Speculative Diffusion is a Tale of Two Tradeoffs

# Speculative Diffusion is a Tale of Two Tradeoffs
Source: [https://arxiv.org/html/2610.31819](https://arxiv.org/html/2610.31819)
Guy Van den Broeck†Aditya Grover†Daniel Israel†Affiliation:∗Independent Researcher†University of CaliforniaLos Angeles

###### Abstract

Speculative decoding is a technique that accelerates the inference speed of autoregressive large language models \(LLMs\) by sampling from a faster, lower quality draft model\. Traditionally, a fundamental characteristic of speculative decoding is that it is lossless with respect to the verifier distribution, guaranteeing speed without sacrificing quality\. This differs from recently developed diffusion LLMs \(dLLMs\), which are notably able to achieve substantial speedups by degrading the quality of their outputs with less diffusion effort\. Because diffusion models are able to sample faster than autoregressive models by generating tokens in parallel, they are considered strong candidates for draft models in speculative diffusion\. While it is widely understood that diffusion models can sacrifice quality for speed, speculative decoding itself can also be generalized to generate lossy samples faster using more lenient acceptance criteria with respect to the verifier distribution\. Presented with the two parameters of diffusion effort and speculative decoding leniency, this paper explores the full achievable Pareto frontier via joint variations of these parameters\. Our experiments show that jointly varying diffusion effort and acceptance leniency can extend the speed–quality Pareto frontier over single step diffusion drafting\.

## 1Introduction

Autoregressive \(AR\) large language models \(LLMs\) generate its output token by token, as the distribution of the next token over the vocabulary depends on all the previously generated tokens\([Vaswani et al\., 2017](https://arxiv.org/html/2610.31819#bib.bib4)\)\. Because each sequential decoding step requires transferring all model parameters and the growing key\-value \(KV\) cache for a single token, inference is severely memory\-bandwidth bound\([Yuan et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib24)\)\. Thus, sequential dependency is a major bottleneck on the inference speed of AR decoding\([Stern et al\., 2018](https://arxiv.org/html/2610.31819#bib.bib6);[Leviathan et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib1)\)\.

In light of this problem, speculative decoding was invented to speed up the inference of AR LLMs while preserving the original output distribution\([Leviathan et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib1);[Chen et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib5)\)\. It employs a much faster model as a drafter\([Xia et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib8)\)to generate a short candidate sequence during each drafting round\. The original LLM acts as a verifier and processes these proposed tokens in parallel via rejection sampling, accepting an initial prefix while strictly guaranteeing that the sampled distribution matches that of the verifier\([Leviathan et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib1);[Chen et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib5)\)\. It is known that the rejection probability at a given position is the total variation \(TV\) distance between the drafter distribution and the verifier distribution at that position\([Leviathan et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib1);[Chen et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib5);[Yin et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib7)\)\. Therefore, a drafter more aligned with the verifier will have a higher expected acceptance rate across the sequence and a longer accepted prefix in each drafting round\. Because only one verifier call is made during each drafting round, a longer accept length will lead to a higher throughput in output tokens per unit time\. Because the resulting sequence is provably drawn from the verifier model’s output distribution, traditional speculative decoding islossless, providing an exact distributional guarantee that yields acceleration without degrading generation quality\.

Figure 1:A conceptual overview\.Diffusion has a speed\-quality tradeoff parameterized by diffusion effortKK, while speculative decoding has a speed\-quality tradeoff parameterized by acceptance leniencyλ\\lambda\. This work argues that speculative diffusion exposes both tradeoffs and therefore reveals optimal configurations that could not be found by varying only a single parameter\.While this exact distributional guarantee ensures that model quality is fully preserved, lossless decoding limits the maximum achievable speedup by how closely the drafter can approximate the verifier\. Various methods explorelossyspeculative decoding, deliberately relaxing the exact distributional guarantee to trade off output quality for faster inference\. For example,[Leviathan et al\. \(2023\)](https://arxiv.org/html/2610.31819#bib.bib1)experimented with a lenience parameter that increases the acceptance rate of draft tokens by a predefined ratio\. Cactus\([Hao and Mou, 2026](https://arxiv.org/html/2610.31819#bib.bib2)\)formalizes a relaxation by constraining the distributional divergence \(such as Kullback\-Leibler divergence\) between the drafter and verifier to remain beneath a specified threshold\. These lossy formulations provide a mechanism to control the trade\-off between output quality and speed\.

In parallel with developments in speculative decoding, diffusion large language models \(dLLMs\) have emerged as an alternative to autoregressive models\. Rather than decoding from left to right, dLLMs iteratively denoise a noise corrupted sequence over multiple steps\. Diffusion language models can be categorized by their noising process into masked diffusion\([Sahoo et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib10);[Nie et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib11);[Ye et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib12)\)and uniform diffusion\([Austin et al\., 2021](https://arxiv.org/html/2610.31819#bib.bib9);[Sahoo et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib14);[DiffusionGemma Team, 2026](https://arxiv.org/html/2610.31819#bib.bib3)\)\. By updating all token positions in parallel during each denoising step, diffusion models avoid the sequentiality demanded by autoregressive decoding\. Crucially, diffusion language models possess an intrinsic speed\-quality trade\-off: decreasing the number of denoising steps increases speed at the cost of quality\.

More recently, an increasing number of diffusion LLMs have been used in speculative decoding\. Due to their ability to generate multiple tokens simultaneously, diffusion LLMs serve as natural candidates as draft models\. We refer to speculative decoding with a diffusion drafter asspeculative diffusion\([Christopher et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib25);[Liu et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib15);[Chen et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib16);[Fu et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib13);[Sandler et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib44)\)\. Speculative diffusion presents two intertwined trade\-offs: the drafter trades draft quality for speed via the number of denoising steps, while the verifier can trade quality for speed via lossy acceptance\. Mapping out this joint optimization space reveals how both parameters interact to yield the final Pareto frontier between generation speed and output quality \(Figure[1](https://arxiv.org/html/2610.31819#S1.F1)\)\.

This paper makes the following contributions to speculative diffusion:

1. \(i\)Theoretical\.We prove that, for a fixed proposal distribution, linear interpolation between the verifier and draft distributions maximizes token acceptance under a total variation bound on deviation from the verifier \(Theorem[1](https://arxiv.org/html/2610.31819#Thmtheorem1)\)\. We also establish conditions under which rejection sampling preserves the verifier distribution for multistep diffusion drafts \(Proposition[2](https://arxiv.org/html/2610.31819#Thmproposition2)\)\. We then provide a theoretical throughput model that explains the interaction between speed and quality in speculative diffusion \(Section[3\.3](https://arxiv.org/html/2610.31819#S3.SS3)\)\.
2. \(ii\)Empirical\.Across masked and uniform diffusion drafters on reasoning and coding tasks, we show that different combinations of denoising effort and acceptance leniency contribute to the speed\-accuracy Pareto frontier\. In particular, multistep denoising under more lenient acceptance enables faster generation at comparable accuracy \(Section[4](https://arxiv.org/html/2610.31819#S4)\)\. We show that joint variation over diffusion effort and acceptance leniency often extends the Pareto frontier of speculative diffusion\.

## 2Background

The contributions of this work depend on diffusion language models and lossy speculative decoding\. We first provide background on these methods\.

##### Diffusion Language Models\.

Diffusion language models iteratively generate text from noise\. For a token vocabulary𝒱\\mathcal\{V\}and sequence ofBBtokens𝐱∈𝒱B\\mathbf\{x\}\\in\\mathcal\{V\}^\{B\}, let𝐳t\\mathbf\{z\}\_\{t\}be its corrupted version at noise levelt∈\[0,1\]t\\in\[0,1\]\. The forward noising process independently sets eachztiz\_\{t\}^\{i\}to a noisy token with probabilitytt\. Masked diffusion sets a special mask symbolzti=\[𝙼𝙰𝚂𝙺\]z\_\{t\}^\{i\}=\\mathtt\{\[MASK\]\}\([Austin et al\., 2021](https://arxiv.org/html/2610.31819#bib.bib9);[Sahoo et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib10);[Shi et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib33)\), while uniform diffusion sampleszti∼Uniform⁡\(𝒱\)z\_\{t\}^\{i\}\\sim\\mathrm\{Uniform\}\(\\mathcal\{V\}\)\([Sahoo et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib14);[Von Rütte et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib35)\)\. The goal of diffusion language models is to learn a denoiserfθf\_\{\\theta\}that can reverse the forward noise process\. In this work, we generate one block ofBBtokens at a time\([Arriola et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib26)\), conditioned on a fixed prefixhh\. The denoiserfθ​\(𝐳t∣h\)f\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\\mid h\)outputs a probability distribution over\|𝒱\|\|\\mathcal\{V\}\|over clean tokens at each block position in parallel\. To generate a block, a dLLM starts from noise and iteratively denoises the block until termination\. We refer to the number of diffusion stepsKKasdiffusion effort\. Similar to “reasoning effort” in reasoning models\([Guo et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib34)\), greater effort applies more computation to generate the text, typically improving quality at the cost of inference speed\.

##### Lossless Speculative Decoding\.

Speculative decoding is a rejection sampling method to draw samples from a distributionppusing a proposal distributionqq\. Letp⁡\(v∣h\)p\(v\\mid h\)andq⁡\(v∣h\)q\(v\\mid h\)denote the verifier and draft distributions over a vocabulary𝒱\\mathcal\{V\}, given a prefixhh\. Speculative decoding accepts a proposaly∼q\(⋅∣h\)y\\sim q\(\\cdot\\mid h\)with probabilitymin⁡\{1,p⁡\(y∣h\)/q⁡\(y∣h\)\}\\min\\\{1,p\(y\\mid h\)/q\(y\\mid h\)\\\}, and upon rejection, samples a replacement with probability proportional to\[p⁡\(v∣h\)−q⁡\(v∣h\)\]\+\[p\(v\\mid h\)\-q\(v\\mid h\)\]\_\{\+\}, where\[a\]\+=max⁡\{a,0\}\[a\]\_\{\+\}=\\max\\\{a,0\\\}\. This rule preserves the verifier distribution\([Leviathan et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib1);[Chen et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib5)\)\. Letα⁡\(h\)\\alpha\(h\)denote the probability of accepting a proposal given accepted prefixhh\. Under lossless verification,

α\(h\)=∑v∈𝒱min\{p\(v∣h\),q\(v∣h\)\}=1−DTV\(p\(⋅∣h\),q\(⋅∣h\)\)\.\\alpha\(h\)=\\sum\_\{v\\in\\mathcal\{V\}\}\\min\\\{p\(v\\mid h\),q\(v\\mid h\)\\\}=1\-D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\.\(1\)
whereDTVD\_\{\\mathrm\{TV\}\}defines the total variation distance betweenppandqq\.

DTV​\(p,q\)=12​∑v∈𝒱\|p⁡\(v\)−q⁡\(v\)\|\.D\_\{\\mathrm\{TV\}\}\(p,q\)=\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\|p\(v\)\-q\(v\)\|\.\(2\)Thus, the acceptance rate of lossless speculative decoding is determined by the alignment of the drafterqqand verifierpp\. In practice, the space of possible prefixes is too large to estimateα⁡\(h\)\\alpha\(h\)for every prefixhh\. We therefore measure the expected acceptance rate over a distribution of prefixesℋ\\mathcal\{H\}\.

α¯=𝔼h∼ℋ​\[α​\(h\)\],\\bar\{\\alpha\}=\\mathbb\{E\}\_\{h\\sim\\mathcal\{H\}\}\[\\alpha\(h\)\],\(3\)which can be estimated with sampling\.

##### Lossy Speculative Decoding\.

Lossy speculative decoding relaxes the requirement of sampling exactly fromppto increase acceptance rate\. The degree of relaxation is theacceptance leniency, as introduced by[Leviathan et al\. \(2023, Appendix A\.5\)](https://arxiv.org/html/2610.31819#bib.bib1)\. In particular, we consider linear interpolation between the verifier and draft distributions\([Fu et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib29);[Hao and Mou, 2026](https://arxiv.org/html/2610.31819#bib.bib2);[Wang et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib28)\)and characterize its optimality under a total variation constraint\.

Suppose the targetppcan be replaced by a modified targetp~\\tilde\{p\}in the rejection\-sampling rule of Section[2](https://arxiv.org/html/2610.31819#S2.SS0.SSS0.Px2)\. Ifp~\(⋅∣h\)\\tilde\{p\}\(\\cdot\\mid h\)is chosen before drawingy∼q\(⋅∣h\)y\\sim q\(\\cdot\\mid h\), rejection sampling can be performed usingp~\(⋅∣h\)\\tilde\{p\}\(\\cdot\\mid h\)with an acceptance rate ofα\(h\)=1−DTV\(p~\(⋅∣h\),q\(⋅∣h\)\)\\alpha\(h\)=1\-D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\}\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\. Observe that the distance betweenp~\\tilde\{p\}andqqdetermines acceptance, while the distance betweenp~\\tilde\{p\}andppcorresponds to the deviation from the original output distributionpp\. Given a boundϵ≥0\\epsilon\\geq 0, the best achievable acceptance can be characterized by a constrained optimization

maxp~\(⋅∣h\)∈Δ\(𝒱\)1−DTV\(p~\(⋅∣h\),q\(⋅∣h\)\)subject toDTV\(p~\(⋅∣h\),p\(⋅∣h\)\)≤ϵ\\max\_\{\\tilde\{p\}\(\\cdot\\mid h\)\\in\\Delta\(\\mathcal\{V\}\)\}1\-D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\}\(\\cdot\\mid h\),\\,q\(\\cdot\\mid h\)\)\\quad\\text\{subject to\}\\quad D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\}\(\\cdot\\mid h\),\\,p\(\\cdot\\mid h\)\)\\leq\\epsilon\(4\)
whereΔ⁡\(𝒱\)\\Delta\(\\mathcal\{V\}\)is the set of probability distributions on the vocabulary\.

###### Theorem 1\(Optimality of Linear Acceptance\)\.

For a fixed prefixhhwithp\(⋅∣h\)≠q\(⋅∣h\)p\(\\cdot\\mid h\)\\neq q\(\\cdot\\mid h\)and a TV budget0≤ϵ≤DTV\(p\(⋅∣h\),q\(⋅∣h\)\)0\\leq\\epsilon\\leq D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\), an optimal target in Equation[4](https://arxiv.org/html/2610.31819#S2.E4)isp~⋆​\(v∣h\)=\(1−λ\)​p​\(v∣h\)\+λ​q​\(v∣h\),λ=ϵDTV\(p\(⋅∣h\),q\(⋅∣h\)\)\\tilde\{p\}^\{\\star\}\(v\\mid h\)=\(1\-\\lambda\)p\(v\\mid h\)\+\\lambda q\(v\\mid h\),\\qquad\\lambda=\\frac\{\\epsilon\}\{D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\}\(5\)and achieves maximum acceptance probabilityα⋆\(h\)=1−DTV\(p\(⋅∣h\),q\(⋅∣h\)\)\+ϵ\\alpha^\{\\star\}\(h\)=1\-D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\+\\epsilon\.

DTV​\(p~,q\)D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\},q\)ϵ\\epsilonDTV​\(p,q\)D\_\{\\mathrm\{TV\}\}\(p,q\)ppqqp~⋆\\tilde\{p\}^\{\\star\}p~\\tilde\{p\}DTV​\(p~,p\)≤ϵD\_\{\\mathrm\{TV\}\}\(\\tilde\{p\},p\)\\leq\\epsilon

\(a\)Optimal relaxed target distribution
\(b\)Motivating example
Figure 2:Illustrative examples\.Left: Visualization of the proof of Theorem[1](https://arxiv.org/html/2610.31819#Thmtheorem1)shows that the linear interpolated distributionp~⋆\\tilde\{p\}^\{\\star\}is the optimal solution to maximize drafterqqand verifierppalignment while staying within a constraint on deviation from the verifierϵ\\epsilon\. The solution to this constrained optimization isp~\\tilde\{p\}\. Right: A motivating example shows that the speed\-quality frontiers independently induced by diffusion effort and acceptance lenience in speculative decoding can occupy separate regions of the Pareto frontier\. We compare diffusion only with AR speculative decoding with leniency\.In the above theorem,λ\\lambdadictates the acceptance leniency by shifting the target toward the drafter\. Lossless sampling from the verifier occurs atλ=0\\lambda=0, whileλ=1\\lambda=1samples from the draft distribution\. Figure[2\(a\)](https://arxiv.org/html/2610.31819#S2.F2.sf1)gives an intuitive visual proof of the theorem\. Total variation distance is a distance metric, so geometrically the optimal path between two distributions will be a linear interpolation\. Refer to Appendix[A](https://arxiv.org/html/2610.31819#A1)or[B](https://arxiv.org/html/2610.31819#A2)for the proof\.

Note that this result characterizes the best acceptance obtainable by choosing a target distributionbeforeobserving the proposed token\. Another point of clarification in the above proof is that it is a statement on the next token distribution, which assumes conditioning on prefixhh\. These bounds are not trivially extendable to the joint distribution ofppandqqover multiple token positions\.

Cactus\([Hao and Mou, 2026](https://arxiv.org/html/2610.31819#bib.bib2)\)is another example that allows lossy acceptance\. Unlike linear acceptance, Cactus controls leniency by modifying the verifier distributionafterobserving a proposed token\. It increases the probability of that token subject to a bound on the modified target’s KL divergence, or any otherff\-divergence, from the verifier, with larger budgets permitting more lenient acceptance\. This dependence on the sampled token distinguishes Cactus from the linear acceptance defined above, whose target is chosen before sampling\.

## 3Speculative Diffusion

Because diffusion models can draft tokens in parallel, speculative diffusion is a promising approach to achieve fast and high quality LLM inference\. As discussed previously, both diffusion and speculative decoding have inherent speed quality tradeoffs\.

### 3\.1A Motivating Example

To motivate varying diffusion denoising effort and speculative decoding leniency jointly, it is useful to examine each parameter independently\. This work is motivated by the following observations\.

First, even with a large leniency under lossy speculative decoding and a very small AR drafter, the range of achievable drafting speeds is much narrower than that of in a dLLM when varying the diffusion effort\. For example, Figure[2\(b\)](https://arxiv.org/html/2610.31819#S2.F2.sf2)shows that ignoring the verifier and completely relying on the AR drafter distribution Qwen2\.5\-1\.5B\-Instruct\([Qwen Team, 2024a](https://arxiv.org/html/2610.31819#bib.bib18)\)achieves a speedup, but does cannot capture the full frontier\. Meanwhile, a diffusion model is capable of can running over 10×\\timesfaster\. This suggests dLLM drafters expose a new axis of variation on its drafting quality and speed in a way that traditional AR drafters cannot\.

Second, even though varying diffusion effort with the dLLM alone can yield a speed and quality on the Pareto frontier, most dLLMs have significantly lower accuracies than those of the state\-of\-the\-art AR LLMs\. For example, in Figure[2\(b\)](https://arxiv.org/html/2610.31819#S2.F2.sf2)the accuracy of Dream 7B plateaus despite running a high number of diffusion steps\. Therefore, a way to uplift the accuracy of the diffusion LLMs with an AR model as a guide would be desirable\. Many recent works echo this sentiment\([Liu et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib43);[Israel et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib41);[Hu et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib42)\)\.

With these observations, it is evident that the speed quality tradeoffs of diffusion and lossy speculative decoding are complementary\. Diffusion alone can achieve higher speeds\. Speculative decoding alone can achieve higher quality\. Because each method is Pareto optimal independently, it is possible that combining the approaches can unlock further improvements in the Pareto frontier of speed and quality\.

### 3\.2Multi\-Step Diffusion Drafting

Many instances of speculative diffusion focus on single step diffusion drafting[Fu et al\. \(2026\)](https://arxiv.org/html/2610.31819#bib.bib13);[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.31819#bib.bib30);[Chen et al\. \(2026\)](https://arxiv.org/html/2610.31819#bib.bib16);[Cheng et al\. \(2026b\)](https://arxiv.org/html/2610.31819#bib.bib17)\. In this work, we use a diffusion drafter for multiple steps per verifier call, as shown in Algorithm[1](https://arxiv.org/html/2610.31819#alg1)\. While diffusion effort can be parameterized directly in the number of stepsKK\(as mentioned in Section[2](https://arxiv.org/html/2610.31819#S2.SS0.SSS0.Px1)\), other criteria can also be used as parametrization to control the diffusion effort\. For example, confidence thresholds\([Nie et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib39)\)or entropy bounds\([Ben\-Hamu et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib38)\)can indirectly affect the number of diffusion steps\.

Algorithm 1Speculative diffusion1:prompt

hh; block size

BB; denoising steps

KK;

2:denoiser

fθf\_\{\\theta\}; verifier

pp; target ruleRelaxTarget

3:while

EOS∉h\\texttt\{EOS\}\\notin hdo

4:

\(y1:B,q1:B\)←DiffusionDraft\(fθ,h,B,K\)\(y\_\{1:B\},q\_\{1:B\}\)\\leftarrow\\textsc\{DiffusionDraft\}\(\\,f\_\{\\theta\},h,B,K\)
5:

p1:B\+1←VerifierForward\(p,h,y1:B\)p\_\{1:B\+1\}\\leftarrow\\textsc\{VerifierForward\}\(p,h,y\_\{1:B\}\)
6:

p~1:B←RelaxTarget\(p1:B,q1:B,y1:B\)\\tilde\{p\}\_\{1:B\}\\leftarrow\\textsc\{RelaxTarget\}\(p\_\{1:B\},q\_\{1:B\},y\_\{1:B\}\)
7:

k←0k\\leftarrow 0
8:do

9:

k←k\+1k\\leftarrow k\+1
10:

u∼𝒰⁡\(0,1\)u\\sim\\mathcal\{U\}\(0,1\)
11:while

k≤Bk\\leq Band

u≤p~k​\(yk\)/qk​\(yk\)u\\leq\\tilde\{p\}\_\{k\}\(y\_\{k\}\)/q\_\{k\}\(y\_\{k\}\)
12:

yk∼\{Normalize​\(\[p~k−qk\]\+\),k≤B,pk,k=B\+1\.\\displaystyle y\_\{k\}\\sim\\begin\{cases\}\\textsc\{Normalize\}\(\[\\tilde\{p\}\_\{k\}\-q\_\{k\}\]\_\{\+\}\),&k\\leq B,\\\\\[2\.0pt\] p\_\{k\},&k=B\+1\.\\end\{cases\}
13:

h←Append\(h,y1:k\)h\\leftarrow\\textsc\{Append\}\(h,y\_\{1:k\}\)
14:endwhile

15:return

truncate​\(h\)\\textsc\{truncate\}\(h\)

In speculative decoding with an autoregressive drafter, when sampling token at positioniiit is conditioned on tokens prior to positioniithat have been accepted by the verifier\. Multi\-step diffusion drafting does not maintain this property\. A final sampled token can be obtained through many possible denoising trajectories\. Evaluating the marginal probability at a position would require summing the probability over an intractable number of trajectories or approximating via an evidence lower bound \(ELBO\)\([Zhao et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib40)\)\. Therefore, it is not immediately clear what proposal probabilities should be used in rejection sampling\.

Since the diffusion trajectory is stochastic, the proposal distribution itself can be seen as a random vector over the probability simplexΔ⁡\(𝒱\)\\Delta\(\\mathcal\{V\}\)\. Proposition[2](https://arxiv.org/html/2610.31819#Thmproposition2)shows that without knowing the underlying distribution of the proposal distribution random vector, rejection sampling can still be performed on the proposal distribution that generated each token, maintaining exact samples from the verifier\. In the context of speculative diffusion, this means the final step probabilities \(e\.g\. final transition probabilities in uniform diffusion or marginals in masked diffusion\) can be used as the proposal probabilities in rejection sampling\.

### 3\.3A Simple Throughput Model

\(a\)Theoretical\(b\)Measured\(c\)Theoretical\(d\)MeasuredFigure 3:A model of speculative diffusion\.Theoretical and measured tradeoffs for Dream\-7B drafting for Qwen2\.5\-32B on GSM8K\. \(a\) The theoretical acceptance rate \(speed\) and verifier agreement \(quality\) under linear acceptance for each ofKKdiffusion steps per round\. \(b\) Empirically measured acceptance rate for eachKK\. \(c\) A transformation from acceptance rate to throughput according to a simple model of throughput\. \(d\) Empirically measured throughput\.A theoretical model can aid our understanding in how speed and generation quality vary as a function of the diffusion effort and leniency\. To this end, the expected acceptance rateα¯=1−DTV​\(p~,q\)\\bar\{\\alpha\}=1\-D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\},q\)can be used to model throughput and the similarity between the verifier distribution and the modified target distribution1−DTV​\(p~,p\)1\-D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\},p\)can be used as a proxy for generation quality, where both distances can be derived from an empirically profiledDTV​\(p,q\)D\_\{\\mathrm\{TV\}\}\(p,q\)under Theorem[1](https://arxiv.org/html/2610.31819#Thmtheorem1)at different effort levels\. For diffusion effort usingKKsteps per drafting round, letℋK\\mathcal\{H\}\_\{K\}be a fixed distribution of prefixes andqKq\_\{K\}the corresponding draft distribution\. Averaging over the distribution of prefixes, we definedK=𝔼h∼ℋK\[DTV\(p\(⋅∣h\),qK\(⋅∣h\)\)\]d\_\{K\}=\\mathbb\{E\}\_\{h\\sim\\mathcal\{H\}\_\{K\}\}\[D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\_\{K\}\(\\cdot\\mid h\)\)\], which is the TV distance between the verifier and drafter at effortKKwith zero leniency\. Applying the linear acceptance rule in Theorem[1](https://arxiv.org/html/2610.31819#Thmtheorem1)overHKH\_\{K\}gives the expected acceptance rate and verifier distance as

α¯K​\(λ\)=1−\(1−λ\)​dKand1−D¯TV​\(p,p~\)=1−λ​dK\.\\bar\{\\alpha\}\_\{K\}\(\\lambda\)=1\-\(1\-\\lambda\)d\_\{K\}\\qquad\\text\{ and \}\\qquad 1\-\\overline\{D\}\_\{\\mathrm\{TV\}\}\(p,\\tilde\{p\}\)=1\-\\lambda d\_\{K\}\.\(6\)
Letcd​\(K\)c\_\{d\}\(K\)andcvc\_\{v\}denote the time it takes to call the drafter and verifier per drafting round respectively\. Empirically,cd​\(K\)≈a\+b​Kc\_\{d\}\(K\)\\approx a\+bKandcvc\_\{v\}is constant\. Proposition[1](https://arxiv.org/html/2610.31819#Thmproposition1)in Appendix[C](https://arxiv.org/html/2610.31819#A3)then gives the theoretical throughput as

𝒯⁡\(K,α¯\)=∑j=0Bα¯jcd​\(K\)\+cv\.\\mathcal\{T\}\(K,\\bar\{\\alpha\}\)=\\frac\{\\sum\_\{j=0\}^\{B\}\\bar\{\\alpha\}^\{\\,j\}\}\{c\_\{d\}\(K\)\+c\_\{v\}\}\.\(7\)
Figures[3\(b\)](https://arxiv.org/html/2610.31819#S3.F3.sf2)and[3\(d\)](https://arxiv.org/html/2610.31819#S3.F3.sf4)directly measure each quantity predicted in Figures[3\(a\)](https://arxiv.org/html/2610.31819#S3.F3.sf1)and[3\(c\)](https://arxiv.org/html/2610.31819#S3.F3.sf3), and the details for obtaining these results can be found in Appendix[F\.4](https://arxiv.org/html/2610.31819#A6.SS4)\. The theoretical model captures two important qualitative features seen in the empirically measured results: \(1\) The tradeoff between acceptance rate and verifier agreement is approximately linear and \(2\) no denoising budget captures the full throughput\-quality Pareto frontier\. The quantitative discrepancy between theoretical and measured results is driven by multiple factors\. First, the assumption of a universal acceptance rateα¯\\bar\{\\alpha\}is incorrect\. Leniency changes which draft tokens are accepted and hence the prefixes on which subsequent predictions are conditioned\. More formally, the prefix distributionℋK\\mathcal\{H\}\_\{K\}encountered during the empirical profiling differs from the prefix distribution at higher leniency, resulting in a different value ofdKd\_\{K\}\. Another difference between theory and practice is that the theory computes acceptance over an entire block, but in practice throughput can be affected by EOS termination effectively reducing drafting block size\.

This simple setup illustrates the qualitative features of speculative diffusion when varying diffusion effort and acceptance leniency\. However, it is not obvious how quality measured by total variation translates to downstream metrics such as reasoning accuracy\. Furthermore, the linear interpolation acceptance criterion enables simpler modeling of acceptance rate, but other acceptance criteria may not behave the same way\. These questions will be addressed in the following experimental section\.

## 4Experiments

Figure 4:Examples of the Pareto frontier\.We provide examples of speculative diffusion for both masked diffusion \(left\) and uniform diffusion \(right\)\. Each point signifies a configuration of acceptance leniency in verification and diffusion effort \(KKsteps\) in drafting\. We observe the same qualitative trend that the speed\-quality Pareto frontier is occupied by many different settings of diffusion effort which require different levels of lenience\.The goal of our experiments is to empirically explore the two tradeoffs of speculative diffusion: diffusion effort and leniency\. We first establish a common experimental setup, then conduct experiments in two different settings\.

### 4\.1Experimental Setup

We first examine masked and uniform diffusion using the following model pairs: Dream\-7B\([Ye et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib12)\)with Qwen2\.5\-32B and DiffusionGemma\-26B\-A4B\([DiffusionGemma Team, 2026](https://arxiv.org/html/2610.31819#bib.bib3)\)with Gemma4\-31B\([Gemma Team, 2026](https://arxiv.org/html/2610.31819#bib.bib20)\)\. We then compare several other recent speculative diffusion models under a common Qwen3\-8B verifier\.

In general, we benchmark quality using GSM8K\([Cobbe et al\., 2021](https://arxiv.org/html/2610.31819#bib.bib21)\), GPQA Diamond\([Rein et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib23)\), and HumanEval\([Chen et al\., 2021](https://arxiv.org/html/2610.31819#bib.bib22)\)for math, reasoning, and code\. In our Dream\-Qwen and DiffusionGemma\-Gemma experiments, some model\-benchmark combinations are excluded based on two criteria that depend on the standalone capabilities of the drafter and verifier\. The first criterion is that if the standalone accuracies of the drafter and verifier models are nearly the same on a benchmark, then there is no benefit to running speculative decoding, and thus the combination will be excluded\. The second criterion is that if the drafter model performs worse than random, then the combination will be excluded\.

Table[1](https://arxiv.org/html/2610.31819#A6.T1)reports the baseline accuracies for both the standalone drafter and verifier of the Dream, SDAR, and DiffusionGemma pairs, on GSM8K, HumanEval, and GPQA Diamond\. Based on these measurements, we evaluate Dream and SDAR on GSM8K and HumanEval, and DiffusionGemma on GPQA\. An evaluation configuration consists of a benchmark, model pair, sampling configuration, diffusion effort, and acceptance leniency\. The detailed experimental configurations can be found in the Appendix[F](https://arxiv.org/html/2610.31819#A6)\.

### 4\.2Joint Exploration Over Two Tradeoffs

Figure[4](https://arxiv.org/html/2610.31819#S4.F4)is obtained by fixing a denoising budgetKKand varying the acceptance leniency\. In this particular plot, we display the results using the modified Cactus acceptance criterion instead of linear acceptance to demonstrate the generality of our findings\. The corresponding linear\-acceptance results are in Figures[7\(b\)](https://arxiv.org/html/2610.31819#A7.F7.sf2)and[8\(b\)](https://arxiv.org/html/2610.31819#A7.F8.sf2)and are qualitatively similar\. See Appendix[E](https://arxiv.org/html/2610.31819#A5)for details\.

Every point in Figure[4](https://arxiv.org/html/2610.31819#S4.F4)represents the speed and accuracy measured from a diffusion effort, acceptance leniency pair\. We then visualize the Pareto frontier over all pairs for latency and throughput\. Note that latency is wall\-clock time to problem completion, while throughput is the number of generated tokens divided by the wall clock time to problem completion\. Both are needed to quantify speed, because throughput can be over optimized by generating unnecessary tokens, yet latency improvements achieved entirely from shorter generation lengths is also not desirable\. The Pareto frontier is visualized as a thick connected line, while hollow markers are measurements not on the frontier\. For visual clarity, we omit the points on the Pareto frontier itself and instead mark them with white ticks on the line\. The Pareto frontier’s color identifies the diffusion effortKKin that region of the frontier\. Between measurements of different colors on the frontier, we apply a color gradient\. This makes it possible to see which denoising budgets contribute to the frontier\.

For both model pairs in Figure[4](https://arxiv.org/html/2610.31819#S4.F4), several values ofKKcontribute to the frontier\. Varying leniency at a single diffusion effort configuration will fail to achieve Pareto\-optimal results\. In general, low diffusion effort with strict acceptance requirements tends to capture the higher accuracy regions of the Pareto frontier, while higher diffusion effort with more lenient acceptance captures the higher speed regions\. This is natural because as previously observed, diffusion has higher speed but lower quality and autoregression has higher quality but lower speed\. The observation that the Pareto frontier contains many configurations of diffusion effort and acceptance leniency persists across drafting block sizes\. In Appendix[G\.2](https://arxiv.org/html/2610.31819#A7.SS2), we ablate drafting block size and show that no single diffusion effort captures the full frontier\.

Figure 5:Comparison of Pareto frontiers\.For a single Qwen3 8B verifier, we benchmark several speculative diffusion methods\. Each line corresponds to the achievable speed\-quality frontier of a specific draft model\. Dashed curves fix one diffusion step and vary leniency, while solid curves are varied over diffusion effort and acceptance lenience\. For the SDAR models that support multi\-step speculative diffusion, varying over the parameters jointly significantly extends the Pareto frontier\.
### 4\.3Comparing Drafters with a Shared Verifier

While the previous experimental setup analyzes speculative diffusion for model pairs that were trained separately, many recent works train specifically for diffusion speculation\. Under a Qwen3 8B\([Yang et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib36)\)verifier, these works include Uno\([Sahoo et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib30)\), DFlash\([Chen et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib16)\), and DSpark\([Cheng et al\., 2026b](https://arxiv.org/html/2610.31819#bib.bib17)\)\. For this experiment we also include SDAR\([Cheng et al\., 2026a](https://arxiv.org/html/2610.31819#bib.bib37)\), which notably was not trained for the task of speculative decoding\.

We compare SDAR\-1\.7B, SDAR\-8B, Uno, DFlash, and DSpark on GSM8K and HumanEval with a shared Qwen3\-8B verifier\. For each model, we apply linear acceptance atλ∈\{0,0\.2,0\.4,0\.6,0\.8,1\}\\lambda\\in\\\{0,0\.2,0\.4,0\.6,0\.8,1\\\}, from lossless verification to full acceptance\. Uno, DFlash, and DSpark are trained specifically for one step diffusion drafting\. DFlash and DSpark apply feature conditioning from the verifier, making multiple diffusion steps per verifier call impossible\. Thus, for these models we only include their Pareto frontier using one step drafting and acceptance lenience\. For Uno, we evaluateK∈\{1,2,4\}K\\in\\\{1,2,4\\\}in Appendix[G\.5](https://arxiv.org/html/2610.31819#A7.SS5), but we find onlyK=1K=1one\-step diffusion occupies its Pareto frontier\. Because SDAR is trained as a standalone diffusion model with a multistep objective, we measure it’s Pareto frontier by varying diffusion steps as well\. We vary SDAR’s adaptive denoising effort using a confidence thresholdτ∈\{0\.3,0\.5,0\.7,0\.9\}\\tau\\in\\\{0\.3,0\.5,0\.7,0\.9\\\}and also evaluate fixed one\-step \(K=1K=1\) drafts\. For each model size, we compare the joint frontier over all these settings with the frontier restricted toK=1K=1\. Appendix[G\.4](https://arxiv.org/html/2610.31819#A7.SS4)shows all measured configurations\. The results are shown in Figure[5](https://arxiv.org/html/2610.31819#S4.F5)\. Joint variation over diffusion effort and acceptance leniency substantially improves SDAR’s speed\-accuracy tradeoff\. While the specialized one\-step drafters remain strong, especially in throughput, joint variation places SDAR on the frontier despite not having specialized training and architecture for speculative diffusion\. These results motivate training more specialized drafters for multi\-step refinement\.

## 5Related Work

##### Diffusion drafters\.

Recent work on diffusion drafting for speculative decoding aims to generate proposals with fewer sequential model evaluations\. SpecDiff introduced diffusion drafters and studied how denoising steps balance drafting cost against acceptance\([Christopher et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib25)\)\. Subsequent work improves draft\-verifier alignment through causal\-consistency path search\([Li et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib27)\)or conditioning diffusion drafts on target\-model features\([Chen et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib16)\), while hybrid models integrate diffusion drafting and autoregressive verification within one model\([Liu et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib15);[Fu et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib13);[Sahoo et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib30)\)\. In general, diffusion drafting in speculative decoding rarely applies multiple steps of diffusion effort because it does not typically yield benefit\([Sandler et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib44), Appendix A\.2\)\. While quadratic decoding\([Samragh et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib31)\)applies multiple steps of diffusion for the purpose of tree\-based verification\([Miao et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib32)\), tree\-based speculative decoding is orthogonal to acceptance leniency and is beyond the scope of this work\.

##### Lossy speculative decoding\.

Multiple works have introduced leniency into speculative decoding acceptance\. It was first introduced in the seminal speculative decoding paper by[Leviathan et al\. \(2023\)](https://arxiv.org/html/2610.31819#bib.bib1)\. Later theoretical work formalized the tradeoff as a linear relationship between rejection rates and total variation loss\([Yin et al\., 2024](https://arxiv.org/html/2610.31819#bib.bib7)\)\. In this paper, we spotlight the linear acceptance criterion which has been mentioned in several previous works under different names: collaborative decoding\([Fu et al\., 2025](https://arxiv.org/html/2610.31819#bib.bib29)\), linear interpolation\([Hao and Mou, 2026](https://arxiv.org/html/2610.31819#bib.bib2)\), and dynamic ensemble verification\([Wang et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib28)\)\. While the method is not original, to the best of our knowledge, our formulation is the first analyze it from the perspective of constrained optimization and show that it is optimal under a total variation constraint\. Furthermore, these works do not investigate leniency in the context of speculative diffusion\. Nemotron\-Labs\-Diffusion briefly describes a verifier that interpolates AR and diffusion distributions\([Fu et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib13), Appendix C\), but it is mentioned as a possible direction without experimental evaluation\.

## 6Conclusion

This work demonstrates that speculative decoding with diffusion drafters, also known as speculative diffusion, is governed by two complementary mechanisms that trade speed for quality\. Specifically, these mechanisms are the denoising effort of the drafter and the acceptance leniency of the verifier\. Our cost model and empirical findings show that neither parameter independently can capture the full Pareto frontier between generation speed and task accuracy\. The speed quality trade\-offs of speculative diffusion expand the design space for large language model serving\. A promising future direction is to improve query routing, where incoming requests are mapped to distinct configurations along the Pareto frontier according to their task difficulty and tolerance for error\. Beyond routing at the query level, future systems could employ dynamic scheduling policies that adjust denoising budgets and verification thresholds at runtime\.

### AI Usage Disclosure

The authors used LLMs to assist in finding relevant papers, plotting results, polishing LaTeX, and drafting code\. We take responsibility for the final content of this work\.

#### Acknowledgments

This work was funded in part by the DARPA CLARA program under Agreement No\. HR0011269E224, NSF CAREER \#2341040, Schmidt Sciences, and gifts from Cisco Research, Qualcomm, and Amazon\.

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## Appendix AProof of Theorem[1](https://arxiv.org/html/2610.31819#Thmtheorem1)

###### Proof\.

For any feasiblep~\(⋅∣h\)\\tilde\{p\}\(\\cdot\\mid h\), the triangle inequality gives

DTV\(p~\(⋅∣h\),q\(⋅∣h\)\)\\displaystyle D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\}\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)≥DTV\(p\(⋅∣h\),q\(⋅∣h\)\)−DTV\(p~\(⋅∣h\),p\(⋅∣h\)\)\\displaystyle\\geq D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\-D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\}\(\\cdot\\mid h\),p\(\\cdot\\mid h\)\)≥DTV\(p\(⋅∣h\),q\(⋅∣h\)\)−ϵ\.\\displaystyle\\geq D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\-\\epsilon\.Hence, the acceptance probability is at most

1−DTV\(p\(⋅∣h\),q\(⋅∣h\)\)\+ϵ\.1\-D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\+\\epsilon\.
We must now show that this bound is attained\. Consider

p~⋆\(⋅∣h\)=\(1−λ\)p\(⋅∣h\)\+λq\(⋅∣h\),λ=ϵDTV\(p\(⋅∣h\),q\(⋅∣h\)\)\.\\tilde\{p\}^\{\\star\}\(\\cdot\\mid h\)=\(1\-\\lambda\)p\(\\cdot\\mid h\)\+\\lambda q\(\\cdot\\mid h\),\\qquad\\lambda=\\frac\{\\epsilon\}\{D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\}\.Under the theorem’s assumptions,λ∈\[0,1\]\\lambda\\in\[0,1\]\. By the definition of total variation distance,

DTV\(p~⋆\(⋅∣h\),p\(⋅∣h\)\)\\displaystyle D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\}^\{\\star\}\(\\cdot\\mid h\),p\(\\cdot\\mid h\)\)=12​∑v∈𝒱\|λ⁡\(q⁡\(v∣h\)−p⁡\(v∣h\)\)\|\\displaystyle=\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|\\lambda\\bigl\(q\(v\\mid h\)\-p\(v\\mid h\)\\bigr\)\\right\|=λDTV\(p\(⋅∣h\),q\(⋅∣h\)\)=ϵ,\\displaystyle=\\lambda D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)=\\epsilon,sop~⋆\\tilde\{p\}^\{\\star\}is feasible\. Similarly,

DTV\(p~⋆\(⋅∣h\),q\(⋅∣h\)\)\\displaystyle D\_\{\\mathrm\{TV\}\}\(\\tilde\{p\}^\{\\star\}\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)=12​∑v∈𝒱\|\(1−λ\)​\(p⁡\(v∣h\)−q⁡\(v∣h\)\)\|\\displaystyle=\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|\(1\-\\lambda\)\\bigl\(p\(v\\mid h\)\-q\(v\\mid h\)\\bigr\)\\right\|=\(1−λ\)DTV\(p\(⋅∣h\),q\(⋅∣h\)\)\\displaystyle=\(1\-\\lambda\)D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)=DTV\(p\(⋅∣h\),q\(⋅∣h\)\)−ϵ\.\\displaystyle=D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\-\\epsilon\.Thusp~⋆\\tilde\{p\}^\{\\star\}attains the lower bound and therefore achieves acceptance rate

1−DTV\(p\(⋅∣h\),q\(⋅∣h\)\)\+ϵ\.1\-D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\+\\epsilon\.∎

## Appendix BAlternative Proof of Theorem[1](https://arxiv.org/html/2610.31819#Thmtheorem1)

This alternative proof helps to provide a more geometric understanding of the result\.

Here we abbreviatep\(⋅∣h\)p\(\\cdot\\mid h\)asppandq\(⋅∣h\)q\(\\cdot\\mid h\)asqq\.

Viewppandqqas probability measures on aσ\\sigma\-algebraℱ\\mathcal\{F\}on a setΩ\\Omega, and define

pr:=p\+r⁡\(q−p\)for​r∈\[0,1\]\.p\_\{r\}\\;:=\\;p\+r\\,\(q\-p\)\\quad\\text\{for \}r\\in\[0,\\,1\]\.
Since the total variation is a norm on the real vector space of finite signed measures, denote it as∥⋅∥TV\\\|\\cdot\\\|\_\{\\mathrm\{TV\}\}, but we shall prove the statement for any norm∥⋅∥\\\|\\cdot\\\|\.

###### Proof\.

We show that

‖pr−q‖≤‖p~−q‖any signed measure​p~​such that​‖p~−p‖≤‖pr−p‖\.\\\|p\_\{r\}\-q\\\|\\;\\leq\\;\\\|\\tilde\{p\}\-q\\\|\\quad\\text\{ any signed measure \}\\tilde\{p\}\\text\{ such that \}\\\|\\tilde\{p\}\-p\\\|\\leq\\\|p\_\{r\}\-p\\\|\.\(8\)
Indeed, by the absolute homogeneity property of norms, we have

‖pr−q‖=‖\(1−r\)​\(p−q\)‖=\(1−r\)​‖p−q‖\\\|p\_\{r\}\-q\\\|\\;=\\;\\\|\(1\-r\)\\,\(p\-q\)\\\|\\;=\\;\(1\-r\)\\,\\\|p\-q\\\|‖pr−p‖=‖r⁡\(q−p\)‖=r​‖p−q‖\\\|p\_\{r\}\-p\\\|\\;=\\;\\\|r\(q\-p\)\\\|\\;=\\;r\\,\\\|p\-q\\\|
Now for anyp~\\tilde\{p\}such that‖p~−p‖≤‖pr−p‖=r​‖q−p‖\\\|\\tilde\{p\}\-p\\\|\\leq\\\|p\_\{r\}\-p\\\|=r\\,\\\|q\-p\\\|, the triangle inequality implies that

‖p~−q‖≥‖q−p‖−‖p~−p‖≥‖q−p‖−r​‖q−p‖=\(1−r\)\|q−p\|=‖pr−q‖\.\\\|\\tilde\{p\}\-q\\\|\\;\\geq\\;\\\|q\-p\\\|\-\\\|\\tilde\{p\}\-p\\\|\\;\\geq\\;\\\|q\-p\\\|\-r\\,\\\|q\-p\\\|\\;=\\;\(1\-r\)\\,\\\|q\-p\\\|\\;=\\;\\\|p\_\{r\}\-q\\\|\.
In words, this is saying that the signed measureprp\_\{r\}is the closest point toqqamong all points that are at most‖pr−p‖\\\|p\_\{r\}\-p\\\|distance away frompp\.

Note that Equation[8](https://arxiv.org/html/2610.31819#A2.E8)is true for any signed measurep~\\tilde\{p\}, so in particular it is true for any probability measurep~\\tilde\{p\}\.

Also note that it is easy to see thatprp\_\{r\}is a probability measure, which means that the probability measureprp\_\{r\}is the closest toqqamong all signed measures within the ball, and thus the closest among all probability measures within the ball\.

Now, with the choice ofrrequal toλ=ϵDTV\(p\(⋅∣h\),q\(⋅∣h\)\)\\lambda=\\frac\{\\epsilon\}\{D\_\{\\mathrm\{TV\}\}\(p\(\\cdot\\mid h\),q\(\\cdot\\mid h\)\)\}, it follows that the optimal solution to Equation[4](https://arxiv.org/html/2610.31819#S2.E4)isp~⋆​\(v∣h\)=\(1−λ\)​p​\(v∣h\)\+λ​q​\(v∣h\)\\tilde\{p\}^\{\\star\}\(v\\mid h\)=\(1\-\\lambda\)p\(v\\mid h\)\+\\lambda q\(v\\mid h\)\. ∎

## Appendix CExpected Number of Generated Tokens per Drafting Round

###### Proposition 1\.

Assume there is a draft sequence of lengthBB\. Let𝟙𝕚\\mathbbold\{1\}\_\{i\}be the indicator variable for whether tokeniiis accepted conditioned on all previous tokens being accepted, wherei∈\[B\]i\\in\[B\]\. Assume\{𝟙𝕚\}\\\{\\mathbbold\{1\}\_\{i\}\\\}are independent and letℙ⁡\(𝟙𝕚=𝟙\)=ℙ𝕚\\mathbb\{P\}\(\\mathbbold\{1\}\_\{i\}=1\)=P\_\{i\}, wherePiP\_\{i\}is another set of i\.i\.d\. random variables\. Letμ:=𝔼​Pi\\mu:=\\mathbb\{E\}P\_\{i\}\. Then the expected number of generated tokens is

∑n=0Bμn=\{1−μB\+11−μfor​μ≠1,1\+Bfor​μ=1\.\\sum\_\{n=0\}^\{B\}\\mu^\{n\}=\\begin\{cases\}\\dfrac\{1\-\\mu^\{B\+1\}\}\{1\-\\mu\}&\\text\{for \}\\mu\\neq 1,\\\\\[6\.45831pt\] 1\+B&\\text\{for \}\\mu=1\.\\end\{cases\}\(9\)

###### Proof\.

The expected number of accepted tokens for a draft sequence of lengthBBis

A\(B\)=𝔼\[𝟙𝟙\+𝟙𝟙𝟙𝟚\+⋯\+𝟙𝟙𝟙𝟚⋯𝟙𝔹\]=∑𝕟=𝟙𝔹𝔼∏𝕚=𝟙𝕟𝟙𝕚\.A\(B\)=\\mathbb\{E\}\\big\[\\mathbbold\{1\}\_\{1\}\+\\mathbbold\{1\}\_\{1\}\\mathbbold\{1\}\_\{2\}\+\\dots\+\\mathbbold\{1\}\_\{1\}\\mathbbold\{1\}\_\{2\}\\cdots\\mathbbold\{1\}\_\{B\}\\big\]=\\sum\_\{n=1\}^\{B\}\\mathbb\{E\}\\prod\_\{i=1\}^\{n\}\\mathbbold\{1\}\_\{i\}\.\(10\)
Since the indicator variables𝟙𝕚\\mathbbold\{1\}\_\{i\}are independent, we have

𝔼\[∏i=1n𝟙𝕚\|ℙ𝟙,…,ℙ𝕟\]=ℙ\(𝟙𝕚=𝟙∀𝕚≤𝕟\)=∏𝕚=𝟙𝕟ℙ𝕚,\\mathbb\{E\}\\Big\[\\prod\_\{i=1\}^\{n\}\\mathbbold\{1\}\_\{i\}\\,\\Big\|\\,P\_\{1\},\\dots,P\_\{n\}\\Big\]=\\mathbb\{P\}\(\\mathbbold\{1\}\_\{i\}=1\\ \\forall i\\leq n\)=\\prod\_\{i=1\}^\{n\}P\_\{i\},\(11\)and

𝔼⁡\[∏i=1n𝟙𝕚\]\\displaystyle\\mathbb\{E\}\\Big\[\\prod\_\{i=1\}^\{n\}\\mathbbold\{1\}\_\{i\}\\Big\]=𝔼\[𝔼\[∏i=1n𝟙𝕚\|ℙ𝟙,…,ℙ𝕟\]\]law of total expectation\\displaystyle=\\mathbb\{E\}\\Big\[\\mathbb\{E\}\\Big\[\\prod\_\{i=1\}^\{n\}\\mathbbold\{1\}\_\{i\}\\,\\Big\|\\,P\_\{1\},\\dots,P\_\{n\}\\Big\]\\Big\]\\quad\\text\{law of total expectation\}=𝔼∏i=1nPiby Equation[11](https://arxiv.org/html/2610.31819#A3.E11)\\displaystyle=\\mathbb\{E\}\\prod\_\{i=1\}^\{n\}P\_\{i\}\\quad\\text\{by Equation~\\ref\{eq:conditional\-product\}\}=∏i=1n𝔼Pias\{Pi\}are i\.i\.d\.\\displaystyle=\\prod\_\{i=1\}^\{n\}\\mathbb\{E\}P\_\{i\}\\quad\\text\{as \}\\\{P\_\{i\}\\\}\\text\{ are i\.i\.d\.\}=μnas​μ=𝔼​Pi,\\displaystyle=\\mu^\{n\}\\quad\\text\{as \}\\mu=\\mathbb\{E\}P\_\{i\},\(12\)Therefore,

A⁡\(B\)=∑n=1B𝔼​∏i=1n𝟙𝕚=∑𝕟=𝟙𝔹μ𝕟\.A\(B\)=\\sum\_\{n=1\}^\{B\}\\mathbb\{E\}\\prod\_\{i=1\}^\{n\}\\mathbbold\{1\}\_\{i\}=\\sum\_\{n=1\}^\{B\}\\mu^\{n\}\.\(13\)
Each drafting round also commits one more token, which is either the correction or the bonus token, so the expected number of generated tokens per drafting round is

1\+A⁡\(B\)=1\+∑n=1Bμn=∑n=0Bμn\.1\+A\(B\)=1\+\\sum\_\{n=1\}^\{B\}\\mu^\{n\}=\\sum\_\{n=0\}^\{B\}\\mu^\{n\}\.\(14\)
∎

## Appendix DRejection Sampling for Multi\-Step Diffusion Drafting

###### Proposition 2\.

LetQ:Ω→Δ⁡\(𝒱\)Q:\\Omega\\to\\Delta\(\\mathcal\{V\}\)be a random vector representing the final probability vector with which the draft token is sampled from\.QQis of unknown distribution because the diffusion trajectory that produced the drafter probability vector is unknown\. LetX:Ω→𝒱X:\\Omega\\to\\mathcal\{V\}be a random variable that represents the outcome of performing rejection sampling with a fixed target distributionp~\\tilde\{p\}and the proposal distributionQQ\. Thenℙ⁡\(X=v\)=p~​\(v\)∀v∈𝒱\\mathbb\{P\}\(X=v\)=\\tilde\{p\}\(v\)\\quad\\forall v\\in\\mathcal\{V\}\(15\)

###### Proof of Proposition[2](https://arxiv.org/html/2610.31819#Thmproposition2)\.

Based on rejection sampling theory\([Leviathan et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib1);[Chen et al\., 2023](https://arxiv.org/html/2610.31819#bib.bib5)\), conditioned on the actual probabilityQQwith which the drafted token is sampled from, we know that

𝔼⁡\(𝟙𝕏=𝕧∣ℚ\)=ℙ⁡\(𝕏=𝕧∣ℚ\)=𝕡~​\(𝕧\)∀𝕧∈𝒱\.\\mathbb\{E\}\(\\mathbbold\{1\}\_\{X=v\}\\mid Q\)=\\mathbb\{P\}\(X=v\\mid Q\)=\\tilde\{p\}\(v\)\\quad\\forall v\\in\\mathcal\{V\}\.\(16\)
So by the law of total expectation, we have

ℙ⁡\(X=v\)=𝔼⁡\(𝟙𝕏=𝕧\)=𝔼𝕢​\[𝔼⁡\(𝟙𝕏=𝕧∣ℚ=𝕢\)\]=𝔼𝕢​\[𝕡~​\(𝕧\)\]=𝕡~​\(𝕧\)∀𝕧∈𝒱\\mathbb\{P\}\(X=v\)=\\mathbb\{E\}\(\\mathbbold\{1\}\_\{X=v\}\)=\\mathbb\{E\}\_\{q\}\\\!\\left\[\\mathbb\{E\}\(\\mathbbold\{1\}\_\{X=v\}\\mid Q=q\)\\right\]=\\mathbb\{E\}\_\{q\}\[\\tilde\{p\}\(v\)\]=\\tilde\{p\}\(v\)\\quad\\forall v\\in\\mathcal\{V\}Thus Equation[15](https://arxiv.org/html/2610.31819#A4.E15)holds\. ∎

## Appendix EModified Cactus

Given a prefix𝐱<t\\mathbf\{x\}\_\{<t\}and a drafted tokenyywith drafter probabilityq⁡\(y∣𝐱<t\)q\(y\\mid\\mathbf\{x\}\_\{<t\}\)and verifier probabilityp⁡\(y∣𝐱<t\)p\(y\\mid\\mathbf\{x\}\_\{<t\}\), Cactus\([Hao and Mou, 2026](https://arxiv.org/html/2610.31819#bib.bib2)\)creates a modified target probability distributionp~\\tilde\{p\}that depends onyythat maximizesp~​\(y∣𝐱<t\)\\tilde\{p\}\(y\\mid\\mathbf\{x\}\_\{<t\}\)subject to the constraintDKL\(p~\(⋅∣𝐱<t\)∥p\(⋅∣𝐱<t\)\)≤δD\_\{\\mathrm\{KL\}\}\(\\tilde\{p\}\(\\cdot\\mid\\mathbf\{x\}\_\{<t\}\)\\,\\\|\\,p\(\\cdot\\mid\\mathbf\{x\}\_\{<t\}\)\)\\leq\\delta\. Specifically, it setsp~​\(y∣𝐱<t\)=γ\\tilde\{p\}\(y\\mid\\mathbf\{x\}\_\{<t\}\)=\\gamma, whereγ\\gammais the solution to Equation[17](https://arxiv.org/html/2610.31819#A5.E17), and rescales the assigned probabilities for the rest of the tokens so that the overall vector remains a probability distribution, i\.e\.,p~​\(v∣𝐱<t\)=1−γ1−p⁡\(y∣𝐱<t\)​p​\(v∣𝐱<t\)\\tilde\{p\}\(v\\mid\\mathbf\{x\}\_\{<t\}\)=\\frac\{1\-\\gamma\}\{1\-p\(y\\mid\\mathbf\{x\}\_\{<t\}\)\}\\,p\(v\\mid\\mathbf\{x\}\_\{<t\}\)forv≠yv\\neq y\. For0<p⁡\(y∣𝐱<t\)<10<p\(y\\mid\\mathbf\{x\}\_\{<t\}\)<1, reduces to

Φ⁡\(γ\):=γ​log⁡γp⁡\(y∣𝐱<t\)\+\(1−γ\)​log⁡1−γ1−p⁡\(y∣𝐱<t\)≤δ\.\\Phi\(\\gamma\):=\\gamma\\log\\frac\{\\gamma\}\{p\(y\\mid\\mathbf\{x\}\_\{<t\}\)\}\+\(1\-\\gamma\)\\log\\frac\{1\-\\gamma\}\{1\-p\(y\\mid\\mathbf\{x\}\_\{<t\}\)\}\\leq\\delta\.\(17\)Cactus approximatesΦ\\Phiby its second\-order Taylor expansion atγ0=p⁡\(y∣𝐱<t\)\\gamma\_\{0\}=p\(y\\mid\\mathbf\{x\}\_\{<t\}\),

Φ⁡\(γ\)≈Φ⁡\(γ0\)\+Φ′​\(γ0\)​\(γ−γ0\)\+Φ′′​\(γ0\)2​\(γ−γ0\)2,\\Phi\(\\gamma\)\\approx\\Phi\(\\gamma\_\{0\}\)\+\\Phi^\{\\prime\}\(\\gamma\_\{0\}\)\(\\gamma\-\\gamma\_\{0\}\)\+\\frac\{\\Phi^\{\\prime\\prime\}\(\\gamma\_\{0\}\)\}\{2\}\(\\gamma\-\\gamma\_\{0\}\)^\{2\},\(18\)and solvesΦ⁡\(γ\)=δ\\Phi\(\\gamma\)=\\deltaunder this approximation, which givesγ=min⁡\{p⁡\(y∣𝐱<t\)\+2​δ​p​\(y∣𝐱<t\)​\(1−p⁡\(y∣𝐱<t\)\),1\}\\gamma=\\min\\\{p\(y\\mid\\mathbf\{x\}\_\{<t\}\)\+\\sqrt\{2\\delta p\(y\\mid\\mathbf\{x\}\_\{<t\}\)\(1\-p\(y\\mid\\mathbf\{x\}\_\{<t\}\)\)\},\\,1\\\}\. Cactus then acceptsyywith probabilitymin⁡\{1,γ/q⁡\(y∣𝐱<t\)\}\\min\\\{1,\\gamma/q\(y\\mid\\mathbf\{x\}\_\{<t\}\)\\\}\.

However, the Taylor approximation is only justified when the desired modified probability for the drafted token is close toγ0=p⁡\(y∣𝐱<t\)\\gamma\_\{0\}=p\(y\\mid\\mathbf\{x\}\_\{<t\}\)\.

In order to support larger leniencyδ\\deltavalues, when the verifier probabilityγ0=p⁡\(y∣𝐱<t\)\\gamma\_\{0\}=p\(y\\mid\\mathbf\{x\}\_\{<t\}\)is small and the drafter probabilityq⁡\(y∣𝐱<t\)q\(y\\mid\\mathbf\{x\}\_\{<t\}\)is large, the solutionγ∗\\gamma^\{\*\}needs to be much larger thanγ0\\gamma\_\{0\}to become comparable to the drafter probability, which will break the Taylor approximation\.

Therefore, in our experiments we use Newton’s method to solve for the optimalγ∗\\gamma^\{\*\}exactly for any given values ofδ\\delta,p⁡\(y∣𝐱<t\)p\(y\\mid\\mathbf\{x\}\_\{<t\}\), andq⁡\(y∣𝐱<t\)q\(y\\mid\\mathbf\{x\}\_\{<t\}\)\. On the GPU, the Newton steps can be run in parallel for the entire drafting block, so the runtime overhead is minimal\.

## Appendix FExperimental Details

All experiments were run on a single H200 GPU\.

Table 1:Baseline task accuracy with one standard error\. The gap is verifier minus drafter accuracy in percentage points\.### F\.1Benchmark Selection

The benchmarks use a random set of 500 GSM8K questions, all GPQA Diamond and all HumanEval problems\. GSM8K and GPQA score the extracted final answer, while HumanEval reports accuracy from execution of the generated code against its tests using pass@1\. We generate exactly one completion for all problems\.

Table[1](https://arxiv.org/html/2610.31819#A6.T1)reports baseline accuracies used to select benchmarks\. Dream uses 512 denoising steps on GSM8K and HumanEval and 1024 on GPQA\. DiffusionGemma uses its native sampler with an upper limit of 48 diffusion steps\. SDAR uses a confidence threshold ofτ=0\.9\\tau=0\.9\. These settings assess drafter quality with high diffusion effort\. We focus on settings where the drafter has useful task accuracy and the verifier offers room for improvement\. Dream and SDAR meet these criteria on GSM8K and HumanEval, but perform worse than random on GPQA\. DiffusionGemma has almost the same accuracy as the verifier model than on GSM8K and Humaneval so it is only run on GPQA Diamond\.

### F\.2Models, Data, and Sampling

Table[2](https://arxiv.org/html/2610.31819#A6.T2)shows the Huggingface checkpoints used for the experiments\. All models, both drafters and verifiers, are run without thinking enabled\. All models have a generation limit of 2048 for GPQA Diamond and 512 on GSM8K and HumanEval\.

Table 2:Hugging Face checkpoints used in the experiments\.The Dream GSM8K experiment uses the model’s chat template without an additional instruction prefix\. On GSM8K, Gemma and the Qwen3 comparisons prepend the instruction to put the final answer inside\\boxed\{\}\. Within each model\-pair comparison, the AR baseline uses the corresponding prompt format\.

SamplerTemperatureTop\-ppTop\-kkDream draft / standalone0\.20\.95–Qwen2\.5\-32B verifier0\.70\.820DiffusionGemma draft / standalone0\.8→0\.40\.8\\to 0\.40\.95–Gemma\-4\-31B verifier1\.00\.9564SDAR draft0\.60\.9520Uno draft1\.00\.9550DFlash draftGreedy––DSpark draft1\.0––Qwen3\-8B verifier / AR1\.00\.9550Table 3:Sampling settings for the reported runs\. A dash indicates no additional truncation recorded for that setting\. DiffusionGemma anneals temperature during denoising and uses 0\.4 for a single denoising step\.Dream 7B uses entropy\-based unmasking with deterministic position selection\. Each drafting round denoises the remaining generation canvas\.

DiffusionGemma uses a block size of 256\. When the DiffusionGemma is run as a standalone model, it uses an entropy bound sampler which terminates the diffusion when a confidence threshold is reached, so the number of diffusion steps is variable, but is bounded by an upper limit\. In our experiments, the values of 1, 2, 4, 8, 16, 32, and 48 are used as the upper bound\. When DiffusionGemma is run as a drafter, we modify it to run a fixed number of diffusion steps \(KK\) instead of using the original entropy bound sampler, because DiffusionGemma has both a stability threshold and a confidence threshold to determine when to terminate the diffusion, along with the entropy bound sampler threshold, which makes it not straightforward to control the diffusion effort with the original sampling approach\.

Table 4:Main experimental configurations\. The final five rows share the Qwen3\-8B verifier\. Block lengths are the configured values, distinct from full\-completion limits\. DFlash’s 16\-position block includes one anchor and 15 draft positions\. SDAR’s confidence thresholdτ\\taucontrols denoising effort adaptively\.SDAR commits tokens whose confidence exceedsτ\\tau, so its number of denoising passes varies with the draft\. We sweepτ∈\{0\.3,0\.5,0\.7,0\.9\}\\tau\\in\\\{0\.3,0\.5,0\.7,0\.9\\\}for both model sizes\. We also run another configuration where the SDAR drafter runs exactly one diffusion step instead of relying on the confidence threshold for termination\. All Qwen3 comparisons useλ∈\{0,0\.2,0\.4,0\.6,0\.8,1\}\\lambda\\in\\\{0,0\.2,0\.4,0\.6,0\.8,1\\\}with the same verifier model and sampling settings\. Table[4](https://arxiv.org/html/2610.31819#A6.T4)gives block sizes and completion limits\. DSpark retains its native confidence threshold of zero\.

Figure[5](https://arxiv.org/html/2610.31819#S4.F5)uses one\-pass Uno drafting\. Appendix[G\.5](https://arxiv.org/html/2610.31819#A7.SS5)reports the additionalK=2,4K=2,4experiments that support this choice\. For DFlash and DSpark, we vary acceptance leniency while keeping their released drafting procedures\.

### F\.3Sweep Grids and Measurement Protocol

Table[5](https://arxiv.org/html/2610.31819#A6.T5)lists the Cactus budgets for the main results, along with the diffusion effortKK\.

On the other hand, for the linear acceptance modifier,λ∈\{0,0\.1,…,1\}\\lambda\\in\\\{0,0\.1,\\ldots,1\\\}is used, along withK∈\{1,2,3,4\}K\\in\\\{1,2,3,4\\\}for the Dream pair on GSM8K,K∈\{1,2,3\}K\\in\\\{1,2,3\\\}for the Dream pair on HumanEval,K∈\{1,2,4,8\}K\\in\\\{1,2,4,8\\\}for DiffusionGemma pair on GPQA, andK∈\{1,2,4,6,10\}K\\in\\\{1,2,4,6,10\\\}for DiffusionGemma pair on HumanEval\.

Model pair / datasetKKCactus budgetsδ\\deltaDream–Qwen / GSM8K10,0\.25,0\.5,1,…,80,0\.25,0\.5,1,\\ldots,820,0\.25,0\.5,1,…,110,0\.25,0\.5,1,\\ldots,1130,0\.25,0\.5,1,…,130,0\.25,0\.5,1,\\ldots,1340,0\.25,0\.5,1,…,150,0\.25,0\.5,1,\\ldots,15DiffusionGemma–Gemma / GPQA10,0\.25,0\.5,1,…,50,0\.25,0\.5,1,\\ldots,520,0\.25,0\.5,1,…,60,0\.25,0\.5,1,\\ldots,640,0\.25,0\.5,1,…,8,100,0\.25,0\.5,1,\\ldots,8,1081,3,5,7,10,151,3,5,7,10,15Dream–Qwen / HumanEval10,0\.25,0\.5,1,…,60,0\.25,0\.5,1,\\ldots,620,0\.25,0\.5,1,…,120,0\.25,0\.5,1,\\ldots,1230,0\.25,0\.5,1,…,130,0\.25,0\.5,1,\\ldots,1340,0\.25,0\.5,1,…,150,0\.25,0\.5,1,\\ldots,15DiffusionGemma–Gemma / HumanEval10,0\.25,0\.5,1,…,80,0\.25,0\.5,1,\\ldots,820,0\.25,0\.5,1,…,120,0\.25,0\.5,1,\\ldots,1240,0\.25,0\.5,1,…,160,0\.25,0\.5,1,\\ldots,1660,0\.25,0\.5,1,…,150,0\.25,0\.5,1,\\ldots,15100\.25,0\.5,1,…,15,200\.25,0\.5,1,\\ldots,15,20Table 5:Evaluated Cactus budgets at each diffusion effort\.
### F\.4Simple Throughput Model Calibration

The calibration for Figure[3](https://arxiv.org/html/2610.31819#S3.F3)usesK∈\{1,2,3,4\}K\\in\\\{1,2,3,4\\\}and runs 50 random questions from the GSM8K dataset\. For eachKK, we compute the average TVD between the drafter and verifier distributions\. We then substitute them into Equation[6](https://arxiv.org/html/2610.31819#S3.E6)and derive the theoretical throughput and quality at different leniency values\. The block sizeBBis set to 512 in Equation[7](https://arxiv.org/html/2610.31819#S3.E7)\.

Based on empirical profiling of the Dream\-7B drafter and Qwen2\.5\-32B verifier, we estimatedKd\_\{K\}to be0\.566,0\.517,0\.424,0\.3600\.566,0\.517,0\.424,0\.360forK=1,2,3,4K=1,2,3,4respectively\. These distances imply lossless acceptance rates1−dK1\-d\_\{K\}of 0\.434, 0\.483, 0\.576, and 0\.640, which determine the starting points of the lines in Figure[3\(a\)](https://arxiv.org/html/2610.31819#S3.F3.sf1)\. Figure[6](https://arxiv.org/html/2610.31819#A6.F6)shows the goodness of fit for the linear regression plot and the constant time for calling the verifier\.

Figure 6:Per\-round latency measurements supporting Figure[3](https://arxiv.org/html/2610.31819#S3.F3)\. Left: drafting costs and their linear fit inKK\. Right: the distribution of verifier costs across configurations, with its mean and one\-standard\-deviation band\.

## Appendix GAdditional Results

### G\.1Linear Acceptance

Figures[7](https://arxiv.org/html/2610.31819#A7.F7)and[8](https://arxiv.org/html/2610.31819#A7.F8)show the linear\-acceptance counterparts of Figure[4](https://arxiv.org/html/2610.31819#S4.F4), alongside Cactus for comparison\.

\(a\)Cactus\(b\)Linear acceptance
Figure 7:Latency and throughput on GSM8K with the Dream\-7B drafter and Qwen2\.5\-32B\-Instruct verifier\(a\)Cactus\(b\)Linear acceptance
Figure 8:Latency and throughput on GPQA Diamond with the DiffusionGemma 26B drafter and Gemma4 31B verifier\.
### G\.2Drafting Block Size Ablation

We vary the drafting block size for Dream\-7B with Qwen2\.5\-32B\-Instruct on the 500 GSM8K questions, using linear acceptance and a 512\-token generation limit\. ForB∈\{32,64,128\}B\\in\\\{32,64,128\\\}, we sweepK∈\{1,2,3,4\}K\\in\\\{1,2,3,4\\\}andλ∈\{0,0\.2,0\.4,0\.6,0\.8,1\}\\lambda\\in\\\{0,0\.2,0\.4,0\.6,0\.8,1\\\}\. We compare these runs with the remaining\-canvas configuration, denotedB≤512B\\leq 512which means drafting over the remaining generation canvas\. We compute a separate empirical Pareto frontier over all measured settings within each block\-size configuration\. Figure[9](https://arxiv.org/html/2610.31819#A7.F9)shows that changing block size affects latency and throughput, while preserving the main observation of Section[4\.2](https://arxiv.org/html/2610.31819#S4.SS2)\. Different diffusion efforts contribute to different regions of the frontier\.

\(a\)Absolute latency and throughput\(b\)Speedup relative to theB≤512B\\leq 512frontier
Figure 9:Block\-size ablation on GSM8K with Dream–Qwen\.
### G\.3HumanEval

Figures[10](https://arxiv.org/html/2610.31819#A7.F10)and[11](https://arxiv.org/html/2610.31819#A7.F11)extend the Dream–Qwen and DiffusionGemma–Gemma comparisons to HumanEval, using both acceptance rules\.

\(a\)Cactus\(b\)Linear acceptance
Figure 10:Latency and throughput on HumanEval with the Dream 7B drafter and Qwen2\.5 32B verifier\.\(a\)Cactus\(b\)Linear acceptance
Figure 11:Latency and throughput on HumanEval with the DiffusionGemma 26B drafter and Gemma4 31B verifier\.
### G\.4Shared\-Verifier Comparison

Figure 12:Complete sweeps for Figure[5](https://arxiv.org/html/2610.31819#S4.F5)\. Colors distinguish SDAR’s effort settings within each model size\. Hollow markers show measured configurations, including dominated points\. Thick curves show the joint and fixed\-K=1K=1frontiers, using the same definitions as the main figure\.
### G\.5Uno at Multiple Diffusion Steps

We evaluate Uno withK∈\{1,2,4\}K\\in\\\{1,2,4\\\}and linear acceptance atλ∈\{0,0\.2,0\.4,0\.6,0\.8,1\}\\lambda\\in\\\{0,0\.2,0\.4,0\.6,0\.8,1\\\}\. We hold the block size at 16 and use the same problems, sampling settings, and 512\-token limit as Figure[5](https://arxiv.org/html/2610.31819#S4.F5)\.

In Figure[13](https://arxiv.org/html/2610.31819#A7.F13),K=1K=1captures the entire latency frontier on both benchmarks and the entire throughput frontier on HumanEval\. On GSM8K, one low\-accuracyK=2K=2configuration also contributes to the measured throughput frontier, whileK=4K=4contributes to neither frontier\. Thus, additional steps do not provide the speed\-quality frontier improvements seen with diffusion drafters that are trained to perform as standalone models\. This pattern is consistent with Uno’s one\-step distillation objective, which trains it to recover a clean block from fully corrupted inputs\([Sahoo et al\., 2026](https://arxiv.org/html/2610.31819#bib.bib30)\)following the base model’s distribution\.

\(a\)GSM8K\(b\)HumanEval
Figure 13:Uno with one, two, and four drafting steps\. Colors indicateKK, with leniency varied within each color\. Thick lines connect nondominated measurements, and hollow markers show the remaining configurations\. The GSM8K sweep lacks completedK=4K=4runs atλ∈\{0\.8,1\}\\lambda\\in\\\{0\.8,1\\\}\.

## Appendix HQualitative Examples

We present qualitative examples of speculative diffusion using Dream\-7B with Qwen2\.5\-32B\-Instruct under linear acceptance\. These selected samples illustrate failure modes of high acceptance leniency\. The original benchmark prompts and selected responses are displayed in full, including repetition, formatting markers, and saved end tokens\. Colored bold text highlights correct calculations or code \(green\), errors or repetition \(red\)\.

### H\.1Math

GSM8K 1116 Full benchmark prompt[⬇](data:text/plain;base64,TGFzdCB5ZWFyIHRoZXJlIHdlcmUgNTAgc3R1ZGVudHMgZW5yb2xsZWQgaW4gYSBjYWxsaWdyYXBoeSBjbGFzcy4gVGhpcyB5ZWFyLCB0aGVyZSB3YXMgYSAyMCUgaW5jcmVhc2UgaW4gZW5yb2xsbWVudC4gSG93IG1hbnkgc3R1ZGVudHMgYXJlIGVucm9sbGVkIHRoaXMgeWVhciBpbiB0aGUgY2FsbGlncmFwaHkgY2xhc3M/)Lastyeartherewere50studentsenrolledinacalligraphyclass\.Thisyear,therewasa20%increaseinenrollment\.Howmanystudentsareenrolledthisyearinthecalligraphyclass?

Reference answer\.50\+0\.20×50=6050\+0\.20\\times 50=60students\.

Fixed effortK=1K=1: increasing acceptance leniency

K=1,λ=0K=1,\\ \\lambda=0Correct[⬇](data:text/plain;base64,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)Todeterminethenumberofstudentsenrolledinthecalligraphyclassthisyear,weneedtocalculatea20%increaseonlastyear'senrollmentof50students\.First,wecalculate20%of50:\\\[20\\%\\text\{of\}50=\\frac\{20\}\{100\}\\times50=0\.2\\times50=10\\\]Next,weaddthisincreasetolastyear'senrollment:\\\[50\+10=60\\\]Thus,thenumberofstudentsenrolledthisyearinthecalligraphyclassis\\boxed\{60\}\.<\|im\_end\|\>

K=1,λ=0\.4K=1,\\ \\lambda=0\.4Incorrect[⬇](data:text/plain;base64,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)Todeterminethenumberofstudentsenrolledthisyearinthecalligraphyclass,weneedtocalculatea20%increaseonlastyear'senrollmentof50students\.0\.First,wecalculate20%of50students:\\\[0\.20\\times50=00=10\\\]Next\.we,addthisincreasetotheoriginalnumberofstudents:\\\[50\+10=60\\\]Therefore,thenumberofofstudentsenrolledthisyearinthecalligraphyclassis$@@ERROR@@\\boxed\{600\}@@END@@$\.<\|im\_end\|\>

K=1,λ=0\.6K=1,\\ \\lambda=0\.6Incorrect[⬇](data:text/plain;base64,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)Todeterminethenumberofstudentsenrolledthisyearinthecalligraphyclass,weneedtocalculatetheincreaseinenrollmentandlastyear'senrollment\.1\.Calculatetheincreaseinenrollment:20%oflastyear'senrollment\.2\.Addtheincreasetolastyear'senrollmenttofindthisyear'senrollment\.Let'sthethisstep\-by\-step:\`\`\`pythonthenumberofstudentsenrolledlastyear\.last\_year\_enrollment=50thepercentageinenrollment\.percentage\_increase=20theincreaseinenrollment\.increase=\(percentage\_increase/100\)\*last\_year\_enrollmenttheyear'senrollment\.this\_year\_enrollment=last\_year\_enrollment\+increaseprint\_year\_year\_en\)\`\`\`\`\`outputSyntaxError:invalidsyntax\`Itlooksliketherewassyntaxerrorinthecode\.LetLetmecorrecttheandcalculatethenumber'senrollmentthisyearyearproperly\.\`\`\`python\#numberofstudentsenrolledlastlastlastlastlastlast===5500000000\#percentageininininininininininininininininininininininininininininininininininininininininininin0inininininininin0ininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininininin

Increasing diffusion effort atλ=0\.4\\lambda=0\.4andλ=0\.6\\lambda=0\.6

K=2,λ=0\.4K=2,\\ \\lambda=0\.4Correct[⬇](data:text/plain;base64,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)Todeterminethenumberofstudentsenrolledthisyearinthecalligraphyclass,weneedtocalculatea20%increaseonlastyear'senrollmentof50students\.A20%increasemeansthatthenumberofstudentsthisyearis120%oflastyear'senrollment\.Tocalculate120%of50,wecanmultiply50by1\.20\(since120%isequivalentto1\.20\)\.Let'sperformthecalculation:\\\[50\\times1\.20=60\\\]Therefore,thenumberofstudentsenrolledthisyearinthecalligraphyclassis$@@GOOD@@\\boxed\{60\}@@END@@$\.

K=4,λ=0\.6K=4,\\ \\lambda=0\.6Correct[⬇](data:text/plain;base64,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)Todeterminethenumberofstudentsenrolledthisyearinthecalligraphyclass,weneedtocalculatetheincreaseinenrollmentandaddittolastyear'senrollment\.Theincreaseinenrollmentis20%oflastyear'senrollment\.Lastyear,therewere50studentsenrolled\.20%of50canbecalculatedasfollows:\\\(50\\times\\frac\{20\}\{100\}=50\\times0\.20=10\\\)\.So,theincreaseinenrollmentis\\\(10\\\)students\.Therefore,thisyear'senrollmentis:\\\(50\+10=\\boxed\{60\\students\}\\\)\.

### H\.2Code

HumanEval/152 Full benchmark prompt[⬇](data:text/plain;base64,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)defcompare\(game,guess\):"""Ithinkweallrememberthatfeelingwhentheresultofsomelong\-awaitedeventisfinallyknown\.Thefeelingsandthoughtsyouhaveatthatmomentaredefinitelyworthnotingdownandcomparing\.Yourtaskistodetermineifapersoncorrectlyguessedtheresultsofanumberofmatches\.Youaregiventwoarraysofscoresandguessesofequallength,whereeachindexshowsamatch\.Returnanarrayofthesamelengthdenotinghowfaroffeachguesswas\.Iftheyhaveguessedcorrectly,thevalueis0,andifnot,thevalueistheabsolutedifferencebetweentheguessandthescore\.example:compare\(\[1,2,3,4,5,1\],\[1,2,3,4,2,\-2\]\)\-\>\[0,0,0,0,3,3\]compare\(\[0,5,0,0,0,4\],\[4,1,1,0,0,\-2\]\)\-\>\[4,4,1,0,0,6\]"""

Fixed effortK=1K=1: increasing acceptance leniency

K=1,λ=0K=1,\\ \\lambda=0Pass[⬇](data:text/plain;base64,ZGVmIGNvbXBhcmUoZ2FtZSwgZ3Vlc3MpOgogICAgIiIiQ29tcGFyZXMgdGhlIGFjdHVhbCBzY29yZXMgb2YgbWF0Y2hlcyB3aXRoIGd1ZXNzZWQgc2NvcmVzIGFuZCByZXR1cm5zIHRoZSBhYnNvbHV0ZSBkaWZmZXJlbmNlcy4iIiIKICAgIHJlc3VsdCA9IFtdCiAgICBAQEdPT0RAQGZvciBpIGluIHJhbmdlKGxlbihnYW1lKSk6QEBFTkRAQAogICAgICAgIHJlc3VsdC5hcHBlbmQoYWJzKGdhbWVbaV0gLSBndWVzc1tpXSkpCiAgICByZXR1cm4gcmVzdWx0)defcompare\(game,guess\):"""Comparestheactualscoresofmatcheswithguessedscoresandreturnstheabsolutedifferences\."""result=\[\]foriinrange\(len\(game\)\):result\.append\(abs\(game\[i\]\-guess\[i\]\)\)returnresult

K=1,λ=0\.6K=1,\\ \\lambda=0\.6Fail[⬇](data:text/plain;base64,QEBFUlJPUkBAYGBgcHl0aG9uIHRoZWRlZgpjb21wYXJlKGdhbWUsIGd1ZXNzKToKCmZvcgppIGluIHJhbmdlKGxlbihsZW4KKSk6QEBFTkRAQAoKICAgICAgICBAQEVSUk9SQEBkaWZmID0gYWJzW2ldZ2FtZVtpXVtpW2lbaVtpW2kgICAgICAgIF0gLSBndWVzc1tpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpW2lbaVtpQEBFTkRAQA==)\`\`\`pythonthedefcompare\(game,guess\):foriinrange\(len\(len\)\):diff=abs\[i\]game\[i\]\[i\[i\[i\[i\[i\]\-guess\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i\[i

Fixed leniencyλ=0\.4\\lambda=0\.4: increasing diffusion effort

K=1,λ=0\.4K=1,\\ \\lambda=0\.4Fail[⬇](data:text/plain;base64,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)\`\`\`pythondefcompare\(game,guess\):"""Ithinkweallrememberthatfeelingwhentheresultofsomelong\-awaitedeventisfinallyknown\.Thefeelingsandthoughtsyouhaveatthatmomentaredefinitelyworthnotingdownandcomparing\.YourtaskistodetermineifapersoncorrectlyguessedtheresultsofanumberofmatchesmatchesYouaregiventwoarraysofscoresandguessesofequallength,whereeachindexshowsamatch\.Returnanarrayofthesamelengthdenotinghowfaroffeachguesswas\.Iftheyhaveguessedcorrectly,thevalueis0,andifnot,thevalueistheabsolutedifferencebetweentheguessandthescore\."""result=\[\]foriinrange\(len\(len\(game\)\)\):result\.append\(abs\(game\[i\]\-guess\[i\]\)\)returnresult

K=3,λ=0\.4K=3,\\ \\lambda=0\.4Pass[⬇](data:text/plain;base64,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)defcompare\(game,guess\):"""Ithinkweallrememberthatfeelingwhentheresultofsomeeventisfinallyknown\.Thefeelingsandthoughtsyouhaveatthatmomentaredefinitelyworthnotingdownandcomparing\.Yourtaskistodetermineifapersoncorrectlyguessedtheresultsofanumberofmatches\.Youaregiventwoarraysofscoresandguessesofequallength,whereeachindexshowsamatch\.Returnanarrayofthesamelengthdenotinghowfaroffeachguesswas\.Iftheyhaveguessedcorrectly,thevalueis0,andifnotthevalueistheabsolutedifferencebetweentheguessguessandthescore\.example:compare\(\[1,2,3,4,5,1\],\[1,2,3,4,2,\-2\]\)\-\>\[0,0,0,0,3,3\]compare\(\[0,5,0,0,0,4\],\[4,1,1,0,0,\-2\]\)\-\>\[4,4,1,0,0,6\]"""return\[abs\(game\[i\]\-guess\[i\]\)foriinrange\(len\(game\)\)\]