3](https://arxiv.org/html/2609.17691#S2.E3)\) gives

btq​\(x\)=−\(1\+ε2\)​v1−t​\(x\)−ε2​xt,\\displaystyle b^\{q\}\_\{t\}\(x\)=\-\\big\(1\+\\varepsilon^\{2\}\\big\)v\_\{1\-t\}\(x\)\-\\varepsilon^\{2\}\\,\\frac\{x\}\{t\},\(35\)and the reverse process \([2](https://arxiv.org/html/2609.17691#S2.E2)\) is

⁡d​Yt=btq​\(Yt\)​⁡d​t\+ε​g1−t​⁡d​Wt,\\mathop\{\}\\\!\\mathrm\{d\}Y\_\{t\}=b^\{q\}\_\{t\}\(Y\_\{t\}\)\\,\\mathop\{\}\\\!\\mathrm\{d\}t\+\\varepsilon\\,g\_\{1\-t\}\\,\\mathop\{\}\\\!\\mathrm\{d\}W\_\{t\},withg1−t=2​\(1−t\)/tg\_\{1\-t\}=\\sqrt\{2\(1\-t\)/t\}\.

##### Transition kernels\.

Applying the Euler–Maruyama discretisation \(4\) on the uniform gridtn=n​γt\_\{n\}=n\\gammayields, conditionally onYn=yY\_\{n\}=y, the Gaussian transitionℚn\(⋅∣y\)=𝒩\(mnq\(y\),σn2𝕀d\)\\mathbb\{Q\}\_\{n\}\(\\cdot\\mid y\)=\\mathcal\{N\}\\big\(m^\{q\}\_\{n\}\(y\),\\sigma\_\{n\}^\{2\}\\mathbb\{I\}\_\{d\}\\big\)of \([5](https://arxiv.org/html/2609.17691#S2.E5)\), with

mnq​\(y\)\\displaystyle m^\{q\}\_\{n\}\(y\)=y−γ⁡\[\(1\+ε2\)​v1−tn​\(y\)\+ε2​ytn\],\\displaystyle=y\-\\gamma\\left\[\\big\(1\+\\varepsilon^\{2\}\\big\)v\_\{1\-t\_\{n\}\}\(y\)\+\\varepsilon^\{2\}\\,\\frac\{y\}\{t\_\{n\}\}\\right\],\(36\)σn\\displaystyle\\sigma\_\{n\}=γ​ε​g1−tn=ε​2​γ​\(1−tn\)tn\.\\displaystyle=\\sqrt\{\\gamma\}\\,\\varepsilon\\,g\_\{1\-t\_\{n\}\}=\\varepsilon\\sqrt\{\\frac\{2\\gamma\\,\(1\-t\_\{n\}\)\}\{t\_\{n\}\}\}\.\(37\)The scale \([37](https://arxiv.org/html/2609.17691#A3.E37)\) is a function of the time grid and the churn level alone, and in particular does not depend on the state\. A draft kernel obtained by substituting a delayed velocity into \([36](https://arxiv.org/html/2609.17691#A3.E36)\) consequently shares the variance of the target kernel and differs from it only in its mean, which is the structure required by \([12](https://arxiv.org/html/2609.17691#S3.E12)\)–\([15](https://arxiv.org/html/2609.17691#S3.E15)\)\. Both RMC and D\-GRS therefore apply to SD3 without modification\.

##### Timestep shift\.

Since SD3 is trained at1024×10241024\\times 1024resolution and its noise schedule is resolution dependent\. Following[Esser et al\. \(2024\)](https://arxiv.org/html/2609.17691#bib.bib38), we adapt it to the512×512512\\times 512generation of our experimental setting by applying the shift

t⟼κ​t1\+\(κ−1\)​t,κ=3\.0,\\displaystyle t\\ \\longmapsto\\ \\frac\{\\kappa\\,t\}\{1\+\(\\kappa\-1\)\\,t\},\\qquad\\kappa=3\.0,\(38\)to the uniform grid before evaluatingvθv\_\{\\theta\}\. The conversion above is unaffected, since \([34](https://arxiv.org/html/2609.17691#A3.E34)\) and \([35](https://arxiv.org/html/2609.17691#A3.E35)\) hold pointwise intt\.

### C\.4 Qualitative Assessment of Correctness

In order to qualitatively asses the correctness of the proposed algorithm D\-GRS and the baseline implementation RMC\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.17691#bib.bib1)\)in the context of image generation, we show 64 of the generated images for the experimental setting reported in Section[5](https://arxiv.org/html/2609.17691#S5)\. Figures[8](https://arxiv.org/html/2609.17691#A3.F8)and[9](https://arxiv.org/html/2609.17691#A3.F9)show the generated images for CIFAR\-10, FFHQ, respectively\. Table[2](https://arxiv.org/html/2609.17691#A3.T2)reports the prompts from COCO 2014 used for the conditional image generation using SD3, and Figure[10](https://arxiv.org/html/2609.17691#A3.F10)shows the generated images\.

Figure 8:64 CIFAR\-10 samples \(32×\\times32\) selected at random generated atK=4K=4,L=4L=4,T=100T=100, for increasing churnε\\varepsilon\. Top row: RMC\. Bottom row: D\-GRS\.Figure 9:First 64 FFHQ samples \(64×\\times64\) generated samples atK=4K=4,L=4L=4,T=100T=100, for increasing churnε\\varepsilon\. Top row: RMC\. Bottom row: D\-GRS\.![Refer to caption](https://arxiv.org/html/2609.17691v1/Figures/experiments/generated-images/sd3-rmc-K7_L2-images.png)\(a\)RMC
![Refer to caption](https://arxiv.org/html/2609.17691v1/Figures/experiments/generated-images/sd3-grc-K7_L2-images.png)\(b\)D\-GRS

Figure 10:First 64 images 512×\\times512 selected at random generated by SD3 conditionally on the prompts reported in Table[2](https://arxiv.org/html/2609.17691#A3.T2), using \(a\) RMC and \(b\) D\-GRS with\(K=1,L=56\)\(K=1,L=56\)and\(K=7,L=2\)\(K=7,L=2\), respectively, in the setting described in Section[5\.3](https://arxiv.org/html/2609.17691#S5.SS3), i\.e\.,T=50T=50, free\-guidance of77, churnε=0\.8\\varepsilon=0\.8\.Table 2:The first 64 prompts from COCO 2014[Lin et al\. \(2014\)](https://arxiv.org/html/2609.17691#bib.bib39)used to conditionally generate the images shown in Figure[10](https://arxiv.org/html/2609.17691#A3.F10), following the experimental setting of Section[5\.3](https://arxiv.org/html/2609.17691#S5.SS3)\.