Revisiting Lossy Verification in Speculative Decoding: Mechanisms, Trade-offs, and Failure Modes

# Revisiting Lossy Verification in Speculative Decoding: Mechanisms, Trade-offs, and Failure Modes
Source: [https://arxiv.org/html/2607.26627](https://arxiv.org/html/2607.26627)
Tianyu Wang1Yuxuan Zhou2Wenbin Wang1 Heng Li1Zikai Xiao3Junyuan Shang2 1Independent Researcher2Baidu Inc\.3Zhejiang University

###### Abstract

Speculative Decoding \(SD\) accelerates large language model inference by allowing a lightweight draft model to propose tokens that are subsequently verified in parallel by a larger target model\. Recent approaches introduce lossy verification schemes to further improve efficiency by relaxing strict distributional matching\. Yet such relaxation silently rewrites the decoding distribution, and the resulting acceleration can come at the cost of unstable, sometimes severely degraded generation quality\. In this work, we present a principled analysis of the distributions induced by lossy verification methods\. We show that many seemingly distinct approaches differ only superficially and can be classified into two categories: truncation\-based verification and collaborative verification\. We further construct a diagnostic evaluation framework across curated benchmarks\. For truncation\-based methods, we identify a fundamental pitfall—performance can degrade significantly compared to the true truncation sampling baseline due to distributional distortion\. For collaborative verification, we uncover a key principle: controlling the overshoot of draft probabilities relative to target probabilities is essential to prevent low\-quality outputs\. Our code is available at[https://github\.com/ZhouYuxuanYX/Fast\-HSD](https://github.com/ZhouYuxuanYX/Fast-HSD)\.

Revisiting Lossy Verification in Speculative Decoding: Mechanisms, Trade\-offs, and Failure Modes

Tianyu Wang1Yuxuan Zhou2††thanks:Corresponding author\.††thanks:Project lead\.Wenbin Wang1Heng Li1Zikai Xiao3Junyuan Shang21Independent Researcher2Baidu Inc\.3Zhejiang University

## 1Introduction

Auto\-regressive Large Language Models \(LLMs\)Achiamet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib41)\); Touvronet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib42)\); Baiet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib43)\)are at the forefront of the AI revolution\. Despite their strong performance, their non\-parallelizable inference presents a significant efficiency bottleneck, particularly for long\-context generation in test\-time scalingOpenAIet al\.\([2026](https://arxiv.org/html/2607.26627#bib.bib44)\); Guoet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib45)\), LLM agentsYaoet al\.\([2022](https://arxiv.org/html/2607.26627#bib.bib12)\); Schicket al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib13)\); Significant\-Gravitas \([2023](https://arxiv.org/html/2607.26627#bib.bib3)\), and multimodal reasoningPenget al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib78)\)\. Speculative Decoding \(SD\)Leviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\)mitigates this challenge by having a lightweight draft model generate candidate tokens, which are then verified by the target model in parallel, reducing expensive forward passes while ensuring the generated distribution matches that of the target model\.

![Refer to caption](https://arxiv.org/html/2607.26627v1/x1.png)Figure 1:Accuracy gap between the lossless baseline and truncation\-based verification widens sharply with task difficulty, increasing from \+0\.38 pp on GSM8K to \+6\.67 pp on AIME\. Here, theTrue baselinemeans the target model with min\-p sampling, while theWrong baselineindicates the target model w/o min\-p sampling\.While theoretical advances have pushed the limits of SDSunet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib38)\); Zhouet al\.\([2026](https://arxiv.org/html/2607.26627#bib.bib67)\), the strict requirements for distribution matching continue to limit the potential for further speedup\. Lossy verification methodsCaiet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib39)\); Narasimhanet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib34)\); Zhouet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib27)\); Fuet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib64)\); Leviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\)relax these requirements for greater acceleration, but existing work overstates their advantages under curated settings \(selective hyperparameters or easy benchmarks\)\. This obscures the true speed–quality trade\-offs arising from distortion of the target distribution and leaves no clear comparison across methods, hampering the broader adoption of these methods and further advancements in the field\.

In this work, we present a principled analysis of the distributions induced by lossy verification methods, resulting in a precise categorization of them\. This analysis reveals that the apparent differences between methods are largely superficial, with most falling into two distinct categories\.

First, methods such as typical acceptance in MedusaCaiet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib39)\)and SpecCascadeNarasimhanet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib34)\)accept draft tokens as long as they fall within the allowed set defined by truncation\-based sampling methods, specificallyη\\eta\-samplingHewittet al\.\([2022](https://arxiv.org/html/2607.26627#bib.bib59)\)and min\-ppsamplingNguyenet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib60)\)\. We refer to these as*truncation\-based verification*methods\. Second, both lenience\-based relaxationLeviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\); Zhouet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib27)\)and Collaborative Decoding via Speculation \(CoS\)Fuet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib64)\)interpolate between the draft and target distributions, with the key distinction being whether the interpolation coefficient is fixed or adaptively adjusted across different ranges\. Following the terminology of CoSFuet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib64)\), we group these as collaborative verification methods, unifying lenience\-based relaxation under this umbrella\.

We first examine truncation\-based verification methods against their corresponding baselines—i\.e\., directly using the same truncation sampling strategies for the target model\. Our experiments show that the seemingly comparable performance reported in prior workCaiet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib39)\); Narasimhanet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib34)\)largely arises from improved baseline performance induced by their adopted truncation sampling methods\. Moreover, the performance gap between truncation\-based verification and the true baseline grows with task difficulty, as shown in[Figure˜1](https://arxiv.org/html/2607.26627#S1.F1)for SpecCascade\. As harder tasks better reflect real\-world scenarios, this widening gap highlights potential pitfalls of truncation\-based verification in practice\. This pitfall is further amplified when lossy verification methods are integrated into the EAGLE\-3 speculative decoding systemLiet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib80)\), where such a performance gap becomes even more pronounced\.

We next turn to collaborative verification, where we identify a key principle for a better speed–quality trade\-off: existing methods interpolate the draft and target distributions either uniformly in CoSFuet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib64)\)or adaptively in lenience\-based relaxationLeviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\), but we find that selectively suppressing the draft at overshoot tokens is enough to preserve generation quality\. This aligns with recent findingsZhouet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib71)\); Yueet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib70)\); Fanet al\.\([2026b](https://arxiv.org/html/2607.26627#bib.bib77)\)that overshoot tokens drive most low\-quality generations, while the rest of the distribution has little effect on quality\.

In summary, we make these contributions:

- •Principled Characterization: Revealing the underlying similarities between seemingly different methods: truncation\-based verification and collaborative verification, each induced by a common underlying mechanism\.
- •Empirical Pitfalls: Identifying a key pitfall in truncation\-based verification: distributional distortion can significantly degrade performance relative to the true truncation sampling baseline\.
- •Governing Principle: Revealing controlling the overshoot of draft probabilities over target probabilities is essential for collaborative verification to achieve acceleration with acceptable quality\.

## 2Preliminaries

We first introduce the speculative decoding framework and its lossless verification mechanism \([Section˜2\.1](https://arxiv.org/html/2607.26627#S2.SS1)\), which forms the foundation for many lossy methods\. We then review two decoding techniques: collaborative decoding \([Section˜2\.2](https://arxiv.org/html/2607.26627#S2.SS2)\) and truncation sampling \([Section˜2\.3](https://arxiv.org/html/2607.26627#S2.SS3)\)\. As shown in[Section˜3](https://arxiv.org/html/2607.26627#S3), these techniques underlie the two categories of lossy verification\.

### 2\.1Speculative Decoding

Letppandqqdenote the next\-token distributions of the target and draft models, respectively, over a shared vocabulary𝒱\\mathcal\{V\}\. In Speculative decoding\(Leviathanet al\.,[2023](https://arxiv.org/html/2607.26627#bib.bib14)\), each draft tokenxxsampled fromqqis accepted with probability:

h​\(x\)=min⁡\(1,p​\(x\)q​\(x\)\)\.h\(x\)=\\min\\\!\\left\(1,\\;\\frac\{p\(x\)\}\{q\(x\)\}\\right\)\.\(1\)If the draft tokenxxis rejected, a replacement token is resampled from the residual distribution\(p​\(x\)−min⁡\{p​\(x\),q​\(x\)\}\)\+\\bigl\(p\(x\)\-\\min\\\{p\(x\),\\,q\(x\)\\\}\\bigr\)\_\{\+\}, renormalized over𝒱\\mathcal\{V\}, ensuring the generated distribution equals the targetppexactly\. Follow\-up work increases acceptance rates by proposing multiple candidates or tree\-structured drafts\(Sunet al\.,[2023](https://arxiv.org/html/2607.26627#bib.bib15); Miaoet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib20); Yanget al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib23); Liet al\.,[2025](https://arxiv.org/html/2607.26627#bib.bib80); Fanet al\.,[2026a](https://arxiv.org/html/2607.26627#bib.bib75)\), while block\-level verification extends lossless guarantees to sequences of tokens\(Sunet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib38); Zhouet al\.,[2026](https://arxiv.org/html/2607.26627#bib.bib67)\)\. As lossless verification approaches its theoretical limits, lossy approaches achieve further speedup by modifying the acceptance criterion: some relax the reference distribution in Eq\. \([1](https://arxiv.org/html/2607.26627#S2.E1)\)\(Leviathanet al\.,[2023](https://arxiv.org/html/2607.26627#bib.bib14); Zhouet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib27); Fuet al\.,[2025](https://arxiv.org/html/2607.26627#bib.bib64)\), while others gate acceptance via a set\-membership criterion derived from truncation sampling\(Caiet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib39); Narasimhanet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib34)\)\. Despite these advances, the underlying mechanisms of lossy methods and their speed–quality trade\-offs remain poorly understood, which we address in[Section˜3](https://arxiv.org/html/2607.26627#S3)\.

### 2\.2Collaborative Decoding

Collaborative decoding combines the predictions of multiple models to form a reshaped next\-token distribution\. Using the target distributionppand draft distributionqqas defined in[Section˜2\.1](https://arxiv.org/html/2607.26627#S2.SS1), and lettingλ∈\[0,1\]\\lambda\\in\[0,1\]denote an interpolation coefficient,Weighted Ensembling \(WE\)Huanget al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib4)\); Yaoet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib5)\)computes the next\-token probability as a convex mixture:

pWE​\(x\)=λ​p​\(x\)\+\(1−λ\)​q​\(x\)\.p\_\{\\mathrm\{WE\}\}\(x\)=\\lambda\\,p\(x\)\+\(1\-\\lambda\)\\,q\(x\)\.\(2\)
Contrastive Decoding \(CD\)Liet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib7)\); O’Brien and Lewis \([2023](https://arxiv.org/html/2607.26627#bib.bib8)\)instead reweightsppusing a contrastive factor derived fromqq:

pCD​\(x\)=p​\(x\)/q​\(x\)λ∑v∈𝒱p​\(v\)/q​\(v\)λ\.p\_\{\\mathrm\{CD\}\}\(x\)\\;=\\;\\frac\{p\(x\)/q\(x\)^\{\\lambda\}\}\{\\sum\_\{v\\in\\mathcal\{V\}\}p\(v\)/q\(v\)^\{\\lambda\}\}\.\(3\)
Whereas standard speculative decoding aims to match the target distribution,Collaborative Decoding via Speculation \(CoS\)Fuet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib64)\)instead uses one of these combined distributions as the verification target, yielding additional speedup\.

### 2\.3Truncation Sampling

Truncation sampling restricts the vocabulary at each decoding step to an allowed set𝒜Θ⊆𝒱\\mathcal\{A\}\_\{\\Theta\}\\subseteq\\mathcal\{V\}determined by a truncation strategyΘ\\Theta, discarding low\-probability tokens to reduce the risk of incoherent outputs\. LetZΘ​\(p\)=∑v∈𝒜Θp​\(v\)Z\_\{\\Theta\}\(p\)=\\sum\_\{v\\in\\mathcal\{A\}\_\{\\Theta\}\}p\(v\)denote the normalization constant\. The resulting distribution is

pΘ​\(x\)=\{p​\(x\)ZΘ​\(p\),x∈𝒜Θ,0,x∉𝒜Θ\.p\_\{\\Theta\}\(x\)=\\begin\{cases\}\\dfrac\{p\(x\)\}\{Z\_\{\\Theta\}\(p\)\},&x\\in\\mathcal\{A\}\_\{\\Theta\},\\\\\[2\.0pt\] 0,&x\\notin\\mathcal\{A\}\_\{\\Theta\}\.\\end\{cases\}\(4\)
Two representative strategies,min‑p samplingNguyenet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib60)\)andη\\eta‑samplingHewittet al\.\([2022](https://arxiv.org/html/2607.26627#bib.bib59)\), are described below:

![Refer to caption](https://arxiv.org/html/2607.26627v1/x2.png)Figure 2:Comparison of distributions induced by collaborative and truncation\-based methods\. Tokens are sorted by decreasing target probability\. Blue and pink bars denote the target distributionppand draft distributionqq, respectively; orange denotes their overlap\. The purple line shows the output distribution\. Gray dashed lines indicate the ceiling\.Min\-p samplingadapts to the top candidate’s confidence by applying a dynamic cutoff relative to the peak probability, wherepbase∈\(0,1\)p\_\{\\text\{base\}\}\\in\(0,1\):

𝒜min\-p=\{x∈𝒱:p​\(x\)≥pbase⋅maxv∈𝒱⁡p​\(v\)\}\\mathcal\{A\}\_\{\\text\{min\-p\}\}=\\bigl\\\{x\\in\\mathcal\{V\}:p\(x\)\\geq p\_\{\\text\{base\}\}\\cdot\\max\_\{v\\in\\mathcal\{V\}\}p\(v\)\\bigr\\\}\(5\)
η\\eta\-samplingadapts to model uncertainty by removing tokens below a threshold derived from the distribution entropyH​\(p\)=−∑v∈𝒱p​\(v\)​log⁡p​\(v\)H\(p\)=\-\\sum\_\{v\\in\\mathcal\{V\}\}p\(v\)\\log p\(v\), whereε,δ\>0\\varepsilon,\\delta\>0:

𝒜η=\{x∈𝒱:p​\(x\)≥min⁡\(ε,δ​e−H​\(p\)\)\}\\mathcal\{A\}\_\{\\eta\}=\\bigl\\\{x\\in\\mathcal\{V\}:p\(x\)\\geq\\min\\\!\\left\(\\varepsilon,\\,\\delta e^\{\-H\(p\)\}\\right\)\\bigr\\\}\(6\)
In[Section˜3](https://arxiv.org/html/2607.26627#S3), we show that state\-of\-the\-art lossy verification methodsNarasimhanet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib34)\); Caiet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib39)\)essentially accept a draft token as long as it falls within the allowed set defined by one of these truncation strategies\.

## 3Mechanisms of Lossy Verification

In this section, we show that existing lossy verification methods fall into two paradigms:collaborative verification, which relaxes verification by matching the combined distribution from collaborative decoding, andtruncation\-based verification, which accepts draft tokens based on the allowed sets induced by truncation sampling\.

### 3\.1Collaborative Verification

Collaborative verification relaxes the acceptance criterion so that the generated distribution becomes a combination of the draft and target distributions, improving decoding speed at the cost of degraded performance\. This view covers both \(i\) methods that explicitly construct the combined distributionFuet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib64)\), and \(ii\) methods whose acceptance rule implicitly induces such a distributionLeviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\)\.

CoSFuet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib64)\)modifies the acceptance probability of the lossless verification to:

h​\(x\)=λ​p​\(x\)\+\(1−λ\)​q​\(x\)q​\(x\)\.h\(x\)\\;=\\;\\frac\{\\lambda\\,p\(x\)\+\(1\-\\lambda\)\\,q\(x\)\}\{q\(x\)\}\.\(7\)
Note that, for simplicity, we focus on the weighted ensemble setup; a similar analysis applies to the contrastive decoding variant\. Under this rule, the resulting token distribution becomes a convex mixture of the target and draft distributions \(see[Figure˜2](https://arxiv.org/html/2607.26627#S2.F2)\(a\)\):

P​\(generate​x\)=λ​p​\(x\)\+\(1−λ\)​q​\(x\)\.P\(\\text\{generate\}~x\)=\\lambda\\,p\(x\)\+\(1\-\\lambda\)\\,q\(x\)\.\(8\)
Lenience\-based relaxationLeviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\)modifies the acceptance rule with a lenience factorℓ∈\(0,1\]\\ell\\in\(0,1\]:

h​\(x\)=min⁡\(1,p​\(x\)ℓ​q​\(x\)\)\.h\(x\)\\;=\\;\\min\\\!\\left\(1,\\frac\{p\(x\)\}\{\\ell\\,q\(x\)\}\\right\)\.\(9\)
The induced distribution admits the following decomposition \(see[Appendix˜A](https://arxiv.org/html/2607.26627#A1)for the proof\):

P​\(generate​x\)=\{Δ​p​\(x\)\+\(1−Δ\)​q​\(x\),q​\(x\)≤p​\(x\),q​\(x\),p​\(x\)<q​\(x\)≤p​\(x\)/ℓ,p​\(x\)/ℓ,q​\(x\)≥p​\(x\)/ℓ\.\\begin\{gathered\}P\(\\mathrm\{generate\}\\,x\)=\\\\ \\left\\\{\\begin\{array\}\[\]\{@\{\}l@\{\\,\}l@\{\}\}\\Delta p\(x\)\+\(1\-\\Delta\)q\(x\),&q\(x\)\\leq p\(x\),\\\\\[3\.0pt\] q\(x\),&p\(x\)<q\(x\)\\leq p\(x\)/\\ell,\\\\\[3\.0pt\] p\(x\)/\\ell,&q\(x\)\\geq p\(x\)/\\ell\.\\end\{array\}\\right\.\\end\{gathered\}\(10\)
where

Δ=12​∑v∈𝒱\|q​\(v\)−p​\(v\)ℓ\|\+12−12​ℓ12​∑v∈𝒱\|q​\(v\)−p​\(v\)\|\.\\Delta=\\frac\{\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|q\(v\)\-\\frac\{p\(v\)\}\{\\ell\}\\right\|\+\\frac\{1\}\{2\}\-\\frac\{1\}\{2\\ell\}\}\{\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|q\(v\)\-p\(v\)\\right\|\}\.\(11\)
This decomposition reveals an adaptive correction mechanism: when the draft*underestimates*the target \(q≤pq\\leq p\), interpolation is applied; moderate overestimation \(p<q≤p/ℓp<q\\leq p/\\ell\) is left unchanged \(pureqq\); and only the*severe*overshoot region \(q≥p/ℓq\\geq p/\\ell\) is corrected \(see[Figure˜2](https://arxiv.org/html/2607.26627#S2.F2)\(b\)\)\.

Intuitively, the ceilingp/ℓp/\\ellterm in[Equation˜10](https://arxiv.org/html/2607.26627#S3.E10)caps the generation probability when the draft distribution is overconfident\. This ceiling prevents the draft from dominating the output at tokens where it assigns probability far beyond what the target distributionppwould allow, even after relaxation by the lenience factorℓ\\ell\. Separately,12​∑v∈𝒱\|q​\(v\)−p​\(v\)\|\\tfrac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\|q\(v\)\-p\(v\)\|is the total variation distance betweenqqandpp, which measures the total mismatched probability mass between the draft and target distributions\. We provide a detailed interpretation in[Appendix˜D](https://arxiv.org/html/2607.26627#A4)\.

In summary, both approaches introduce a control parameter that tunes the degree of interpolation\. Lenience\-based relaxation, however, constrains extreme deviation from the target distribution through two mechanisms: an adaptive coefficient tied to distributional divergence in the underestimated region \(q≤pq\\leq p\), and a hard threshold based on local probability differences in the overshoot region \(q≥p/ℓq\\geq p/\\ell\)\.

### 3\.2Truncation\-based Verification

We formally define truncation\-based verification, a mechanism that underlies many lossy verification methods\. Within this framework, we distinguish two principal approaches:SpecCascadeNarasimhanet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib34)\)andtypical acceptanceCaiet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib39)\)\.

###### Definition 1\(Truncation\-based verification\)\.

Let𝒜Θ⊆𝒱\\mathcal\{A\}\_\{\\Theta\}\\subseteq\\mathcal\{V\}denote the allowed set of a truncation strategyΘ\\Theta\. Truncation\-based verification replaces the acceptance rule in[Equation˜1](https://arxiv.org/html/2607.26627#S2.E1)with

h​\(x\)=\{1,x∈𝒜Θ,0,x∉𝒜Θ\.h\(x\)=\\begin\{cases\}1,&x\\in\\mathcal\{A\}\_\{\\Theta\},\\\\\[4\.0pt\] 0,&x\\notin\\mathcal\{A\}\_\{\\Theta\}\.\\end\{cases\}\(12\)A draft token is accepted if it lies in𝒜Θ\\mathcal\{A\}\_\{\\Theta\}; tokens outside this set are rejected with no resampling\.

The resulting distribution is the renormalized draft distribution restricted to𝒜Θ\\mathcal\{A\}\_\{\\Theta\}, whereZΘ​\(q\)=∑v∈𝒜Θq​\(v\)Z\_\{\\Theta\}\(q\)=\\sum\_\{v\\in\\mathcal\{A\}\_\{\\Theta\}\}q\(v\)is the normalization constant:

P​\(generate​x\)=\{q​\(x\)/ZΘ​\(q\),x∈𝒜Θ,0,x∉𝒜Θ\.P\(\\mathrm\{generate\}\\,x\)=\\begin\{cases\}q\(x\)/Z\_\{\\Theta\}\(q\),&x\\in\\mathcal\{A\}\_\{\\Theta\},\\\\\[2\.0pt\] 0,&x\\notin\\mathcal\{A\}\_\{\\Theta\}\.\\end\{cases\}

\(13\)
Specifically, SpecCascade\(Narasimhanet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib34)\)and Medusa\(Caiet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib39)\)are examples of truncation\-based verification, with allowed sets defined by min\-p \([Equation˜5](https://arxiv.org/html/2607.26627#S2.E5)\) andη\\eta\-sampling \([Equation˜6](https://arxiv.org/html/2607.26627#S2.E6)\), respectively\. Their reported gains over standard speculative decoding conflate the effect of verification with the effect of truncation sampling itself\. When evaluated against the appropriately truncated target, the actual performance gap becomes clear, as shown in[Section˜5](https://arxiv.org/html/2607.26627#S5)\.

Having characterized existing lossy verification into collaborative and truncation\-based verification, we now evaluate both paradigms empirically\. As foreshadowed in the introduction, benchmark difficulty is itself diagnostic: easy tasks such as GSM8K mask the distortions introduced by lossy verification, while extremely hard tasks such as AIMEBalunovićet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib74)\)are too difficult to yield meaningful signal; between these extremes, the gap against the true baseline widens with difficulty \([Figure˜1](https://arxiv.org/html/2607.26627#S1.F1)\)\. We therefore evaluate on four harder benchmarks spanning distinct domains: MATHHendryckset al\.\([2021](https://arxiv.org/html/2607.26627#bib.bib68)\)for mathematical reasoning, MBPP\+Liuet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib72)\)for code generation, INCLUDE\(Romanouet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib66)\)for multilingual understanding, and BFCL\(Patilet al\.,[2025](https://arxiv.org/html/2607.26627#bib.bib76)\)for agentic tool use\. For efficiency we report Block Efficiency \(BE\), the number of tokens accepted per step, and Decoding Speed \(DS\), tokens per second; for task quality we report Accuracy on MATH, INCLUDE, and BFCL and Pass@1 on MBPP\+\. Configurations are given in[Appendix˜G](https://arxiv.org/html/2607.26627#A7)\.

![Refer to caption](https://arxiv.org/html/2607.26627v1/x3.png)Figure 3:Efficiency and task performance tradeoff for collaborative verification across four benchmarks\. Each marker represents one hyperparameter setting, and lines connect settings in sweep order\. The star and dashed reference lines denote the lossless speculative\-decoding baseline\. See[Appendix˜C](https://arxiv.org/html/2607.26627#A3)for complete results\.

## 4Identifying the Key Factor in Collaborative Verification

As analyzed in[Section˜3\.1](https://arxiv.org/html/2607.26627#S3.SS1), both lenience relaxationLeviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\)and CoSFuet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib64)\)fall under collaborative verification, whereas CoS exhibits a clear tradeoff between efficiency and task performance, as shown in[Figure˜3](https://arxiv.org/html/2607.26627#S3.F3)\. As illustrated by[Equation˜8](https://arxiv.org/html/2607.26627#S3.E8), the generated distribution increasingly leans toward the draft distribution, and the relaxed acceptance rule leads to degraded task performance\. In contrast, we observe that lenience\-based relaxation maintains strong task performance while achieving slightly better efficiency than the baseline \([Figure˜3](https://arxiv.org/html/2607.26627#S3.F3)\)\. Sinceq​\(x\)q\(x\)is effectively the worst\-case of interpolation, the gain from lenience must come from either the adaptive interpolation coefficientΔ\\Deltain the undershooting region or the thresholding in the overshooting region\.

To disentangle these contributions, we further conduct an ablation study on MBPP\+ to investigate the key factor behind the effectiveness of lenience\-based relaxation\. We selectively replace the generated distribution of CoS in theq​\(x\)<p​\(x\)q\(x\)<p\(x\)andq​\(x\)\>p​\(x\)/λq\(x\)\>p\(x\)/\\lambdaregions with either ceiling the overshoot region or the adaptive interpolation in the underestimated region, and then measure the changes in efficiency and task performance tradeoff\.

For the adaptive interpolation, we retain the adaptive coefficientΔ\\Deltawhen the draft underestimates the target, while using the CoS interpolation rule in the overshoot region:

P​\(x\)=\{q​\(x\)\+Δ​\(p​\(x\)−q​\(x\)\),q​\(x\)≤p​\(x\),λ​p​\(x\)\+\(1−λ\)​q​\(x\),q​\(x\)\>p​\(x\)\.P\(x\)=\\begin\{cases\}q\(x\)\+\\Delta\\\!\\left\(p\(x\)\-q\(x\)\\right\),&q\(x\)\\leq p\(x\),\\\\\[2\.0pt\] \\lambda p\(x\)\+\(1\-\\lambda\)q\(x\),&q\(x\)\>p\(x\)\.\\end\{cases\}

For the overshoot ceiling, we retain the CoS mixture below the ceiling and clip excessive draft probability atp​\(x\)/λp\(x\)/\\lambda:

P​\(x\)=\{λ​p​\(x\)\+\(1−λ\)​q​\(x\),q​\(x\)≤p​\(x\)/λ,p​\(x\)/λ,q​\(x\)\>p​\(x\)/λ\.P\(x\)=\\begin\{cases\}\\lambda p\(x\)\+\(1\-\\lambda\)q\(x\),&q\(x\)\\leq p\(x\)/\\lambda,\\\\\[2\.0pt\] p\(x\)/\\lambda,&q\(x\)\>p\(x\)/\\lambda\.\\end\{cases\}

As shown in[Table˜1](https://arxiv.org/html/2607.26627#S4.T1), introducing adaptive interpolation in the undershoot region has similar severe tradeoff as CoS\. In contrast, ceiling the overshooting region alone achieves task performance comparable to lossless verification, suggesting that overshoot ceiling is its primary contribution\.

In summary, the ablation identifies the overshoot ceiling, not the adaptive interpolation, as the source of lenience\-based relaxation’s effectiveness: with the overshoot ceiling alone, task performance matches lossless verification while efficiency still improves, whereas the adaptive interpolation in the undershoot region inherits CoS’s severe trade\-off\. This aligns with recent findings that low\-quality generations arise primarily from overshoot tokensZhouet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib71)\); Yueet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib70)\); Fanet al\.\([2026b](https://arxiv.org/html/2607.26627#bib.bib77)\), and points to selective overshoot suppression, rather than uniform interpolation, as the key design principle for effective collaborative verification\.

Table 1:Ablation of the two mechanisms in lenience\-based relaxation on MBPP\+\.![Refer to caption](https://arxiv.org/html/2607.26627v1/x4.png)Figure 4:Net change inΔ​BE\\Delta\\mathrm\{BE\}under different draft–target alignment ratios for Min\-ppandη\\eta\-sampling\. Each triplet reports the ratios of matching, partially overlapping, and unrelated candidate tokens, respectively\.
## 5Revealing the Pitfall in Truncation\-Based Verification

We re\-evaluate two representative truncation\-based verification methods, i\.e\., typical acceptance\(Caiet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib39)\)and SpecCascade\(Narasimhanet al\.,[2024](https://arxiv.org/html/2607.26627#bib.bib34)\), against their matched truncation sampling baselines \(η\\eta\-sampling and Min\-ppsampling, respectively\) directly applied on the target model with lossless verification\(Leviathanet al\.,[2023](https://arxiv.org/html/2607.26627#bib.bib14)\)\. This setup isolates whether their claimed comparable performance the reported gains stem from the verification mechanism itself or from the underlying truncation sampling\. Single\-draft experiments are conducted within the standard SD frameworkLeviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\), and multi\-draft experiments within EAGLE\-3Liet al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib80)\)\.

Table 2:Evaluation results using Qwen2\.5\-72B and Qwen2\.5\-0\.5B pair under standard speculative decoding averaged over all hyperparameter settings\. To ensure a fair comparison, each verification method is compared only with its matched truncation sampling baseline using the same allowed set\.Bolddenotes the better task performance within each matched pair\.Graydenotes the SD baseline, which is not directly comparable\.### 5\.1Standard SD

![Refer to caption](https://arxiv.org/html/2607.26627v1/x5.png)Figure 5:Block efficiency of truncation\-based methods across hyperparameter settings on four benchmarks\.Top row: Min\-ppsampling and SpecCascade aspbasep\_\{\\mathrm\{base\}\}varies\.Bottom row:η\\eta\-sampling and typical acceptance asϵ\\epsilonvaries\. Lines show the mean and shaded regions indicate one standard deviation\. The dashed line denotes the lossless speculative\-decoding baseline under the default Qwen2\.5 sampling parameters\.Across multiple benchmarks \([Table˜2](https://arxiv.org/html/2607.26627#S5.T2)\), truncation\-based verification exhibits a clear performance gap relative to its matched baseline under standard speculative decoding\. Min\-p sampling andη\\eta\-sampling outperform SpecCascade and typical acceptance respectively on every benchmark\. As formalized in[Equation˜12](https://arxiv.org/html/2607.26627#S3.E12), this degradation arises because tokens within the allowed set are always accepted, producing a distribution that mirrors the draft model rather than the target distribution \(see[Figure˜2](https://arxiv.org/html/2607.26627#S2.F2)\)\. Overall, these results reinforce our claim that the apparent gains of truncation\-based verification are largely driven by truncation itself, while the verification mechanism can further distort the decoding distribution and reduce task performance\.

Beyond the performance gaps, we investigate truncation sampling effect in Lemma[1](https://arxiv.org/html/2607.26627#Thmlemma1)\. It decomposes the truncation\-induced change in acceptance probability into a non\-negative*gain*term, accumulated over the retained supportAA, and a*loss*term, accumulated over the discarded region at the tail \(see[Appendix˜H](https://arxiv.org/html/2607.26627#A8)for detailed proof\)\. The sign ofΔ​BE\\Delta\\mathrm\{BE\}thus determines whether truncation sampling helps or hurts SD efficiency\.

###### Lemma 1\(Truncation sampling efficiency effect\)\.

Denote allowed set𝒜Θ⊆𝒱\\mathcal\{A\}\_\{\\Theta\}\\subseteq\\mathcal\{V\}determined by a truncation strategyΘ\\Theta,z=∑v∈𝒜Θp​\(v\)z=\\textstyle\\sum\_\{v\\in\\mathcal\{A\}\_\{\\Theta\}\}p\(v\)\. The change in single\-token acceptance probability induced by truncation sampling satisfies

Δ​BE=∑x∈𝒜Θmin​\{\(q​\(x\)−p​\(x\)\)\+−ℒtrunc,\(1z−1\)​p​\(x\)−ℒtrunc,p​\(x\)z−ℒtrunc\}\.w​i​t​h​ℒtrunc=∑x∉𝒜Θmin⁡\(p​\(x\),q​\(x\)\)\.\\begin\{aligned\} &\\Delta\\mathrm\{BE\}=\\sum\_\{x\\in\\mathcal\{A\}\_\{\\Theta\}\}\\min\\\!\\left\\\{\\begin\{array\}\[\]\{l\}\(q\(x\)\-p\(x\)\)\_\{\+\}\-\\mathcal\{L\}\_\{\\mathrm\{trunc\}\},\\\\\[2\.0pt\] \\left\(\\dfrac\{1\}\{z\}\-1\\right\)p\(x\)\-\\mathcal\{L\}\_\{\\mathrm\{trunc\}\},\\\\\[2\.0pt\] \\dfrac\{p\(x\)\}\{z\}\-\\mathcal\{L\}\_\{\\mathrm\{trunc\}\}\\end\{array\}\\right\\\}\.\\\\ &with\\,\\mathcal\{L\}\_\{\\mathrm\{trunc\}\}=\\sum\_\{x\\notin\\mathcal\{A\}\_\{\\Theta\}\}\\min\\\!\\bigl\(p\(x\),q\(x\)\\bigr\)\.\\end\{aligned\}

\(14\)

To make this concrete,[Figure˜4](https://arxiv.org/html/2607.26627#S4.F4)plots the two terms from Lemma[1](https://arxiv.org/html/2607.26627#Thmlemma1)across a range of distribution pairs\. Under min\-ppsampling, the gain dominates at smallerpbasep\_\{\\text\{base\}\}, but the margin narrows aspbasep\_\{\\text\{base\}\}approaches0\.90\.9, where the discarded mass begins to outweigh the redistributed gain\. Underη\\eta\-sampling,Δ​BE\>0\\Delta\\mathrm\{BE\}\>0uniformly across all distribution pairs andϵ\\epsilon, and the net gain grows monotonically withϵ\\epsilondespite wider confidence bands\. This indicates thatη\\eta\-sampling reliably converts mass from the truncated tail into additional acceptance probability on the retained support, whereas min\-ppdoes not\.

Table 3:Evaluation results using LLaMA\-3\.1 8B and its official released pair under EAGLE\-3 speculative decoding, averaged across the parameter grid for each method \(T=0\.7T\{=\}0\.7, block size=7=7\)\.Bolddenotes the better task performance within each matched pair\.Graydenotes the EAGLE\-3 baseline, which is not directly comparable\.![Refer to caption](https://arxiv.org/html/2607.26627v1/x6.png)Figure 6:Efficiency and task performance tradeoff for truncation methods across hyperparameter settings on four benchmarks\. Each point represents a method’s BE and corresponding accuracy or Pass@1, under certain hyper\-parameter\. Baseline refers to the default decoding configuration without min\-p orη\\etasampling\.The contrasting trends in[Figure˜5](https://arxiv.org/html/2607.26627#S5.F5)further show that the benchmark experiments are consistent with both the theoretical analysis in Lemma[1](https://arxiv.org/html/2607.26627#Thmlemma1)and the simulation in[Figure˜4](https://arxiv.org/html/2607.26627#S4.F4)\. The lemma predicts that the efficiency effect of truncation depends on the balance between the gain and the loss\. This balance is reflected in the simulation: min\-ppsampling yields positive gains at moderate thresholds but becomes less favorable aspbasep\_\{\\text\{base\}\}approaches0\.90\.9, whereasη\\eta\-sampling maintains positiveΔ​BE\\Delta\\mathrm\{BE\}across the tested range\. The benchmark results follow the same pattern: min\-ppsampling rises to a peak and then declines, whileη\\eta\-sampling increases more consistently, with only minor deviations\. Overall, these aligned trends support the theoretical decomposition and the controlled simulation\.

### 5\.2EAGLE\-3

Beyond standard single\-draft SD, EAGLE\-3 is of particular interest because its verification operates over multiple drafts arranged in a tree\. We are therefore intrigued to investigate whether lossy verification behaves similarly or differently in this setting\. To this end, we begin by characterizing the per\-token divergence from the matched truncation sampling target under both schemes\.

###### Lemma 2\.

Characterization of the draft\-target divergence\.For a strategyΘ\\Thetawith allowed set𝒜Θ\\mathcal\{A\}\_\{\\Theta\}, writeZΘ​\(p\)=∑x∈𝒜Θp​\(x\)Z\_\{\\Theta\}\(p\)=\\sum\_\{x\\in\\mathcal\{A\}\_\{\\Theta\}\}p\(x\)for the retained target mass andZΘ​\(q\)=∑x∈𝒜Θq​\(x\)Z\_\{\\Theta\}\(q\)=\\sum\_\{x\\in\\mathcal\{A\}\_\{\\Theta\}\}q\(x\)for the retained draft mass\. The per\-token KL divergence between the distributions generated by truncation\-based verification and truncation sampling is, for standard SD and EAGLE\-3 \(see[Appendix˜H](https://arxiv.org/html/2607.26627#A8)for proof\):

KLSD=𝔼​\[∑x∈𝒜Θq​\(x\)ZΘ​\(q\)​log⁡q​\(x\)​ZΘ​\(p\)ZΘ​\(q\)​p​\(x\)\],KLEAGLE=𝔼​\[log⁡ZΘ​\(p\)p​\(x\)\]\.\\begin\{gathered\}\\mathrm\{KL\}\_\{\\mathrm\{SD\}\}=\\mathbb\{E\}\\\!\\left\[\\sum\_\{x\\in\\mathcal\{A\}\_\{\\Theta\}\}\\frac\{q\(x\)\}\{Z\_\{\\Theta\}\(q\)\}\\log\\frac\{q\(x\)\\,Z\_\{\\Theta\}\(p\)\}\{Z\_\{\\Theta\}\(q\)\\,p\(x\)\}\\right\],\\\\\[3\.0pt\] \\mathrm\{KL\}\_\{\\mathrm\{EAGLE\}\}=\\mathbb\{E\}\\\!\\left\[\\log\\frac\{Z\_\{\\Theta\}\(p\)\}\{p\(x\)\}\\right\]\.\\end\{gathered\}

These two divergences respond differently as the draft improves, as the following proposition shows\.

###### Proposition 1\.

Behavior of the gapWithKLSD\\mathrm\{KL\}\_\{\\mathrm\{SD\}\}andKLEAGLE\\mathrm\{KL\}\_\{\\mathrm\{EAGLE\}\}as in Lemma[2](https://arxiv.org/html/2607.26627#Thmlemma2)and\|𝒜Θ\|\>1\|\\mathcal\{A\}\_\{\\Theta\}\|\>1,

limq→pKLSD=0,limq→pKLEAGLE=𝔼​\[log⁡ZΘ​\(p\)p​\(x\)\]\>0\.\\begin\{gathered\}\\lim\_\{q\\to p\}\\mathrm\{KL\}\_\{\\mathrm\{SD\}\}=0,\\\\ \\lim\_\{q\\to p\}\\mathrm\{KL\}\_\{\\mathrm\{EAGLE\}\}=\\mathbb\{E\}\\\!\\left\[\\log\\frac\{Z\_\{\\Theta\}\(p\)\}\{p\(x\)\}\\right\]\>0\.\\end\{gathered\}

In particularKLSD<KLEAGLE\\mathrm\{KL\}\_\{\\mathrm\{SD\}\}<\\mathrm\{KL\}\_\{\\mathrm\{EAGLE\}\}for allqqsufficiently close topp: improving the draft drives the per\-token distortion to zero under standard SD but not under EAGLE\-3\. This theory predicts that under EAGLE\-3, the task\-performance gap between truncation\-based verification and the matched truncation sampling baseline widens\.

Consistent with the theoretical prediction, every matched pair differs by at most1\.41\.4points in task performance \([Table˜2](https://arxiv.org/html/2607.26627#S5.T2)\) under standard SD; while under EAGLE\-3 \([Table˜3](https://arxiv.org/html/2607.26627#S5.T3)and[Figure˜6](https://arxiv.org/html/2607.26627#S5.F6)\), where Proposition[1](https://arxiv.org/html/2607.26627#Thmproposition1)predicts a non\-vanishing divergence, the average gap to the matched truncation sampling baseline widens from−0\.38\-0\.38to−1\.68\-1\.68points for SpecCascade and from−0\.32\-0\.32to−6\.32\-6\.32points for typical acceptance, reaching−8\.8\-8\.8points on INCLUDE \(see[Table˜2](https://arxiv.org/html/2607.26627#S5.T2)and[Table˜3](https://arxiv.org/html/2607.26627#S5.T3)\)\. The degradation is severe enough that typical acceptance falls below even the EAGLE\-3 baseline on all four benchmarks and SpecCascade on two, whereas truncation sampling stays at or above it throughout\. The only thing tree verification buys in return is a slight block\-efficiency gain \([Appendix˜H](https://arxiv.org/html/2607.26627#A8)\), which does not offset the distortion\.

## 6Conclusion

We provide a mechanistic characterization of lossy verification in speculative decoding, showing that existing methods fall into two paradigms: truncation\-based verification and collaborative verification\. Our analysis reveals a critical pitfall of truncation\-based methods: distributional distortion can cause substantial performance degradation relative to the matched truncation\-sampling baseline, particularly under EAGLE\-3\. For collaborative verification, we identify overshoot control as a key mechanism for preserving generation quality\. Together, these findings show that lossy verification should be evaluated against distribution\-matched baselines rather than default decoding alone\. More broadly, our results provide principled guidance for designing speculative decoding algorithms that achieve meaningful acceleration while preserving generation quality\.

## Limitations

Our study has several limitations\. First, our empirical analysis focuses on the Qwen2\.5 and Llama\-3\.1 model families and a fixed pairing of target and draft models; whether the magnitude of the truncation\-based pitfall and the overshoot principle transfer quantitatively to other architectures, model scales, or draft–target ratios remains to be verified\. Second, our evaluation centers on reasoning, code, multilingual, and function\-calling benchmarks \(e\.g\., MATH, MBPP\+, INCLUDE, BFCL\); open\-ended generation and dialogue settings, where quality is harder to measure automatically, are outside our current scope\. Third, our theoretical results characterize the per\-token and per\-position distributional gap under standard SD and EAGLE\-3 tree verification, but the analysis assumes the specific truncation and collaborative rules we formalize and may not directly cover every future lossy verification scheme\. Finally, wall\-clock speedups depend on the hardware and serving stack; while we report block efficiency as a hardware\-agnostic proxy, the precise speed–quality trade\-off will vary across deployment environments\.

## Ethics Statement

This work analyzes verification schemes for speculative decoding and does not involve human subjects, private data, or the collection of new datasets\. All experiments use publicly available, openly licensed models and benchmarks, and are used in accordance with their intended research use\. Our findings aim to improve the reliability of accelerated LLM inference by surfacing quality\-degradation failure modes that are otherwise easy to overlook; we are not aware of any direct risks of misuse arising from this analysis\. We report the models, benchmarks, and hardware used to support reproducibility\.

## References

- J\. Achiam, S\. Adler, S\. Agarwal, L\. Ahmad, I\. Akkaya, F\. L\. Aleman, D\. Almeida, J\. Altenschmidt, S\. Altman, S\. Anadkat,et al\.\(2023\)Gpt\-4 technical report\.arXiv preprint arXiv:2303\.08774\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- J\. Bai, S\. Bai, Y\. Chu, Z\. Cui, K\. Dang, X\. Deng, Y\. Fan, W\. Ge, Y\. Han, F\. Huang,et al\.\(2023\)Qwen technical report\.arXiv preprint arXiv:2309\.16609\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- M\. Balunović, J\. Dekoninck, I\. Petrov, N\. Jovanović, and M\. Vechev \(2025\)Matharena: evaluating llms on uncontaminated math competitions\.arXiv preprint arXiv:2505\.23281\.Cited by:[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p4.1)\.
- T\. Cai, Y\. Li, Z\. Geng, H\. Peng, J\. D\. Lee, D\. Chen, and T\. Dao \(2024\)Medusa: simple llm inference acceleration framework with multiple decoding heads\.InProceedings of the International Conference on Machine Learning \(ICML\),Cited by:[Table 4](https://arxiv.org/html/2607.26627#A1.T4.207.205.205.13.1.1.1),[§1](https://arxiv.org/html/2607.26627#S1.p2.1),[§1](https://arxiv.org/html/2607.26627#S1.p4.2),[§1](https://arxiv.org/html/2607.26627#S1.p5.1),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9),[§2\.3](https://arxiv.org/html/2607.26627#S2.SS3.p6.1),[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p1.1),[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p3.1),[Table 2](https://arxiv.org/html/2607.26627#S5.T2.2.2.7.5.1),[Table 3](https://arxiv.org/html/2607.26627#S5.T3.6.2.7.5.1),[§5](https://arxiv.org/html/2607.26627#S5.p1.2)\.
- Z\. Chen, X\. Yang, J\. Lin, C\. Sun, K\. Chang, and J\. Huang \(2024\)Cascade speculative drafting for even faster llm inference\.Advances in Neural Information Processing Systems \(NeurIPS\)\.Cited by:[Table 4](https://arxiv.org/html/2607.26627#A1.T4.86.84.84.13.1.1.1)\.
- J\. Fan, D\. Cao, X\. Luo, J\. Fu, C\. Liu, and X\. Yang \(2026a\)Flatter tokens are more valuable for speculative draft model training\.arXiv preprint arXiv:2601\.18902\.Cited by:[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9)\.
- J\. Fan, D\. Cao, X\. Luo, J\. Fu, C\. Liu, and X\. Yang \(2026b\)Flatter tokens are more valuable for speculative draft model training\.External Links:2601\.18902,[Link](https://arxiv.org/abs/2601.18902)Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p6.1),[§4](https://arxiv.org/html/2607.26627#S4.p6.1)\.
- E\. Frantar, S\. Ashkboos, T\. Hoefler, and D\. Alistarh \(2022\)Gptq: accurate post\-training quantization for generative pre\-trained transformers\.arXiv preprint arXiv:2210\.17323\.Cited by:[Appendix G](https://arxiv.org/html/2607.26627#A7.SS0.SSS0.Px1.p1.1)\.
- J\. Fu, Y\. Jiang, J\. Chen, J\. Fan, X\. Geng, and X\. Yang \(2025\)Fast large language model collaborative decoding via speculation\.arXiv preprint arXiv:2502\.01662\.Cited by:[Table 4](https://arxiv.org/html/2607.26627#A1.T4.315.313.313.13.1.1.1),[§1](https://arxiv.org/html/2607.26627#S1.p2.1),[§1](https://arxiv.org/html/2607.26627#S1.p4.2),[§1](https://arxiv.org/html/2607.26627#S1.p6.1),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9),[§2\.2](https://arxiv.org/html/2607.26627#S2.SS2.p3.1),[§3\.1](https://arxiv.org/html/2607.26627#S3.SS1.p1.1),[§3\.1](https://arxiv.org/html/2607.26627#S3.SS1.p2.1),[§4](https://arxiv.org/html/2607.26627#S4.p1.2)\.
- D\. Guo, D\. Yang, H\. Zhang, J\. Song, R\. Zhang, R\. Xu, Q\. Zhu, S\. Ma, P\. Wang, X\. Bi,et al\.\(2025\)Deepseek\-r1: incentivizing reasoning capability in llms via reinforcement learning\.arXiv preprint arXiv:2501\.12948\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- D\. Hendrycks, C\. Burns, S\. Kadavath, A\. Arora, S\. Basart, E\. Tang, D\. Song, and J\. Steinhardt \(2021\)Measuring mathematical problem solving with the math dataset\.arXiv preprint arXiv:2103\.03874\.Cited by:[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p4.1)\.
- J\. Hewitt, C\. D\. Manning, and P\. Liang \(2022\)Truncation sampling as language model desmoothing\.arXiv preprint arXiv:2210\.15191\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p4.2),[§2\.3](https://arxiv.org/html/2607.26627#S2.SS3.p2.1),[Table 2](https://arxiv.org/html/2607.26627#S5.T2.2.2.2.1),[Table 3](https://arxiv.org/html/2607.26627#S5.T3.6.2.2.1)\.
- Y\. Huang, X\. Feng, B\. Li, Y\. Xiang, H\. Wang, T\. Liu, and B\. Qin \(2024\)Ensemble learning for heterogeneous large language models with deep parallel collaboration\.Advances in Neural Information Processing Systems \(NeurIPS\)\.Cited by:[§2\.2](https://arxiv.org/html/2607.26627#S2.SS2.p1.3)\.
- Y\. Leviathan, M\. Kalman, and Y\. Matias \(2023\)Fast inference from transformers via speculative decoding\.InProceedings of the International Conference on Machine Learning \(ICML\),Cited by:[Table 4](https://arxiv.org/html/2607.26627#A1.T4.14.12.12.13),[Table 5](https://arxiv.org/html/2607.26627#A1.T5.24.20.20.10.1),[Appendix A](https://arxiv.org/html/2607.26627#A1.p2.5),[§1](https://arxiv.org/html/2607.26627#S1.p1.1),[§1](https://arxiv.org/html/2607.26627#S1.p2.1),[§1](https://arxiv.org/html/2607.26627#S1.p4.2),[§1](https://arxiv.org/html/2607.26627#S1.p6.1),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.5),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9),[§3\.1](https://arxiv.org/html/2607.26627#S3.SS1.p1.1),[§3\.1](https://arxiv.org/html/2607.26627#S3.SS1.p4.1),[§4](https://arxiv.org/html/2607.26627#S4.p1.2),[Table 2](https://arxiv.org/html/2607.26627#S5.T2.2.2.5.3.1),[§5](https://arxiv.org/html/2607.26627#S5.p1.2)\.
- X\. L\. Li, A\. Holtzman, D\. Fried, P\. Liang, J\. Eisner, T\. B\. Hashimoto, L\. Zettlemoyer, and M\. Lewis \(2023\)Contrastive decoding: open\-ended text generation as optimization\.InProceedings of the 61st Annual Meeting of the Association for Computational Linguistics \(Volume 1: Long Papers\),Cited by:[§2\.2](https://arxiv.org/html/2607.26627#S2.SS2.p2.2)\.
- Y\. Li, F\. Wei, C\. Zhang, and H\. Zhang \(2025\)Eagle\-3: scaling up inference acceleration of large language models via training\-time test\.arXiv preprint arXiv:2503\.01840\.Cited by:[Appendix G](https://arxiv.org/html/2607.26627#A7.SS0.SSS0.Px1.p1.1),[§H\.1](https://arxiv.org/html/2607.26627#A8.SS1.p1.5),[§1](https://arxiv.org/html/2607.26627#S1.p5.1),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9),[Table 3](https://arxiv.org/html/2607.26627#S5.T3.6.2.5.3.1),[§5](https://arxiv.org/html/2607.26627#S5.p1.2)\.
- J\. Liu, C\. S\. Xia, Y\. Wang, and L\. Zhang \(2023\)Is your code generated by chatgpt really correct? rigorous evaluation of large language models for code generation\.Advances in Neural Information Processing Systems \(NeurIPS\)\.Cited by:[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p4.1)\.
- X\. Miao, G\. Oliaro, Z\. Zhang, X\. Cheng, Z\. Wang, Z\. Zhang, R\. Y\. Y\. Wong, A\. Zhu, L\. Yang, X\. Shi,et al\.\(2024\)Specinfer: accelerating large language model serving with tree\-based speculative inference and verification\.InProceedings of the ACM International Conference on Architectural Support for Programming Languages and Operating Systems \(ASPLOS\),Cited by:[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9)\.
- H\. Narasimhan, W\. Jitkrittum, A\. S\. Rawat, S\. Kim, N\. Gupta, A\. K\. Menon, and S\. Kumar \(2024\)Faster cascades via speculative decoding\.arXiv preprint arXiv:2405\.19261\.Cited by:[Table 5](https://arxiv.org/html/2607.26627#A1.T5.71.67.67.13.1),[§1](https://arxiv.org/html/2607.26627#S1.p2.1),[§1](https://arxiv.org/html/2607.26627#S1.p4.2),[§1](https://arxiv.org/html/2607.26627#S1.p5.1),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9),[§2\.3](https://arxiv.org/html/2607.26627#S2.SS3.p6.1),[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p1.1),[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p3.1),[Table 2](https://arxiv.org/html/2607.26627#S5.T2.2.2.6.4.1),[Table 3](https://arxiv.org/html/2607.26627#S5.T3.6.2.6.4.1),[§5](https://arxiv.org/html/2607.26627#S5.p1.2)\.
- M\. N\. Nguyen, A\. Baker, C\. Neo, A\. Roush, A\. Kirsch, and R\. Shwartz\-Ziv \(2024\)Turning up the heat: min\-p sampling for creative and coherent llm outputs\.arXiv preprint arXiv:2407\.01082\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p4.2),[§2\.3](https://arxiv.org/html/2607.26627#S2.SS3.p2.1),[Table 2](https://arxiv.org/html/2607.26627#S5.T2.1.1.1.1),[Table 3](https://arxiv.org/html/2607.26627#S5.T3.5.1.1.1)\.
- S\. O’Brien and M\. Lewis \(2023\)Contrastive decoding improves reasoning in large language models\.arXiv preprint arXiv:2309\.09117\.Cited by:[§2\.2](https://arxiv.org/html/2607.26627#S2.SS2.p2.2)\.
- OpenAI, :, A\. Jaech, A\. Kalai, A\. Lerer, A\. Richardson, A\. El\-Kishky, A\. Low, A\. Helyar, A\. Madry, A\. Beutel, A\. Carney, A\. Iftimie, A\. Karpenko, A\. T\. Passos, A\. Neitz, A\. Prokofiev, A\. Wei, A\. Tam, A\. Bennett, A\. Kumar, A\. Saraiva, A\. Vallone, A\. Duberstein, A\. Kondrich, A\. Mishchenko, A\. Applebaum, A\. Jiang, A\. Nair, B\. Zoph, B\. Ghorbani, B\. Zhang, B\. Rossen, B\. Sokolowsky, B\. Barak, B\. McGrew, B\. Minaiev, B\. Hao, B\. Baker, B\. Houghton, B\. McKinzie, B\. Eastman, C\. Lugaresi, C\. Bassin, C\. Hudson, C\. M\. Li, C\. de Bourcy, C\. Voss, C\. Shen, C\. Zhang, C\. Koch, C\. Orsinger, C\. Hesse, C\. Fischer, C\. Chan, D\. Roberts, D\. Kappler, D\. Levy, D\. Selsam, D\. Dohan, D\. Farhi, D\. Mely, D\. Robinson, D\. Tsipras, D\. Li, D\. Oprica, E\. Freeman, E\. Zhang, E\. Wong, E\. Proehl, E\. Cheung, E\. Mitchell, E\. Wallace, E\. Ritter, E\. Mays, F\. Wang, F\. P\. Such, F\. Raso, F\. Leoni, F\. Tsimpourlas, F\. Song, F\. von Lohmann, F\. Sulit, G\. Salmon, G\. Parascandolo, G\. Chabot, G\. Zhao, G\. Brockman, G\. Leclerc, H\. Salman, H\. Bao, H\. Sheng, H\. Andrin, H\. Bagherinezhad, H\. Ren, H\. Lightman, H\. W\. Chung, I\. Kivlichan, I\. O’Connell, I\. Osband, I\. C\. Gilaberte, I\. Akkaya, I\. Kostrikov, I\. Sutskever, I\. Kofman, J\. Pachocki, J\. Lennon, J\. Wei, J\. Harb, J\. Twore, J\. Feng, J\. Yu, J\. Weng, J\. Tang, J\. Yu, J\. Q\. Candela, J\. Palermo, J\. Parish, J\. Heidecke, J\. Hallman, J\. Rizzo, J\. Gordon, J\. Uesato, J\. Ward, J\. Huizinga, J\. Wang, K\. Chen, K\. Xiao, K\. Singhal, K\. Nguyen, K\. Cobbe, K\. Shi, K\. Wood, K\. Rimbach, K\. Gu\-Lemberg, K\. Liu, K\. Lu, K\. Stone, K\. Yu, L\. Ahmad, L\. Yang, L\. Liu, L\. Maksin, L\. Ho, L\. Fedus, L\. Weng, L\. Li, L\. McCallum, L\. Held, L\. Kuhn, L\. Kondraciuk, L\. Kaiser, L\. Metz, M\. Boyd, M\. Trebacz, M\. Joglekar, M\. Chen, M\. Tintor, M\. Meyer, M\. Jones, M\. Kaufer, M\. Schwarzer, M\. Shah, M\. Yatbaz, M\. Y\. Guan, M\. Xu, M\. Yan, M\. Glaese, M\. Chen, M\. Lampe, M\. Malek, M\. Wang, M\. Fradin, M\. McClay, M\. Pavlov, M\. Wang, M\. Wang, M\. Murati, M\. Bavarian, M\. Rohaninejad, N\. McAleese, N\. Chowdhury, N\. Chowdhury, N\. Ryder, N\. Tezak, N\. Brown, O\. Nachum, O\. Boiko, O\. Murk, O\. Watkins, P\. Chao, P\. Ashbourne, P\. Izmailov, P\. Zhokhov, R\. Dias, R\. Arora, R\. Lin, R\. G\. Lopes, R\. Gaon, R\. Miyara, R\. Leike, R\. Hwang, R\. Garg, R\. Brown, R\. James, R\. Shu, R\. Cheu, R\. Greene, S\. Jain, S\. Altman, S\. Toizer, S\. Toyer, S\. Miserendino, S\. Agarwal, S\. Hernandez, S\. Baker, S\. McKinney, S\. Yan, S\. Zhao, S\. Hu, S\. Santurkar, S\. R\. Chaudhuri, S\. Zhang, S\. Fu, S\. Papay, S\. Lin, S\. Balaji, S\. Sanjeev, S\. Sidor, T\. Broda, A\. Clark, T\. Wang, T\. Gordon, T\. Sanders, T\. Patwardhan, T\. Sottiaux, T\. Degry, T\. Dimson, T\. Zheng, T\. Garipov, T\. Stasi, T\. Bansal, T\. Creech, T\. Peterson, T\. Eloundou, V\. Qi, V\. Kosaraju, V\. Monaco, V\. Pong, V\. Fomenko, W\. Zheng, W\. Zhou, W\. Zhan, W\. McCabe, W\. Zaremba, Y\. Dubois, Y\. Lu, Y\. Chen, Y\. Cha, Y\. Bai, Y\. He, Y\. Zhang, Y\. Wang, Z\. Shao, and Z\. Li \(2026\)OpenAI o1 system card\.External Links:2412\.16720,[Link](https://arxiv.org/abs/2412.16720)Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- S\. G\. Patil, H\. Mao, C\. Cheng\-Jie Ji, F\. Yan, V\. Suresh, I\. Stoica, and J\. E\. Gonzalez \(2025\)The berkeley function calling leaderboard \(bfcl\): from tool use to agentic evaluation of large language models\.InForty\-second International Conference on Machine Learning,Cited by:[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p4.1)\.
- Y\. Peng, G\. Zhang, M\. Zhang, Z\. You, J\. Liu, Q\. Zhu, K\. Yang, X\. Xu, X\. Geng, and X\. Yang \(2025\)LMM\-r1: empowering 3b lmms with strong reasoning abilities through two\-stage rule\-based rl\.External Links:2503\.07536,[Link](https://arxiv.org/abs/2503.07536)Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- A\. Romanou, N\. Foroutan, A\. Sotnikova, Z\. Chen, S\. H\. Nelaturu, S\. Singh, R\. Maheshwary, M\. Altomare, M\. A\. Haggag, A\. Amayuelas,et al\.\(2024\)Include: evaluating multilingual language understanding with regional knowledge\.arXiv preprint arXiv:2411\.19799\.Cited by:[§3\.2](https://arxiv.org/html/2607.26627#S3.SS2.p4.1)\.
- T\. Schick, J\. Dwivedi\-Yu, R\. Dessì, R\. Raileanu, M\. Lomeli, E\. Hambro, L\. Zettlemoyer, N\. Cancedda, and T\. Scialom \(2023\)Toolformer: language models can teach themselves to use tools\.Advances in Neural Information Processing Systems \(NeurIPS\)\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- Significant\-Gravitas \(2023\)AutoGPT\.Note:Accessed: 2026\-01\-22External Links:[Link](https://github.com/Significant-Gravitas/AutoGPT)Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- Z\. Sun, U\. Mendlovic, Y\. Leviathan, A\. Aharoni, A\. Beirami, J\. H\. Ro, and A\. T\. Suresh \(2024\)Block verification accelerates speculative decoding\.arXiv preprint arXiv:2403\.10444\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p2.1),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9)\.
- Z\. Sun, A\. T\. Suresh, J\. H\. Ro, A\. Beirami, H\. Jain, and F\. Yu \(2023\)Spectr: fast speculative decoding via optimal transport\.Advances in Neural Information Processing Systems \(NeurIPS\)\.Cited by:[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9)\.
- H\. Touvron, L\. Martin, K\. Stone, P\. Albert, A\. Almahairi, Y\. Babaei, N\. Bashlykov, S\. Batra, P\. Bhargava, S\. Bhosale,et al\.\(2023\)Llama 2: open foundation and fine\-tuned chat models\.arXiv preprint arXiv:2307\.09288\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- A\. Yang, B\. Yang, B\. Zhang, B\. Hui, B\. Zheng, B\. Yu, C\. Li, D\. Liu, F\. Huang, H\. Wei, H\. Lin, J\. Yang, J\. Tu, J\. Zhang, J\. Yang, J\. Yang, J\. Zhou, J\. Lin, K\. Dang, K\. Lu, K\. Bao, K\. Yang, L\. Yu, M\. Li, M\. Xue, P\. Zhang, Q\. Zhu, R\. Men, R\. Lin, T\. Li, T\. Xia, X\. Ren, X\. Ren, Y\. Fan, Y\. Su, Y\. Zhang, Y\. Wan, Y\. Liu, Z\. Cui, Z\. Zhang, and Z\. Qiu \(2025\)Qwen2\.5 technical report\.arXiv preprint arXiv:2412\.15115\.Cited by:[Appendix G](https://arxiv.org/html/2607.26627#A7.SS0.SSS0.Px1.p1.1)\.
- S\. Yang, S\. Huang, X\. Dai, and J\. Chen \(2024\)Multi\-candidate speculative decoding\.arXiv preprint arXiv:2401\.06706\.Cited by:[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9)\.
- S\. Yao, J\. Zhao, D\. Yu, N\. Du, I\. Shafran, K\. R\. Narasimhan, and Y\. Cao \(2022\)React: synergizing reasoning and acting in language models\.InProceedings of the International Conference on Learning Representations \(ICLR\),Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p1.1)\.
- Y\. Yao, H\. Wu, M\. Liu, S\. Luo, X\. Han, J\. Liu, Z\. Guo, and L\. Song \(2024\)Determine\-then\-ensemble: necessity of top\-k union for large language model ensembling\.arXiv preprint arXiv:2410\.03777\.Cited by:[§2\.2](https://arxiv.org/html/2607.26627#S2.SS2.p1.3)\.
- Y\. Yue, Z\. Chen, R\. Lu, A\. Zhao, Z\. Wang, Y\. Yue, S\. Song, and G\. Huang \(2025\)Does reinforcement learning really incentivize reasoning capacity in llms beyond the base model?\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p6.1),[§4](https://arxiv.org/html/2607.26627#S4.p6.1)\.
- Y\. Zhou, K\. Lyu, A\. S\. Rawat, A\. K\. Menon, A\. Rostamizadeh, S\. Kumar, J\. Kagy, and R\. Agarwal \(2024\)DistillSpec: improving speculative decoding via knowledge distillation\.InProceedings of the International Conference on Learning Representations \(ICLR\),Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p2.1),[§1](https://arxiv.org/html/2607.26627#S1.p4.2),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9)\.
- Y\. Zhou, F\. Huang, H\. Li, F\. Wu, T\. Wang, J\. Zhang, J\. Lin, and Z\. Cheng \(2026\)Overcoming joint intractability with lossless hierarchical speculative decoding\.arXiv preprint arXiv:2601\.05724\.Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p2.1),[§2\.1](https://arxiv.org/html/2607.26627#S2.SS1.p1.9)\.
- Y\. Zhou, M\. Keuper, and M\. Fritz \(2025\)Balancing diversity and risk in llm sampling: how to select your method and parameter for open\-ended text generation\.InProceedings of the Annual Meeting of the Association for Computational Linguistics \(ACL\),Cited by:[§1](https://arxiv.org/html/2607.26627#S1.p6.1),[§4](https://arxiv.org/html/2607.26627#S4.p6.1)\.

## Appendix ADerivations for Lenience\-based Collaborative Verification

This section derives the closed\-form expression for the generation distribution induced by lenience\-based relaxation \([Equation˜10](https://arxiv.org/html/2607.26627#S3.E10)\) and for the mixing coefficientΔ\\Delta\([Equation˜10](https://arxiv.org/html/2607.26627#S3.E10)\)\.

In tokenwise speculative sampling\(Leviathanet al\.,[2023](https://arxiv.org/html/2607.26627#bib.bib14)\), each tokenxtx\_\{t\}is drafted fromq​\(xt\)q\(x\_\{t\}\)and verified againstp​\(xt\)p\(x\_\{t\}\)\. It is accepted with probabilityh​\(xt\)=min⁡\{1,p​\(xt\)/ℓ​q​\(xt\)\}h\(x\_\{t\}\)=\\min\\\{1,\\,p\(x\_\{t\}\)/\\ell q\(x\_\{t\}\)\\\}, or rejected and replaced fromPres​\(xt\)P\_\{\\text\{res\}\}\(x\_\{t\}\)\. Thus the yield probability is:

P​\(generate​xt\)\\displaystyle P\(\\text\{generate\}\\,x\_\{t\}\)=P​\(draft and accept​xt\)\\displaystyle=P\(\\text\{draft and accept\}\\,x\_\{t\}\)\+P​\(draft and reject​v,resampled​xt\)\\displaystyle\\quad\+P\(\\text\{draft and reject\}\\,v,\\,\\text\{resampled\}\\,x\_\{t\}\)=q​\(xt\)​h​\(xt\)\+\[∑v∈𝒱q​\(v\)​\(1−h​\(v\)\)\]\\displaystyle=q\(x\_\{t\}\)h\(x\_\{t\}\)\+\\Bigl\[\\sum\_\{v\\in\\mathcal\{V\}\}q\(v\)\(1\-h\(v\)\)\\Bigr\]×p​\(xt\)−min⁡\{p​\(xt\),q​\(xt\)\}∑v∈𝒱\(p​\(v\)−min⁡\{p​\(v\),q​\(v\)\}\)\\displaystyle\\quad\\times\\frac\{p\(x\_\{t\}\)\-\\min\\\{p\(x\_\{t\}\),q\(x\_\{t\}\)\\\}\}\{\\textstyle\\sum\_\{v\\in\\mathcal\{V\}\}\\bigl\(p\(v\)\-\\min\\\{p\(v\),q\(v\)\\\}\\bigr\)\}=min⁡\{q​\(xt\),p​\(xt\)ℓ\}\\displaystyle=\\min\\Bigl\\\{q\(x\_\{t\}\),\\,\\frac\{p\(x\_\{t\}\)\}\{\\ell\}\\Bigr\\\}\+∑v∈𝒱\(q​\(v\)−min⁡\{q​\(v\),p​\(v\)/ℓ\}\)∑v∈𝒱\(q​\(v\)−min⁡\{q​\(v\),p​\(v\)\}\)\\displaystyle\\quad\+\\frac\{\\textstyle\\sum\_\{v\\in\\mathcal\{V\}\}\\bigl\(q\(v\)\-\\min\\\{q\(v\),\\,p\(v\)/\\ell\\\}\\bigr\)\}\{\\textstyle\\sum\_\{v\\in\\mathcal\{V\}\}\\bigl\(q\(v\)\-\\min\\\{q\(v\),\\,p\(v\)\\\}\\bigr\)\}×\[p​\(xt\)−min⁡\{p​\(xt\),q​\(xt\)\}\]\\displaystyle\\qquad\\times\\Bigl\[p\(x\_\{t\}\)\-\\min\\\{p\(x\_\{t\}\),q\(x\_\{t\}\)\\\}\\Bigr\]=min⁡\{q​\(xt\),p​\(xt\)ℓ\}\\displaystyle=\\min\\Bigl\\\{q\(x\_\{t\}\),\\,\\frac\{p\(x\_\{t\}\)\}\{\\ell\}\\Bigr\\\}\+Δ​\[p​\(xt\)−min⁡\{p​\(xt\),q​\(xt\)\}\]\\displaystyle\\quad\+\\Delta\\Bigl\[p\(x\_\{t\}\)\-\\min\\\{p\(x\_\{t\}\),q\(x\_\{t\}\)\\\}\\Bigr\]=\{q​\(xt\)\+Δ​\(p​\(xt\)−q​\(xt\)\),q​\(xt\)≤p​\(xt\),q​\(xt\),p​\(xt\)≤q​\(xt\)≤p​\(xt\)/ℓ,p​\(xt\)/ℓ,q​\(xt\)≥p​\(xt\)/ℓ\.\\displaystyle=\\begin\{cases\}q\(x\_\{t\}\)\+\\Delta\(p\(x\_\{t\}\)\-q\(x\_\{t\}\)\),&q\(x\_\{t\}\)\\leq p\(x\_\{t\}\),\\\\ q\(x\_\{t\}\),&p\(x\_\{t\}\)\\leq q\(x\_\{t\}\)\\leq p\(x\_\{t\}\)/\\ell,\\\\ p\(x\_\{t\}\)/\\ell,&q\(x\_\{t\}\)\\geq p\(x\_\{t\}\)/\\ell\.\\end\{cases\}
#### Derivation forΔ\\Delta

Δ\\displaystyle\\Delta=∑v∈𝒱\(q​\(v\)−min⁡\{q​\(v\),p​\(v\)ℓ\}\)∑v∈𝒱\(q​\(v\)−min⁡\{q​\(v\),p​\(v\)\}\)\.\\displaystyle=\\frac\{\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-\\min\\left\\\{q\(v\),\\,\\frac\{p\(v\)\}\{\\ell\}\\right\\\}\\right\)\}\{\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-\\min\\left\\\{q\(v\),\\,p\(v\)\\right\\\}\\right\)\}\.\(15\)
Using the identity

a−min⁡\{a,b\}=\|a−b\|\+\(a−b\)2,\\displaystyle a\-\\min\\\{a,b\\\}=\\frac\{\|a\-b\|\+\(a\-b\)\}\{2\},\(16\)the numerator becomes

∑v∈𝒱\(q​\(v\)−min⁡\{q​\(v\),p​\(v\)ℓ\}\)\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-\\min\\left\\\{q\(v\),\\,\\frac\{p\(v\)\}\{\\ell\}\\right\\\}\\right\)=12​∑v∈𝒱\|q​\(v\)−p​\(v\)ℓ\|\\displaystyle=\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|q\(v\)\-\\frac\{p\(v\)\}\{\\ell\}\\right\|\+12​∑v∈𝒱\(q​\(v\)−p​\(v\)ℓ\)\.\\displaystyle\\quad\+\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-\\frac\{p\(v\)\}\{\\ell\}\\right\)\.\(17\)
Sinceppandqqare probability distributions on𝒱\\mathcal\{V\},

∑v∈𝒱q​\(v\)=1,∑v∈𝒱p​\(v\)=1,\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}q\(v\)=1,\\qquad\\sum\_\{v\\in\\mathcal\{V\}\}p\(v\)=1,\(18\)we have

∑v∈𝒱\(q​\(v\)−p​\(v\)ℓ\)=1−1ℓ\.\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-\\frac\{p\(v\)\}\{\\ell\}\\right\)=1\-\\frac\{1\}\{\\ell\}\.\(19\)Therefore,

∑v∈𝒱\(q​\(v\)−min⁡\{q​\(v\),p​\(v\)ℓ\}\)\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-\\min\\left\\\{q\(v\),\\,\\frac\{p\(v\)\}\{\\ell\}\\right\\\}\\right\)=12​∑v∈𝒱\|q​\(v\)−p​\(v\)ℓ\|\+12−12​ℓ\.\\displaystyle=\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|q\(v\)\-\\frac\{p\(v\)\}\{\\ell\}\\right\|\+\\frac\{1\}\{2\}\-\\frac\{1\}\{2\\ell\}\.\(20\)
Similarly, the denominator satisfies

∑v∈𝒱\(q​\(v\)−min⁡\{q​\(v\),p​\(v\)\}\)\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-\\min\\left\\\{q\(v\),\\,p\(v\)\\right\\\}\\right\)=12​∑v∈𝒱\|q​\(v\)−p​\(v\)\|\+12​∑v∈𝒱\(q​\(v\)−p​\(v\)\)\.\\displaystyle=\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|q\(v\)\-p\(v\)\\right\|\+\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-p\(v\)\\right\)\.\(21\)
Again using

∑v∈𝒱q​\(v\)=∑v∈𝒱p​\(v\)=1,\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}q\(v\)=\\sum\_\{v\\in\\mathcal\{V\}\}p\(v\)=1,\(22\)we obtain

∑v∈𝒱\(q​\(v\)−p​\(v\)\)=0,\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-p\(v\)\\right\)=0,\(23\)and hence

∑v∈𝒱\(q​\(v\)−min⁡\{q​\(v\),p​\(v\)\}\)\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}\\left\(q\(v\)\-\\min\\left\\\{q\(v\),\\,p\(v\)\\right\\\}\\right\)=12​∑v∈𝒱\|q​\(v\)−p​\(v\)\|\.\\displaystyle=\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|q\(v\)\-p\(v\)\\right\|\.\(24\)
Substituting these two expressions back intoΔ\\Deltayields

Δ\\displaystyle\\Delta=12​∑v∈𝒱\|q​\(v\)−p​\(v\)ℓ\|\+12−12​ℓ12​∑v∈𝒱\|q​\(v\)−p​\(v\)\|\.\\displaystyle=\\frac\{\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|q\(v\)\-\\frac\{p\(v\)\}\{\\ell\}\\right\|\+\\frac\{1\}\{2\}\-\\frac\{1\}\{2\\ell\}\}\{\\frac\{1\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}\\left\|q\(v\)\-p\(v\)\\right\|\}\.\(25\)
Table 4:Verification method results across the four benchmarks \(MATH, MBPP\+, INCLUDE, BFCL\)\. For each method we sweep its characteristic hyperparameter \(Param\) and report block efficiency \(BE\), decoding speed \(DS\), and accuracy / Pass@1; all entries are mean±\\pmstd over three seeds\.Table 5:EAGLE\-3 speculative decoding results across four benchmarks \(LLaMA\-3\.1 8B, temperature=0\.7=0\.7, block size=7=7\)\.Boldmarks the best value per column across all methods;underlinemarks the second best\.MATHMBPP\+INCLUDEBFCLMethodParamBEDSAcc \(%\)BEDSPass@1 \(%\)BEDSAcc \(%\)BEDSAcc \(%\)Baseline—3\.763\.76140\.21140\.2173\.2073\.204\.704\.70167\.83167\.8359\.3059\.300\.670\.6745\.7245\.7235\.502\.572\.5785\.2785\.2786\.0086\.00LenienceLeviathanet al\.\([2023](https://arxiv.org/html/2607.26627#bib.bib14)\)0\.24\.494\.49161\.29161\.2971\.2071\.205\.175\.17181\.9561\.110\.860\.8650\.6150\.6130\.9130\.912\.632\.6387\.3183\.5083\.500\.44\.344\.34157\.03157\.0376\.2076\.205\.025\.02177\.13177\.1359\.7959\.790\.800\.8048\.9348\.9332\.7332\.732\.612\.6186\.1485\.5085\.500\.64\.194\.19152\.85152\.8576\.0076\.004\.964\.96175\.43175\.4359\.5259\.520\.730\.7347\.0347\.0334\.5534\.552\.592\.5985\.7785\.7786\.0086\.000\.84\.074\.07149\.39149\.3975\.2075\.204\.924\.92174\.25174\.2559\.7959\.790\.720\.7246\.7346\.7333\.6433\.642\.592\.5985\.5685\.5684\.0084\.00SpecCascadeNarasimhanet al\.\([2024](https://arxiv.org/html/2607.26627#bib.bib34)\)0\.14\.474\.47159\.83159\.8373\.0073\.005\.165\.16180\.81180\.8159\.7959\.790\.920\.9251\.7651\.7630\.9130\.912\.622\.6285\.8285\.8284\.5084\.500\.34\.284\.28154\.42154\.4277\.405\.035\.03176\.79176\.7961\.380\.790\.7948\.2648\.2631\.8231\.822\.592\.5985\.1885\.1884\.5084\.500\.54\.184\.18151\.56151\.5677\.604\.994\.99175\.66175\.6660\.8560\.850\.760\.7647\.4347\.4331\.8231\.822\.582\.5885\.0385\.0386\.5086\.500\.74\.124\.12149\.71149\.7176\.2076\.204\.934\.93173\.70173\.7061\.380\.730\.7346\.7446\.7436\.822\.582\.5884\.9984\.9985\.5085\.500\.94\.064\.06148\.20148\.2076\.6076\.604\.894\.89172\.60172\.6060\.3260\.320\.720\.7246\.4546\.4535\.0035\.002\.582\.5885\.0085\.0086\.0086\.00Min\-p Smpl\.\+ SD0\.13\.893\.89140\.72140\.7275\.2075\.204\.844\.84166\.79166\.7961\.9061\.900\.540\.5445\.3445\.3440\.002\.582\.5884\.4584\.4586\.0086\.000\.33\.963\.96142\.85142\.8577\.604\.894\.89168\.29168\.2960\.3260\.320\.550\.5545\.6545\.6536\.8236\.822\.582\.5884\.1384\.1386\.0086\.000\.53\.983\.98143\.42143\.4277\.404\.914\.91168\.98168\.9860\.8560\.850\.560\.5645\.9145\.9135\.9135\.912\.582\.5884\.0984\.0986\.5086\.500\.74\.00143\.8877\.404\.92169\.2362\.4362\.430\.560\.5645\.9635\.9135\.912\.592\.5984\.2184\.2187\.500\.94\.05145\.2476\.8076\.804\.93169\.5863\.4963\.490\.5746\.1336\.3636\.362\.592\.5984\.4284\.4287\.00η\\etaSampl\.\+ SD0\.053\.853\.85137\.92137\.9275\.2075\.204\.844\.84164\.79164\.7959\.7959\.790\.550\.5545\.4245\.4235\.0035\.002\.582\.5883\.6383\.6387\.500\.103\.903\.90139\.33139\.3377\.0077\.004\.864\.86165\.32165\.3261\.1161\.110\.540\.5445\.0645\.0641\.822\.592\.5983\.9283\.9286\.0086\.000\.153\.933\.93140\.21140\.2177\.2077\.204\.884\.88165\.95165\.9561\.9061\.900\.540\.5445\.1345\.1335\.0035\.002\.592\.5983\.9183\.9186\.5086\.500\.203\.963\.96140\.97140\.9776\.0076\.004\.894\.89166\.23166\.2364\.290\.540\.5445\.1445\.1434\.0934\.092\.572\.5783\.5483\.5485\.0085\.000\.253\.993\.99141\.83141\.8375\.2075\.204\.884\.88165\.77165\.7763\.760\.5745\.9245\.9237\.7337\.732\.582\.5863\.8563\.8586\.5086\.50Typical Sampling0\.054\.82168\.7766\.0066\.005\.29182\.8655\.2955\.291\.2459\.8029\.5529\.552\.6686\.1186\.1178\.0078\.000\.104\.66164\.1968\.2068\.205\.21180\.54180\.5456\.0856\.081\.1557\.2029\.5529\.552\.6585\.8585\.8579\.5079\.500\.154\.584\.58161\.86161\.8672\.2072\.205\.165\.16178\.87178\.8757\.4157\.411\.051\.0554\.7654\.7625\.9125\.912\.642\.6485\.6585\.6583\.0083\.000\.204\.544\.54160\.65160\.6569\.8069\.805\.165\.16178\.94178\.9458\.4758\.470\.980\.9852\.7352\.7327\.7327\.732\.632\.6385\.4085\.4083\.5083\.500\.254\.464\.46158\.17158\.1773\.0073\.005\.175\.17179\.31179\.3157\.1457\.141\.001\.0053\.2153\.2126\.8226\.822\.622\.6285\.0585\.0583\.0083\.00

## Appendix BVerification Analysis

This section complements the EAGLE\-3 analysis in[Section˜5](https://arxiv.org/html/2607.26627#S5)with the standard\-SD setting \(Qwen2\.5\-72B target, 0\.5B draft\)\. The same patterns hold, with smaller absolute gaps, aligning with our claim that EAGLE\-3 amplifies the pitfall\.

#### Efficiency–performance trade\-off\.

[Figure˜7](https://arxiv.org/html/2607.26627#A2.F7)sweeps each method’s characteristic hyperparameter \(pbasep\_\{\\text\{base\}\}for min\-ppand SpecCascade;ϵ\\epsilonforη\\eta\-sampling and Medusa\)\. Truncation sampling \(filled markers\) clusters tightly at or above the lossless baseline, while truncation\-based verification \(open markers\) reaches similar BE at the cost of more scattered task performance\. The dispersion is largest for typical acceptance, most visibly on MBPP\+ and INCLUDE\. No verification setting Pareto\-dominates its matched truncation\-sampling baseline\.

#### Hyperparameter sensitivity\.

[Figure˜5](https://arxiv.org/html/2607.26627#S5.F5)traces BE as the controlling hyperparameter is swept\. Min\-ppsampling and SpecCascade \(top row\) produce smooth, near\-monotonic curves, making hyperparameter selection straightforward\. In contrast,η\\eta\-sampling and Medusa \(bottom row\) are non\-monotonic and noisy acrossϵ\\epsilon, with Medusa swinging by more than0\.10\.1BE between adjacent values on MATH\. This is consistent with the entropy\-driven cutoff in[Equation˜6](https://arxiv.org/html/2607.26627#S2.E6): small changes inϵ\\epsiloncan flip whether the thresholdδ​e−H​\(p\)\\delta\\,e^\{\-H\(p\)\}binds, leading to discrete jumps in the allowed set\.

![Refer to caption](https://arxiv.org/html/2607.26627v1/verif_scatter_truncation.png)Figure 7:Efficiency and task performance tradeoff for truncation methods across hyperparameter settings on three benchmarks\. Each point shows a method’s BE and corresponding accuracy or Pass@1, under certain hyper\-parameter\. Baseline refers to the default decoding configuration without min\-p orη\\etasampling\. Detailed results can be found in[Appendix˜C](https://arxiv.org/html/2607.26627#A3)\.

## Appendix CExtended Results

#### Verification\.

This section reports the full per\-benchmark numbers underlying the aggregate comparisons in the main text\. We evaluate every method on four benchmarks chosen to span quantitative reasoning \(MATH\), code generation \(MBPP\+\), multilingual knowledge \(INCLUDE\), and tool use \(BFCL\)\. To keep verification cost tractable while preserving signal, we use the first 500 problems of MATH, the full MBPP\+ test set, 5 problems per language from INCLUDE \(220 in total, balanced across all languages\), and theparallel\_multiplesplit of BFCL\_v3 restricted to 200 calls\. All numbers are reported as mean±\\pmstd over three random seeds; methods sharing a backbone \(e\.g\., Min\-p Smpl\. \+ tokenwise SpD vs\. Cascade\) reuse the same draft trajectories, which makes the std an estimate of verifier\-induced variance rather than full end\-to\-end variance\.

#### EAGLE\-3\.

[Table˜5](https://arxiv.org/html/2607.26627#A1.T5)reports EAGLE\-3 speculative decoding results on MATH, MBPP\+, INCLUDE, and BFCL, revealing a consistent tradeoff between draft throughput and task accuracy: aggressive methods \(Typical Sampling, Lenience, SpecCascade\) achieve the highest block efficiency and decode speed but degrade accuracy, particularly on precision\-sensitive tasks such as BFCL and INCLUDE\. In contrast, entropy\-aware draft filtering methods \(Min\-p Sampling \+ SD andη\\etaSampling \+ SD\) deliver the most balanced profile, matching or exceeding baseline accuracy on every benchmark, including the best scores on MATH \(77\.60%\), MBPP\+ \(64\.29%\), INCLUDE \(41\.82%\), and BFCL \(87\.50%\), while still improving over the baseline in block efficiency and decode speed\.

## Appendix DLenience Relaxation Interpretation

![Refer to caption](https://arxiv.org/html/2607.26627v1/delta_tv_vs_delta.png)\(a\)Adaptive interpolation\.
![Refer to caption](https://arxiv.org/html/2607.26627v1/ablation_b_only.png)\(b\)Overshoot ceiling\.

Figure 8:Two views of lenience\-based relaxation\. Fig\.[8\(a\)](https://arxiv.org/html/2607.26627#A4.F8.sf1)illustrates how the interpolation strength adapts to the draft–target divergence on a per\-token basis\. Fig\.[8\(b\)](https://arxiv.org/html/2607.26627#A4.F8.sf2)contrasts a global linear mixture with a pointwise ceiling atp/ℓp/\\ell, highlighting the asymmetric treatment of overshoot\.Lenience\-based relaxation trades a controlled amount of distributional distortion for a higher acceptance rate, governed by the lenience parameterℓ\\ell\. Figure[8](https://arxiv.org/html/2607.26627#A4.F8)examines two facets of this trade\-off:*how much*relaxation to apply, and*where*it takes effect\.

#### Adaptive interpolation\.

Fig\.[8\(a\)](https://arxiv.org/html/2607.26627#A4.F8.sf1)plots the interpolation strengthΔi\\Delta\_\{i\}against the draft–target divergenceTV​\(p,qi\)\\mathrm\{TV\}\(p,q\_\{i\}\)\. Rather than applying a fixed amount of leniency, the relaxation adapts to local disagreement: when the draft already agrees with the target \(q5q\_\{5\}, smallTV\\mathrm\{TV\}\) we haveΔi≈0\\Delta\_\{i\}\\approx 0and almost no relaxation is applied, whereas a more divergent draft \(q1q\_\{1\}\) induces a largerΔi\\Delta\_\{i\}\. The concave profile further shows that the additional leniency saturates rather than growing without bound as divergence increases, so relaxation is spent precisely where draft and target diverge and withheld where they coincide\.

#### Overshoot ceiling\.

Fig\.[8\(b\)](https://arxiv.org/html/2607.26627#A4.F8.sf2)contrasts a global linear mixtureℓ​p\+\(1−ℓ\)​q\\ell p\+\(1\-\\ell\)q, which blends draft and target uniformly across the vocabulary, with a pointwise ceiling atp/ℓp/\\ell\(reference capping\), which accepts the draft wherever it stays below the ceiling and clips it only where it exceeds it\. Two regimes emerge: forq≤p/ℓq\\leq p/\\ellthe draft lies under the ceiling \(underestimation or moderate overestimation\) and is accepted, while forq\>p/ℓq\>p/\\ellthe draft*overshoots*the relaxed target and is capped\. The shaded region marks exactly where the two schemes disagree—the overshoot mass that the ceiling clips but the linear baseline retains—highlighting the asymmetric treatment of overshoot\.

## Appendix EExtended Experiments for Identifying the Key Factor in SD

The ablation in[Table˜6](https://arxiv.org/html/2607.26627#A5.T6)supports our claim that adaptive interpolation is not the primary source of improvement in lenience\-based relaxation: removing the adaptive interpolation while retaining thep/ℓp/\\ellcap preserves most of the benefit\. The overshoot region is precisely where the target and draft distributions disagree most strongly, and the draft tends to over\-allocate probability to low\-quality tokens there\. Capping atp/ℓp/\\ellin this region directly suppresses this failure mode\.

We further investigate the overlap between the overshoot region and the set of tokens discarded by min\-ppsampling, motivated by the intuition that accepting more tokens from the discarded set can improve efficiency\. This suggests combining the two mechanisms: use the min\-ppallowed set𝒜Θ\\mathcal\{A\}\_\{\\Theta\}to gate acceptance, and apply thep/ℓp/\\ellcap to control overshoot outside it\. Concretely, tokens in𝒜Θ\\mathcal\{A\}\_\{\\Theta\}are yielded with draft probabilityq​\(x\)q\(x\), while tokens outside are yielded withmin⁡\{q​\(x\),p​\(x\)/ℓ\}\\min\\\{q\(x\),\\,p\(x\)/\\ell\\\}, so the rule relies on the draft distribution except in the overlap between the overshoot region and the discarded set\. As shown in the last row of[Table˜6](https://arxiv.org/html/2607.26627#A5.T6), this rule delivers a3\.7%3\.7\\%gain in BE over lossless SD while matching its Pass@1 exactly, outperforming both SpecCascade and lenience\-based relaxation\. The discarded set thus appears to be a promising target for further acceleration at negligible quality cost\.

Table 6:Effect of truncation and overshoot capping on MBPP\+, averaged overℓ∈\{0\.2,0\.4,0\.6,0\.8\}\\ell\\in\\\{0\.2,0\.4,0\.6,0\.8\\\}\.pbase=0\.5p\_\{\\text\{base\}\}=0\.5is used in𝒜Θ\\mathcal\{A\}\_\{\\Theta\}\.
## Appendix FQualitative Examples

We hereby include some samples revealing the failure mode of truncation\-based verifications against their matched baselines\.

MBPP\+ task99— wrong sentinel returnPrompt:*Write a python function to find the minimum number of rotations \(greater than0\) required to get the same string\.*Matched baseline \(min\-pp\):for i in range\(1, n\): if temp\[i:i\+n\] == s: return i returnn✓ SpecCascade \(min\-ppallowed set\):for i in range\(1, n\): if temp\[i:i\+n\] == s: return i return0×\\times*Fingerprint:*both versions share identical body and loop logic; they disagree only on the post\-loop sentinel\. The draft defaults to0, which violates the “greater than0” specification; the target returnsn, the trivial full\-rotation period\.

BFCLparallel\_multiple\_195— math vs\. Python notationGround\-truth slot:calculate\_area\_under\_curve\(function="x\*\*2",…\) Matched baseline \(min\-pp\):function="x\*\*2"✓ SpecCascade \(min\-ppallowed set\):function="xˆ2"×\\times*Fingerprint:*the draft writes the math\-class formxˆ2; the target writes the Python formx\*\*2that the BFCL grader requires\. This divergence recurs1414times across the\(pbase,seed\)\(p\_\{\\text\{base\}\},\\,\\text\{seed\}\)grid, always in the same direction\.

MBPP\+ task637637— function signature mishreadPrompt:*Write a function to check whether the given amount has no profit and no loss\.*Hidden testscall the function asnoprofit\_noloss\(cost\_price, selling\_price\)\.Matched baseline \(min\-pp\):def noprofit\_noloss\(cost\_price, selling\_price\): return cost\_price == selling\_price✓ SpecCascade \(min\-ppallowed set\):def noprofit\_noloss\(amount\): return amount == 0×\\times*Fingerprint:*the draft latches onto the salient word*“amount”*in the prompt and emits a plausible\-looking single\-argument signature; the target reads the two\-quantity nature of the problem and emits the matching two\-argument signature\.

## Appendix GExperimental Setup

#### Models\.

We utilize models from the Qwen2\.5 familyYanget al\.\([2025](https://arxiv.org/html/2607.26627#bib.bib65)\)that are instruction tuned and quantized to have 8\-bit precisions with GPTQFrantaret al\.\([2022](https://arxiv.org/html/2607.26627#bib.bib73)\)\. Specifically, we chooseQwen2\.5\-72B\-Instruct\-GPTQ\-Int8as the target model andQwen2\.5\-0\.5B\-Instruct\-GPTQ\-Int8as the draft model\. For the EAGLE\-3 experiment in[Table˜5](https://arxiv.org/html/2607.26627#A1.T5), we useLlama\-3\.1\-8B\-Instructas the target model, paired with the corresponding draft model from the official EAGLE\-3 release\(Liet al\.,[2025](https://arxiv.org/html/2607.26627#bib.bib80)\)\.

#### Hardware Platform\.

The experiments in Figure[1](https://arxiv.org/html/2607.26627#S1.F1)are conducted on a single NVIDIA H200 GPU with 140 GB of VRAM\. The EAGLE\-3 experiments in Table[3](https://arxiv.org/html/2607.26627#S5.T3)are conducted on a single NVIDIA A6000 GPU with 48 GB of VRAM\. All other experiments are conducted on two NVIDIA A100 GPUs, each with 80 GB of VRAM\.

## Appendix HBlock Efficiency and Distributional Gap under Tree Verification

Table 7:Gap of truncation\-based verification relative to its matched truncation sampling \+ SD baseline \(Δ=\\Delta=lossy−\-lossless\),*without*EAGLE\-3 \(single\-draft SD,[Table˜2](https://arxiv.org/html/2607.26627#S5.T2)\) and*with*EAGLE\-3 \([Table˜3](https://arxiv.org/html/2607.26627#S5.T3)\)\. EAGLE\-3 amplifies the average task\-performance deficit by roughly4×4\\times\(SpecCascade\) and20×20\\times\(typical acceptance\)\.This section gives the explicit single\-draft versus EAGLE\-3 gap comparison \([Table˜7](https://arxiv.org/html/2607.26627#A8.T7)\), records the tree\-verification procedure as implemented \([Section˜H\.1](https://arxiv.org/html/2607.26627#A8.SS1)\), and proves the distributional gap of Lemma[2](https://arxiv.org/html/2607.26627#Thmlemma2)and the block efficiency of Proposition[2](https://arxiv.org/html/2607.26627#Thmproposition2)through a per\-position analysis \(Lemma[3](https://arxiv.org/html/2607.26627#Thmlemma3), Corollary[1](https://arxiv.org/html/2607.26627#Thmcorollary1)\)\. Throughout,pp,qq,pΘp\_\{\\Theta\},zz,ZΘZ\_\{\\Theta\}, and𝒜Θ\\mathcal\{A\}\_\{\\Theta\}refer to the current decoding step and are conditioned on all previously generated tokens, as in[Section˜2\.3](https://arxiv.org/html/2607.26627#S2.SS3)\.

### H\.1Tree verification as implemented

EAGLE\-3 drafts a token tree and verification accepts a single root\-to\-leaf path\(Liet al\.,[2025](https://arxiv.org/html/2607.26627#bib.bib80)\)\. Greedy drafting fixes a deterministic candidate setx\(1\),…,x\(D\)x^\{\(1\)\},\\dots,x^\{\(D\)\}at each position \(DDfixed by the drafting configuration\), examined in this order, with draft probabilities playing no role in verification\. With a truncation warper active the verifier scores candidates by the truncated targetpΘp\_\{\\Theta\}\([Equation˜13](https://arxiv.org/html/2607.26627#S3.E13)\), accepting each iffr≤h​\(x\(i\)\)r\\leq h\(x^\{\(i\)\}\),r∼Unif​\[0,1\]r\\sim\\mathrm\{Unif\}\[0,1\]:

- •Truncation sampling \+ SD \(lossless\)\. h​\(x\)=min⁡\{pΘ​\(x\)/q,1\}\|q=1=pΘ​\(x\),h\(x\)=\\min\\\!\\Bigl\\\{p\_\{\\Theta\}\(x\)/q,\\,1\\Bigr\\\}\\big\|\_\{q=1\}=p\_\{\\Theta\}\(x\),\(26\)with zero\-and\-renormalize on rejection, reproducingpΘp\_\{\\Theta\}at each position \(Lemma[3](https://arxiv.org/html/2607.26627#Thmlemma3)\(i\)\)\.
- •SpecCascade\. h​\(x\)=𝟏​\[pΘ​\(x\)≥pbase​maxv⁡pΘ​\(v\)\]=𝟏​\[x∈𝒜min\-​p\]\.\\begin\{gathered\}h\(x\)=\\mathbf\{1\}\\\!\\left\[p\_\{\\Theta\}\(x\)\\geq p\_\{\\text\{base\}\}\\,\\max\_\{v\}p\_\{\\Theta\}\(v\)\\right\]\\\\ =\\mathbf\{1\}\[x\\in\\mathcal\{A\}\_\{\\text\{min\-\}p\}\]\.\\end\{gathered\}\(27\)
- •Typical acceptance\. h​\(x\)=min⁡\{pΘ​\(x\)/τ,1\}=𝟏​\[x∈𝒜η\],τ=min⁡\(ε,ε​e−H​\(pΘ\)\)\.\\begin\{gathered\}h\(x\)=\\min\\\!\\Bigl\\\{p\_\{\\Theta\}\(x\)/\\tau,\\,1\\Bigr\\\}=\\mathbf\{1\}\[x\\in\\mathcal\{A\}\_\{\\eta\}\],\\\\ \\tau=\\min\\\!\\bigl\(\\varepsilon,\\,\\sqrt\{\\varepsilon\}\\,e^\{\-H\(p\_\{\\Theta\}\)\}\\bigr\)\.\\end\{gathered\}\(28\)

Equations[27](https://arxiv.org/html/2607.26627#A8.E27)and[28](https://arxiv.org/html/2607.26627#A8.E28)are the indicator of Definition[1](https://arxiv.org/html/2607.26627#Thmdefinition1)withΘ=\\Theta=min\-pp\([Equation˜5](https://arxiv.org/html/2607.26627#S2.E5)\) andΘ=η\\Theta=\\eta\([Equation˜6](https://arxiv.org/html/2607.26627#S2.E6),δ=ε\\delta=\\sqrt\{\\varepsilon\}\): off\-set candidates are rejected and in\-set candidates accepted with probability one\. If a candidate is accepted the walk descends into its subtree; otherwise the path terminates with a bonus token frompΘp\_\{\\Theta\}\. The accepted lengthLLcounts accepted drafted tokens only \(the bonus excluded\) and equals the block efficiency in[Tables˜3](https://arxiv.org/html/2607.26627#S5.T3)and[5](https://arxiv.org/html/2607.26627#A1.T5)\.

### H\.2Per\-position analysis

Writea​\(C\)=Pr⁡\[some​x∈C​accepted\]a\(C\)=\\Pr\[\\text\{some \}x\\in C\\text\{ accepted\}\]for the per\-position acceptance probability of a rule, given the prefix\.

###### Lemma 3\.

Per\-position generation and acceptanceLetC=\{x\(1\),…,x\(D\)\}C=\\\{x^\{\(1\)\},\\dots,x^\{\(D\)\}\\\}be the candidates at the current position, examined in order, withpΘ​\(C\)=∑x∈CpΘ​\(x\)p\_\{\\Theta\}\(C\)=\\sum\_\{x\\in C\}p\_\{\\Theta\}\(x\)\. Under[Section˜H\.1](https://arxiv.org/html/2607.26627#A8.SS1):

1. \(i\)Truncation sampling \+ SD emitspΘp\_\{\\Theta\}exactly, with Pr⁡\[accept​x\(i\)\]=pΘ​\(x\(i\)\),alossless​\(C\)=pΘ​\(C\)\.\\begin\{gathered\}\\Pr\[\\text\{accept \}x^\{\(i\)\}\]=p\_\{\\Theta\}\(x^\{\(i\)\}\),\\\\ a^\{\\mathrm\{lossless\}\}\(C\)=p\_\{\\Theta\}\(C\)\.\\end\{gathered\}\(29\)
2. \(ii\)Any rule withp~=pΘ\\tilde\{p\}=p\_\{\\Theta\}accepts only tokens inC∩𝒜ΘC\\cap\\mathcal\{A\}\_\{\\Theta\}, with alossy​\(C\)=𝟏​\[C∩𝒜Θ≠∅\]​\(SpecCasc\.\);alossy​\(C\)∈\(0,1\],with\>0⇔C∩𝒜Θ≠∅\(typical acc\.\)\.\\begin\{gathered\}a^\{\\mathrm\{lossy\}\}\(C\)=\\mathbf\{1\}\[C\\cap\\mathcal\{A\}\_\{\\Theta\}\\neq\\varnothing\]\\ \\text\{\(SpecCasc\.\)\};\\\\\[2\.0pt\] a^\{\\mathrm\{lossy\}\}\(C\)\\in\(0,1\],\\ \\text\{with \}\{\>\}0\\\\ \\iff C\\cap\\mathcal\{A\}\_\{\\Theta\}\\neq\\varnothing\\ \\ \\text\{\(typical acc\.\)\}\.\\end\{gathered\}\(30\)

###### Proof\.

*\(i\)*Initializep~1=pΘ\\tilde\{p\}\_\{1\}=p\_\{\\Theta\}; the zero\-and\-renormalize update givesp~i=pΘ\(⋅∣𝒱∖\{x\(1\),…,x\(i−1\)\}\)\\tilde\{p\}\_\{i\}=p\_\{\\Theta\}\(\\,\\cdot\\mid\\mathcal\{V\}\\setminus\\\{x^\{\(1\)\},\\dots,x^\{\(i\-1\)\}\\\}\)\. Then

Pr⁡\[reach​x\(i\)\]\\displaystyle\\Pr\[\\text\{reach \}x^\{\(i\)\}\]=∏k=1i−1\(1−p~k​\(x\(k\)\)\)\\displaystyle=\\prod\_\{k=1\}^\{i\-1\}\\bigl\(1\-\\tilde\{p\}\_\{k\}\(x^\{\(k\)\}\)\\bigr\)=∏k=1i−11−∑l≤kpΘ​\(x\(l\)\)1−∑l≤k−1pΘ​\(x\(l\)\)\\displaystyle=\\prod\_\{k=1\}^\{i\-1\}\\frac\{1\-\\sum\_\{l\\leq k\}p\_\{\\Theta\}\(x^\{\(l\)\}\)\}\{1\-\\sum\_\{l\\leq k\-1\}p\_\{\\Theta\}\(x^\{\(l\)\}\)\}=1−∑l<ipΘ​\(x\(l\)\),\\displaystyle=1\-\\sum\_\{l<i\}p\_\{\\Theta\}\(x^\{\(l\)\}\),Pr⁡\[accept​x\(i\)\]\\displaystyle\\Pr\[\\text\{accept \}x^\{\(i\)\}\]=Pr⁡\[reach​x\(i\)\]⋅p~i​\(x\(i\)\)\\displaystyle=\\Pr\[\\text\{reach \}x^\{\(i\)\}\]\\cdot\\tilde\{p\}\_\{i\}\(x^\{\(i\)\}\)=\(1−∑l<ipΘ​\(x\(l\)\)\)\\displaystyle=\\Bigl\(1\-\\\!\\textstyle\\sum\_\{l<i\}p\_\{\\Theta\}\(x^\{\(l\)\}\)\\Bigr\)×pΘ​\(x\(i\)\)1−∑l<ipΘ​\(x\(l\)\)\\displaystyle\\quad\\times\\dfrac\{p\_\{\\Theta\}\(x^\{\(i\)\}\)\}\{1\-\\sum\_\{l<i\}p\_\{\\Theta\}\(x^\{\(l\)\}\)\}=pΘ​\(x\(i\)\),\\displaystyle=p\_\{\\Theta\}\(x^\{\(i\)\}\),⇒alossless​\(C\)=∑ipΘ​\(x\(i\)\)=pΘ​\(C\)\\Rightarrow a^\{\\mathrm\{lossless\}\}\(C\)=\\sum\_\{i\}p\_\{\\Theta\}\(x^\{\(i\)\}\)=p\_\{\\Theta\}\(C\)\. On all\-reject,

Pr⁡\[emit​x\]=\(1−pΘ​\(C\)\)​pΘ​\(x\)1−pΘ​\(C\)=pΘ​\(x\),\\Pr\[\\text\{emit \}x\]=\\bigl\(1\-p\_\{\\Theta\}\(C\)\\bigr\)\\frac\{p\_\{\\Theta\}\(x\)\}\{1\-p\_\{\\Theta\}\(C\)\}=p\_\{\\Theta\}\(x\),forx∉Cx\\notin C, so the position emitspΘp\_\{\\Theta\}on all of𝒱\\mathcal\{V\}\.

*\(ii\)*Withp~=pΘ\\tilde\{p\}=p\_\{\\Theta\},[Equation˜27](https://arxiv.org/html/2607.26627#A8.E27)and[Equation˜28](https://arxiv.org/html/2607.26627#A8.E28)give

x∉𝒜Θ⇒p~​\(x\)=0⇒h​\(x\)=0,x\\notin\\mathcal\{A\}\_\{\\Theta\}\\ \\Rightarrow\\ \\tilde\{p\}\(x\)=0\\ \\Rightarrow\\ h\(x\)=0,so accepted tokens lie inC∩𝒜ΘC\\cap\\mathcal\{A\}\_\{\\Theta\}\. SpecCascade hash≡1h\\equiv 1on𝒜Θ\\mathcal\{A\}\_\{\\Theta\}\(⇒\\Rightarrowfirst in\-set candidate accepted surely\); typical acceptance hash​\(x\)=min⁡\{pΘ​\(x\)/τ,1\}\>0h\(x\)=\\min\\\{p\_\{\\Theta\}\(x\)/\\tau,1\\\}\>0on𝒜Θ\\mathcal\{A\}\_\{\\Theta\}\. This is[Equation˜30](https://arxiv.org/html/2607.26627#A8.E30)\. ∎

###### Corollary 1\.

Tree verification accepts at least as oftenAt every position,

alossless​\(C\)=pΘ​\(C\)=∑v∈C∩𝒜Θp​\(v\)z≤1​\[C∩𝒜Θ≠∅\]=aSpecCascadelossy​\(C\),\\begin\{gathered\}a^\{\\mathrm\{lossless\}\}\(C\)=p\_\{\\Theta\}\(C\)=\\frac\{\\sum\_\{v\\in C\\cap\\mathcal\{A\}\_\{\\Theta\}\}p\(v\)\}\{z\}\\\\ \\leq\\ \\mathbf\{1\}\[C\\cap\\mathcal\{A\}\_\{\\Theta\}\\neq\\varnothing\]=a^\{\\mathrm\{lossy\}\}\_\{\\mathrm\{SpecCascade\}\}\(C\),\\end\{gathered\}\(31\)strict whenever0<pΘ​\(C\)<10<p\_\{\\Theta\}\(C\)<1; for typical acceptancealossless≤alossya^\{\\mathrm\{lossless\}\}\\leq a^\{\\mathrm\{lossy\}\}holds accumulated along the path \(the positiveΔ\\DeltaBE columns of[Table˜7](https://arxiv.org/html/2607.26627#A8.T7)\)\.

###### Proof\.

By Lemma[3](https://arxiv.org/html/2607.26627#Thmlemma3),alossless​\(C\)=pΘ​\(C\)∈\[0,1\]a^\{\\mathrm\{lossless\}\}\(C\)=p\_\{\\Theta\}\(C\)\\in\[0,1\]andpΘ​\(C\)\>0⇔C∩𝒜Θ≠∅p\_\{\\Theta\}\(C\)\>0\\iff C\\cap\\mathcal\{A\}\_\{\\Theta\}\\neq\\varnothing, sopΘ​\(C\)≤𝟏​\[C∩𝒜Θ≠∅\]p\_\{\\Theta\}\(C\)\\leq\\mathbf\{1\}\[C\\cap\\mathcal\{A\}\_\{\\Theta\}\\neq\\varnothing\], with equality iffpΘ​\(C\)∈\{0,1\}p\_\{\\Theta\}\(C\)\\in\\\{0,1\\\}\. ∎

### H\.3Proof of Lemma[2](https://arxiv.org/html/2607.26627#Thmlemma2)and Proposition[1](https://arxiv.org/html/2607.26627#Thmproposition1)

###### Proof\.

WritepΘ​\(x\)=p​\(x\)/ZΘ​\(p\)p\_\{\\Theta\}\(x\)=p\(x\)/Z\_\{\\Theta\}\(p\)forx∈𝒜Θx\\in\\mathcal\{A\}\_\{\\Theta\}\(zero otherwise\), the truncated target; by Lemma[3](https://arxiv.org/html/2607.26627#Thmlemma3)\(i\) truncation sampling emitspΘp\_\{\\Theta\}at every position\.

*Statement 1 \(KLSD\\mathrm\{KL\}\_\{\\mathrm\{SD\}\}\)\.*Truncation\-based verification under standard SD generatesq/ZΘ​\(q\)q/Z\_\{\\Theta\}\(q\)on𝒜Θ\\mathcal\{A\}\_\{\\Theta\}\([Equation˜13](https://arxiv.org/html/2607.26627#S3.E13)\)\. The per\-token term is

DKL​\(q/ZΘ​\(q\)∥pΘ\)\\displaystyle D\_\{\\mathrm\{KL\}\}\\\!\\bigl\(q/Z\_\{\\Theta\}\(q\)\\,\\big\\\|\\,p\_\{\\Theta\}\\bigr\)=∑x∈𝒜Θq​\(x\)ZΘ​\(q\)​log⁡q​\(x\)/ZΘ​\(q\)p​\(x\)/ZΘ​\(p\)\\displaystyle=\\sum\_\{x\\in\\mathcal\{A\}\_\{\\Theta\}\}\\frac\{q\(x\)\}\{Z\_\{\\Theta\}\(q\)\}\\log\\frac\{q\(x\)/Z\_\{\\Theta\}\(q\)\}\{p\(x\)/Z\_\{\\Theta\}\(p\)\}=∑x∈𝒜Θq​\(x\)ZΘ​\(q\)​log⁡q​\(x\)​ZΘ​\(p\)ZΘ​\(q\)​p​\(x\),\\displaystyle=\\sum\_\{x\\in\\mathcal\{A\}\_\{\\Theta\}\}\\frac\{q\(x\)\}\{Z\_\{\\Theta\}\(q\)\}\\log\\frac\{q\(x\)\\,Z\_\{\\Theta\}\(p\)\}\{Z\_\{\\Theta\}\(q\)\\,p\(x\)\},theKLSD\\mathrm\{KL\}\_\{\\mathrm\{SD\}\}of Lemma[2](https://arxiv.org/html/2607.26627#Thmlemma2), finite sincep\>0p\>0on𝒜Θ\\mathcal\{A\}\_\{\\Theta\}, and

q=p⇒ZΘ​\(q\)=ZΘ​\(p\)⇒q/ZΘ​\(q\)=pΘ⇒DKL=0,\\begin\{gathered\}q=p\\ \\Rightarrow\\ Z\_\{\\Theta\}\(q\)=Z\_\{\\Theta\}\(p\)\\\\ \\Rightarrow\\ q/Z\_\{\\Theta\}\(q\)=p\_\{\\Theta\}\\ \\Rightarrow\\ D\_\{\\mathrm\{KL\}\}=0,\\end\{gathered\}soKLSD→0\\mathrm\{KL\}\_\{\\mathrm\{SD\}\}\\to 0asq→pq\\to p\(Proposition[1](https://arxiv.org/html/2607.26627#Thmproposition1)\)\.

*Statement 2 \(KLEAGLE\\mathrm\{KL\}\_\{\\mathrm\{EAGLE\}\}\)\.*By Lemma[3](https://arxiv.org/html/2607.26627#Thmlemma3)\(ii\) the accepted token is the first candidate in𝒜Θ\\mathcal\{A\}\_\{\\Theta\}, deterministic given the prefix, withpΘ​\(x\)=p​\(x\)/ZΘ​\(p\)\>0p\_\{\\Theta\}\(x\)=p\(x\)/Z\_\{\\Theta\}\(p\)\>0and the accept test governed bypΘp\_\{\\Theta\}, notqq\. The position therefore emits the point mass atxx, whose divergence frompΘp\_\{\\Theta\}is

log⁡1pΘ​\(x\)=log⁡ZΘ​\(p\)p​\(x\)=KLEAGLE,\\log\\frac\{1\}\{p\_\{\\Theta\}\(x\)\}=\\log\\frac\{Z\_\{\\Theta\}\(p\)\}\{p\(x\)\}\\;=\\;\\mathrm\{KL\}\_\{\\mathrm\{EAGLE\}\},for SpecCascade and typical acceptance alike\. Finally,

\|𝒜Θ\|\>1⇒pΘ​\(x\)≤maxv∈𝒜Θ⁡pΘ​\(v\)<1⇒log⁡ZΘ​\(p\)p​\(x\)≥log⁡1maxv∈𝒜Θ⁡pΘ​\(v\)\>0,\\begin\{gathered\}\|\\mathcal\{A\}\_\{\\Theta\}\|\>1\\ \\Rightarrow\\ p\_\{\\Theta\}\(x\)\\leq\\max\_\{v\\in\\mathcal\{A\}\_\{\\Theta\}\}p\_\{\\Theta\}\(v\)<1\\\\ \\Rightarrow\\ \\log\\frac\{Z\_\{\\Theta\}\(p\)\}\{p\(x\)\}\\geq\\log\\frac\{1\}\{\\max\_\{v\\in\\mathcal\{A\}\_\{\\Theta\}\}p\_\{\\Theta\}\(v\)\}\>0,\\end\{gathered\}for every draft and uniformly inqq, soKLEAGLE\>0\\mathrm\{KL\}\_\{\\mathrm\{EAGLE\}\}\>0does not vanish asq→pq\\to p\(Proposition[1](https://arxiv.org/html/2607.26627#Thmproposition1)\)\. ∎

### H\.4Block Efficiency Analysis

###### Proposition 2\.

Block efficiency of tree verificationWithLLas above,

BE=𝔼​\[L\]=∑d≥1∏j=1dPr⁡\[L≥j∣L≥j−1\],\\mathrm\{BE\}=\\mathbb\{E\}\[L\]=\\sum\_\{d\\geq 1\}\\ \\prod\_\{j=1\}^\{d\}\\Pr\[L\\geq j\\mid L\\geq j\-1\],\(32\)

In[Equation˜32](https://arxiv.org/html/2607.26627#A8.E32), every factor is at least as large under truncation\-based verification as under truncation sampling \+ SD; hence its block efficiency is weakly higher\.*\(See[Appendix˜H](https://arxiv.org/html/2607.26627#A8)for the proof\.\)*However, the slight gain in BE is at the cost of even more severe performance degradation under multi\-draft setting\. \(see[Table˜7](https://arxiv.org/html/2607.26627#A8.T7)\)

###### Proof of Proposition[2](https://arxiv.org/html/2607.26627#Thmproposition2)\.

BE=𝔼​\[L\]=∑d≥1Pr⁡\[L≥d\]=∑d≥1∏j=1dPr⁡\[L≥j∣L≥j−1\],\\begin\{gathered\}\\mathrm\{BE\}=\\mathbb\{E\}\[L\]=\\sum\_\{d\\geq 1\}\\Pr\[L\\geq d\]\\\\ =\\sum\_\{d\\geq 1\}\\prod\_\{j=1\}^\{d\}\\Pr\[L\\geq j\\mid L\\geq j\-1\],\\end\{gathered\}by[Equation˜32](https://arxiv.org/html/2607.26627#A8.E32), each factor beinga​\(C\)a\(C\)of Lemma[3](https://arxiv.org/html/2607.26627#Thmlemma3)\. By Corollary[1](https://arxiv.org/html/2607.26627#Thmcorollary1),alossy≥alosslessa^\{\\mathrm\{lossy\}\}\\geq a^\{\\mathrm\{lossless\}\}\(per\-position and strict for SpecCascade when0<pΘ​\(C\)<10<p\_\{\\Theta\}\(C\)<1; accumulated along the path for typical acceptance\), so every factor is weakly larger andBE\\mathrm\{BE\}is weakly higher\. ∎