polybasic Speculative Decoding Through a Theoretical Perspective
Authors: Ruilin Wang
Huixia Li
Yuexiao Ma
Xiawu Zheng
Fei Chao
Xuefeng Xiao
Rongrong Ji
Venue: ICML 2025 poster (2025)
Keywords: speculative decoding
Primary area: deep_learning->large_language_models

Abstract: Inference latency stands as a critical bottleneck in the large-scale deployment of Large Language Models (LLMs). Speculative decoding methods have recently shown promise in accelerating inference without compromising the output distribution. However, existing work typically relies on a dualistic draft-verify framework and lacks rigorous theoretical grounding. In this paper, we introduce a novel \emph{polybasic} speculative decoding framework, underpinned by a comprehensive theoretical analysis. Specifically, we prove a fundamental theorem that characterizes the optimal inference time for multi-model speculative decoding systems, shedding light on how to extend beyond the dualistic approach to a more general polybasic paradigm. Through our theoretical investigation of multi-model token generation, we expose and optimize the interplay between model capabilities, acceptance lengths, and overall computational cost. Our framework supports both standalone implementation and integration with existing speculative techniques, leading to accelerated performance in practice. Experimental results across multiple model families demonstrate that our approach yields speedup ratios ranging from $3.31\times$ to $4.01\times$ for LLaMA2-Chat 7B, up to $3.87 \times$ for LLaMA3-8B, up to $4.43 \times$ for Vicuna-7B and up to $3.85 \times$ for Qwen2-7B---all while preserving the original output distribution. We release our theoretical proofs and implementation code to facilitate further investigation into polybasic speculative decoding.

Polybasic Speculative Decoding Through a Theoretical Perspective
Ruilin Wang' Huixia Li
Yuexiao Ma
Xiawu Zheng 134
Fei Chao'
Xuefeng Xiao?
Rongrong Ji
Abstract
1. Introduction
Large Language Models (LLMs) have substantially ad- vanced Natural Language Processing (NLP), achieving lead- ing performance in wide range of tasks. Yet their excep- tional capabilities are tempered by significant computational demands, particularly in low-latency scenarios. Among mul- tiple acceleration techniques, speculative decoding (Stern et al., 2018; Leviathan et al., 2023; Xia et al., 2023; Chen et al., 2023a; Anonymous, 2025b;a; Teng et al., 2024; Zhang et al., 2024b) has emerged as a strategy to speed up inference while preserving output fidelity.
Inference latency stands as a critical bottleneck in the large-scale deployment of Large Language Models (LLMs). Speculative decoding methods have recently shown promise accelerating in- ference without compromising the output distri- bution. However, existing work typically relies a dualistic draft-verify framework and lacks rigorous theoretical grounding. In this paper, we introduce a novel polybasic speculative decod- ing framework, underpinned by a comprehensive theoretical analysis. Specifically, we prove a fun- damental theorem that characterizes the optimal inference time for multi-model speculative de- coding systems, shedding light on how to extend beyond the dualistic approach to a more general polybasic paradigm. Through our theoretical in- vestigation of multi-model token generation, we expose and optimize the interplay between model capabilities, acceptance lengths, and overall com- putational cost. Our framework supports both standalone implementation and integration with existing speculative techniques, leading to accel- erated performance in practice. Experimental re- sults across multiple model families demonstrate that our approach yields speedup ratios ranging from 3.31 to 4. x for LLaMA2-Char 7B, up to 3.87x for LLaMA3-SB, up 4.43x for Vicuna- 7B and up 3.85x for Qwen2-7B-B all while preserving the original output distribution. We release our theoretical proofs and implementation code to facilitate further investigation into poly- basic speculative decoding.
The current speculative decoding ecosystem largely hinges draft-then-verify paradigms, which spawn various sub- directions such as the design of lightweight draft mod- els (Leviathan et al., 2023; Xia et al., 2023; Chen et al., 2023a; Kim et al., 2024a; Svirschevski et al., 2024; Yin al., 2024; Sadhukhan et al., 2024), hierarchical token struc- tures (Stern et al., 2018; Miao al., 2024; Du et al., 2024), and unified architectures (Yi et al., 2024; Cai et al., 2024). Verification strategies typically follow three approaches: greedy sampling, speculative sampling (Leviathan et al., 2023), and typical acceptance (Cai et al., 2024). Despite these efforts, current strategies remain limited by a dualis- tic relationship between draft and target models (Qin et al., 2024; Liu al., 2024; Gui et al., 2024; Khisti et al., 2024), affecting key parameters such as acceptance length due to inherent capacity disparities between the two models. While some recent works (Chen et al., 2023b; Kim et al.; Spector & Ré, 2023) investigate multi-level drafts, they still employ a singular top-level target model. Moreover, the field has hith- erto lacked an overarching theoretical framework to guide system design and provide robust performance guarantees.
In this paper, introduce a principled polybasic specula- tive decoding framework that uses multiple interconnected models, grounded in thorough theoretical analysis. Our investigation yields two central insights. First, we derive a fundamental relationship between the number of forward passes and average acceptance lengths that dictates opti- mal system-level inference speed. This relationship allows to precisely quantify potential inference speedups when adding additional models. Second, we establish the capacity of speculative sampling to enhance stability in acceptance lengths. By optimizing sampling parameters, we can reduce
'Key Laboratory of Multimedia Trusted Perception and Efli- cient Computing, Ministry of Education of China, Xiamen Univer- sity, 361005, P.R. China BByteDance IInstitute of Artificial Intel- ligence, Xiamen University *Peng Cheng Laboratory, Shenzhen, China. Correspondence to: Rongrong Ji <rrj @ xmu.edu.en>.
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Polybasic Speculative Decoding Through a Theoretical Perspective
variance in token acceptance to achieve more predictable
to propose candidate tokens, later verified by a larger model. Methods range from training specialized drafters (Leviathan et al., 2023; Xia et al., 2023; Chen et al., 2023a; Kim et al., 2024a; Metel et al., 2024) to zero-shot usage of pre-existing models (Spector & Ré, 2023). Self-drafting employs the same model at intermediate stages, as in blockwise decod- ing (Stern et al., 2018; Xiao et al., 2024), early exiting (Yang et al., 2023), or mask-predict (Zhao al., 2024), aiming to amortize computation within the same architecture.
performance.
Building on these insights, we propose unified architec- ture for polybasic speculative decoding wherein multiple draft models coordinate with each other and with a single target model. Our implementation guidelines detail how to select models, set speculation lengths, and implement multi-stage verification procedures that maximize through- put. Experimental evaluations demonstrate that this ap- proach outperforms typical dualistic methods on diverse tasks including MT-bench (Zheng et al., 2023), translation, summarization, QA, mathematical reasoning, and Retrieval- Augmented Generation (RAG). Our empirical results indi- cate that the proposed polybasic framework maintains out- put fidelity while delivering speedup ratios from 3x over 4x across a range of widely used LLMs (e.g., Vicuna-7B, LLaMA2-Chat 7B, ILL.M3-8B).
Verification Mechanisms. Verification primarily ensures that proposed tokens maintain consistency with the target distribution. Greedy verification (Kim al., 2024a; Xia et al., 2023; Agrawal et al., 2024) is conceptually straight- forward but may hinder speedups for certain tasks. Specu- lative sampling (Leviathan et al., 2023) introduces a proba- bilistic acceptance rule that adaptively filters tokens while retaining a high acceptance length. liken-treeebaace fication (Miao et al., 2024; Spector & Ré, 2023; Lu et al., 2024; Gao et al., 2024) provides hierarchical checks, which can be beneficial for highly parallel architectures.
veri-
The main contributions of this work are summarized as follows:
We develop a formal theoretical framework for polyba- sic speculative decoding, identifying the system-level dependencies between model forward-pass cost, accep- tance lengths, and stable acceleration performance.
Recent Advances and Limitations. Recent work on cas- cade or multi-level drafting (Chen et al., 2023b; Sun et al., 2024) has partially moved beyond the dualistic target scheme. TRIFORCE (Sun et al., 2024) tackles long se- quence generation by introducing two-level hierarchy with retrieval-based drafting and partial KV cache as an interme- diate layer, achieving up to 2.31 x speedup for Llama2-7B- 128K. CS Drafting (Chen et al., 2023b), on the other hand, employs vertical and horizontal caseades to eliminate neural autoregressive generation and optimize time allocation in drafting, resulting in up to 81% additional speedup over standard speculative decoding. While TRIFORCE focuses on memory efficiency in long-context scenarios through KV cache optimization, CS Drafting targets general infer- ence optimization through cascade structures and statistical drafting.
We prove a fundamental theorem that provides rig- orous expression for optimal inference time in multi- model speculative decoding, highlighting conditions under which additional models improve speedups.
We show that speculative sampling significantly re- duces variance in token acceptance lengths for multi- model settings, increasing stability and improving in- ference throughput.
Our empirical investigation demonstrates the effec- tiveness of the proposed polybasic approach, achiey- ing notable speedups (up to 4.43x) on widely used LLMs across various tasks, while preserving the target model's output distribution.
However, these approaches usually rely on empirical heuris- tics without unified theoretical framework guide model selection, acceptance-length control, and stability analysis. Our work explicitly addresses these gaps by introducing a comprehensive theoretical treatment of polybasic specu- lative decoding and validating the resulting system design empirically.
2. Related Work
The concept of speculative decoding originated from block- wise parallel decoding (Stern et al., 2018), showcasing the viability of partially parallel language generation. More recently, research on speculative decoding (Stewart et al., 2024; 7afrir et al., 2024) has coalesced around two dimen- sions: drafting strategies for token prediction and verifica- tion mechanisms for ensuring correctness.
3. Polybasia Speculative Decoding Framework
Although speculative decoding has been demonstrated as an effective technique for single-model verification, its acceler- ation potential remains capped by the inherent draft-target capacity gap in dualistic paradigms. We propose polyba- sic speculative decoding framework, which systematically
Drafting Strategies. Drafting approaches typically follow either independent or self-draftin protocols. Independent drafting involves utilizing smaller or more efficient models
2

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Polybasic Speculative Decoding Through Theoretical Perspective
(a) Dualistic speculative decoding
(b) Polybasic speculative decoding
Target model
1st draft model
draft model
Target model
Draft model
Speedup ratio: 2-3
Generalization:
Speedup ratio: 4
Generalization
U
Average acceptance length: 3-4
Average acceptance length: 8-10
U
Figure 1. Comparison speculative decoding frameworks. (a) Traditional dualistic approach with a single draft model. (b) Our polybasic framework with multiple draft models achieves superior performance (4x speedup and 8- 10 tokens acceptance length) while maintaining good generalization ability. The framework demonstrates significant improvements over the dualistic baseline.
employs multiple models to increase parallelism and ac- ceptance length while preserving fidelity to the final target distribution. Below, we detail the core problem setting, theoretical foundations, and practical instantiations.
impact computational cost and acceptance lengths. Our anal- ysis focuses on two main aspects: (i) optimal inference time and (ii) stability of acceptance lengths.
Optimal Inference Time. In a conventional dualistic
3.1. Problem Formulation
system with one draft model (M2) and single target (M1), the total inference time is approximately equal to N Ti YN T2, where N is the sequence length and ssytee-ddependee
Let us consider a chain of models M [M Mr where M1 is the final target model we wish replicate in distribution, and M2...., M,, sively lower capacity (higher index indicates smaller faster model). Let V be the vocabulary, and pz(z [gg))= M, the distribution over V given context Zgg.
scaling factor reflecting the final draft model's capability. In polybasic setting with els, additional drafting layers bring more tokens per verification cycle if acceptance lengths between interme- diate pairs are high. However, each additional model also introduces its own forward-pass costs. Formally, we have:
drafters progres-
mod-
At each decoding step 1, M, drafts a block of K tokens, verified ascending order by Mn-1, Mn-2..... M To- kens are accepted if they do not exceed certain mismatch criterion, reflecting speculative sampling, greedy matching. another verification rule. Define:
Lemma 3.1 (Optimal Inference Time), For n-model polybasic system generating N tokens, the total inference time T can be expressed as:
Lj E of consecutive tokens accepted by M.),, (1)
T
T
N Li
T
N
T.
(3)
Ln-1
i.e. the expected block length accepted when verifying with model M, Denoting by Fi the number of forward passes that M, must perform, goal is to minimize the total inference time
where L; the expected acceptance length for verification by M, and B is a system-dependden scaling factor reflecting the final draft model's capability.
T= Σ F.
(2)
Sketch of Proof. We segment the total generation length N into accepted blocks validated by pairs (M, M4+1). Each model M, must run as many forward passes needed to accept N tokens in total. A more detailed version of the proof incorporates the block acceptance process for each ad- jacency (M, M,+1). culminating in the time decomposition of Equation (3).
where T, is the cost of single forward pass for model Mr-
3.2. Theoretical Foundations
We establish fundamental properties of multi-model (polyba- sic) speculative decoding that govern how additional models
3

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This result generalizes to inserting new model at any po- sition in a polybasic system. When inserting model Maow between M, and M,+1, we treat all models before the in- sertion point (Mi through M,) as single composite model, and all models after the insertion point (M,+1 through M2) as another composite model. This reduces the general case to the three-model case proven above, where the composite Polybasic Speculative Decoding Through Theoretical Perspective
Model Selection Criterion. The next question is whether introducing new model Nnw between M; and M,+1 im- proves T' The improvement depends whether the re- duced cost from higher acceptance length outweighs the additional forward-pass overhead. Formally:
Condition 2: Sum of the last two terms is less than
N 3 N..
1
T2 +
T2 < <
L2
1/2
L1
T2
L12
< B
-
T2
11
Theorem 3.2 (Model Insertion Efficiency). Adding Meew between M, and M++1 decreases total inference time if and only if it achieves sufficiently large increase in accep- tance lengths, balanced against its forward-pas. cost neww- Concretely, if Lnew is the acceptance length when verifying tokens from Maew against M,. and L,41 is the acceptance length from M,+1² 's perspective, then improvement occurs if:
Tuew T; < Lnew 1
Li Limnow) ) or Luew-(iit2) -1)
model before insertion corresponds to M, , the new model corresponds to M2. and the composite model after insertion corresponds M2-
I
Tnow
< 3
T:+1
Tri
Therefore, the same conditions for efficiency improvement apply at any insertion point in the model sequence, subject to the specified constraints on acceptance lengths.
Proof. First, we prove the case for three models:
For i =2: 2:
Stability Analysis. Beyond achieving higher acceptance (4) lengths, stability in acceptance is crucial for consistent speedups. We analyze speculative sampling with probability p; = 1 a to accept a token from M,+1 if is likely un- der M'S distribution. Let of be the variance of acceptance lengths. As shown below, for multi-model chaining, accep- tance length variance grows with smaller pi, implying that high acceptance probability supports stable performance:
N
N
T
Ti
B
T2
L1
For iori=33 3: 3:
N
N L/
N L/2
T
Ti
T2 + 3-
T3
(5)
Theorem 3.3 (Sampling Stability). In the model chain using speculative sampling with acceptance probability p: a, the variance in acceptance length satisfies:
where T. is the inference time the i-th model, a is considered to be equal in both equations, and Tg Tg
1 L/ Lt-
0² a [1 - (n² - 1)a" (n² 1)a
We can calculate the difference between Equation 4 and Equation 5:
(1 a)²
1
N
Proof. Let p 1 - be the probability of accepting a token. For a truncated geometric distribution of maximum n trials, define:
Ti
+
T L2
L2 L
( 1
+ N.. N.
T2 < 0
L/2
L
n-1 S
Σ k (1 (1- k=1
(1-
The expression is less than 0 if either of the following con- ditions is met: Condition 1: Sum of the first two terms is less than O
Using standard manipulation (method of differences), one can show:
S - 1 (--1p))~1 n(1 - -- + (1 (1-p)"
1
N T2 <0 0
N.
Ti
p2
L'
L1
L2
Ty
1
Hence, the expectation of the acceptance length, allowing up to n tokens, is
< I/z-
Ti
L1
L
E|N 1- 1- (1- p)"
OR
P

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Polybasic Speculative Decoding Through Theoretical Perspective
We similarly compute
Algorithm Polybasic Speculative Decoding (Three Mod- cls)
n-1 E[N2] Σ 12 pp. p. (1- 11-p) p)= + +2² .22.() -p
Input: Target model M,, intermediate model M2, draft
model M3
2: Input: Context ægg,, total length N
After careful algebra (omitted for brevity), one obtains:
3: Input: Draft length K, threshold μ
E[N2]
4: Initialize: L 4- |argel, accepted t- cnt <-
5: while t < N do
(1 + 2n - 1) + 2(1 - p)= (n 1)
Draft and verify with M3 and M2 Fi.K M2(agg) // Draft
p²
Thus,
8: PI:K + M2(IS.,T1.K) // Verify
Var(N) E[N (E[N))? leading to the formula stated in Theorem 3.3.
for i 1 to K do
10:
if VERIFY p:) then accepted.appeedd(f; ent + cnt +1
11:
12:
Collectively, these results establish a principled foundation for polybasic speculative decoding. Given model inference
13:
else
14:
break end if
times T; and acceptance probabilities, one can estimate
15:
the optimal system layout via Equation (3), gauge whether new model confers net benefit (Theorem 3.2), and use speculative sampling to ensure stable acceptance lengths (Theorem 3.3).
16:
end for
17:
// Check if threshold reached for M1 verification
18:
if ent 2 then
19:
U <- M1 ( accepted) // Verify
20:
if VERIFY BBRIIY v) then
3.3. Three-Model System Design
21:
T++1:t+ent + accepted
To illustrate practical deployment, we describe a three-
22:
+ t ent
model system (M1: target, M2: intermediate, M3:
23:
else
lightweight) that exemplifies the design choices guided by our theory.
24:
T++1 SampleOne(v1) // fallback acceptance
25:
t + +++1
26:
end
Architecture. Our reference system includes:
27: accepted < 0
28: ent
Mi (Target): A high-capacity model such as Vicuna- 7B or LLaMA2-Chat 7B.
29: end if
30:
end while
M2 (Intermediate): A quantized 4-bit version of M1 or a comparable mid-size model to bridge the capacity gap.
Algorithm. Algorithm I details generic procedure for polybasic speculative decoding with three models. The sys- tem accumulates tokens verified by M2 until a threshold, then triggers verification by Mj. Upon acceptance or par- tial acceptance, appends tokens to the growing output sequence and advances t. Simple or more sophisticated error handling (e.g.. partial rollback) can be adopted Mj rejects tokens.
M3 (Draft): A lightweight, fast model (e.g., EA- GLE2(Li et al., 2024)) for initial token proposals.
By Theorem 3.2, M2 should be inserted if it raises the acceptance length enough to offset its own forward-pass cost.
Such staged design is representative rather than exhaus- tive. Model scaling, verification strategies, and drafting lengths can be adapted to different resource constraints or performance targets. Our theoretical framework offers ex- plicit performance bounds, making the design space more transparent.
Staged Verification. Tokens first pass from M3 to M2, whose verification is relatively fast. Accepted tokens are then periodically verified by M1. This two-stage verifica- tion acts as filter, rapidly discarding problematic tokens at the cheaper M2 stage. The threshold for passing tokens to M, is set to accumulate a small block (e.g.. tokens) to amortize Mi's forward-pan pass overhead. This setup capital- izes on the fact that M3 can generate numerous tentative tokens quickly, while Mj only checks consolidated blocks of already moderately validated tokens.
3.4. Generalization to Self-Drafting Methods
Our polybasic speculative decoding framework provides a general theoretical foundation that naturally extends self-
5

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Polybasic Speculative Decoding Through a Theoretical Perspective
drafting approaches, demonstrating the broad applicability of our theoretical principles. This generalization reveals new opportunities for designing efficient speculative decoding systems using model-intenen components.
also adopted this setting.
Performance Metrics. Following previous speculative de- coding studies, we focus on two metrics. Walltime speedup ratio ratio of actual decoding time in our system vs. stan- dard autoregressive decoding. Average acceptance length pe: mean number of consecutively accepted tokens per for- ward pass by the final (largest) model.
In self-drafting methods like FFN heads approaches (Cai et al., 2024) (Ankner et al., 2024) (Kim et al., 2024b) and early exiting (Elhoushi et al., 2024) techniques, multiple prediction sources are derived from the same model architec- ture. These can be viewed as the fundamental draft models in polybasic system, with the original model serving as the target. For instance, in an FFN heads approach, each head functions as a base draft model capable of generating tokens in parallel, forming the lowest tier in our polybasic hierarchy. Similarly, early exit points at different layers can be treated as draft models with varying capabilities.
Quantization and Training Details. For the intermediate model, we adopt 4-bit quantization (Ma et al., 2024) with a group size of 128, balancing reduced inference cost against quality. Draft models are built following EAGLEE2, trained on ShareGPT data. Our experiments run on NVIDIA A800 80G GPUs.
The optimization principles established in Section 3.2 re- main applicable in this context, though they require careful algorithmic design to account for the shared computational paths in self-drafting approaches. The relationship between forward pass frequency and acceptance length provides guid- ance for optimal configuration of these systems, while our stability analysis (Theorem 3.3) informs the design of verifi- cation strategies.
4.2. Theoretical Validation
To empirically validate Theorem 3.2, we conducted two targeted experiments evaluating the impact of inserting ad- ditional draft models into polybasic system. We measured Tnew Loew. and the resultant speedup ratio. Results are summarized in Table 4.2.
Case 1: Non- Com-Compli Insertion We inserted a lightweight Vicuna- 1B model between Vicuna-7B (target) and EAGLE2 (baseline drafter). Here, Tnew Tú 0.80, while the acceptance-length improvement factor Lnew: (1 TT4 1/ 1/Tx-oow) = 0.117. Since 0.80 > 0.117, Theo- rem 3.2 predicts a performance degradation. Empirically, the speedup ratio dropped from 2.61 x to 1.08x, confirming the theoretical prediction. This highlights the necessity of balancing model capacity and computational overhead when expanding the polybasic hierarchy.
This generalization demonstrates that our polybasic frame- work provides a unified theoretical foundation for specula- tive decoding, encompassing both independent draft models and self-drafting approaches. This suggests promising direc- tions for developing new algorithmic techniques that fully exploit the parallel prediction capabilities inherent in these methods while maintaining the theoretical guarantees of our framework.
4. Experiments
Case 2: Compliant Insertion We inserted quantized Vicuna-7B (W4A16) model between the original Vicuna- 7B and EAGLE2. Here, Twww/T; = 0.318 and Inew (1 TL4 - 1/T4 = 0.330. Since 0.318 < 0.330, The- orem 3.2 predicts a speedup improvement. Experimentallyy the system achieved 3.48x speedup. up from 2.61x. This demonstrates the theorem's 's utility in guiding effective model selection.
We conduct comprehensive evaluations to validate our theo- retical claims and to benchmark the proposed polybasic sys- tem against traditional dualistic strategies. Our experiments span various LLMs, tasks, and hyperparameter settings.
4.1. Setup and Metrics
Models and Tasks. We conducted experiments Vicuna- 7B, LLaMA2-chat-7BB and 10-MMM--7B-Insttruee We eval- uated our muulti-model speculative system in SpecBench(Xia et al., 2024), across multiple tasks including multi-turn con- versation, translation, summarization, question answering, mathematical reasoning, and retrieval-augmented genera- tion, employing the MT-bench (Zheng et al., 2023), WMT14 DE-EN, CNN/Daily Mail (Nallapati et al., 2016), Natural Questions (Kwiatkowski et al., 2019), GSMSK (Cobbe et al., 2021), and DPR (Karpukhin et al., 2020). Speculative sam- pling (Leviathan et al., 2023) conducted experiments with a batch size of 1, similarly, the majority of our experiments
Case 3: Generalization To substantiate the universal the- oretical guidance of Theorem 3.2 across diverse cascaded speculative sampling methodologies, we reproduced Cas- cade Speculative Drafting (Chen et al., 2023b) and con- ducted rigorous evaluations spanning multiple model scales including FLAN-T5-XXLL Base, and Small variants. We in- serted LLLA-T55-base between FLAN-T5- XXL and FLAN- TS-small The configuration yields T /T; = 0.403 with acceptance metric Laow - (1/L, L/Lwn)ww = 0.461, satis- fying the acceleration criterion 0.403 < 0.461 as shown in Table 1. The system exhibits statistically significant speedup improvement from 3. 3.19x to 3.88x.
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Polybasic Speculative Decoding Through Theoretical Perspective
Table I. Theoretical Validation via Model Insertion
Case
T; (ms)
Li-new
Teew (ms)
Lnew
T:+1 (ms)
L
Speedup
Non-compliant
22
3.83
17.61
3.77
4
4.34
2.61 x - 1.08x
Compliant
22
6.26
7.00
4.67
4
4.34
2.61x - 33.88x
CS Drafting
47.52
3.50
19.16
3.02
12.42
2.28
3.19x-33.m
These experiments directly corroborate Theorem 3.2, show- ing that the theoretical conditions on Tnew and Inew are nec- essary for improving system performance. The results em- phasize the framework's ability to rigorously guide model selection and hierarchy design, moving beyond heuristic- driven approaches.
4.4. Scalability to Larger Models
To demonstrate the generalizability of our framework across model scales, we conducted additional experiments with Vicuna- 13B and
models. As shown in Table 3, our polybasic approach maintains significant advan- tages over EAGLE baseline even when scaled to larger mod- els. Specifically, we achieve 2.69x speedup with Vicuna- 13B (vs. EAGLE2's 2.30x) and 2.92 for LLaMA-70B (vs. 2.46x), while maintaining substantially higher average acceptance lengths. These results confirm that our method's benefits are not limited to smaller models but extend to larger-scale LLMs. The slightly reduced absolute speedup ratios compared to 7B models align with expectations, as larger models naturally incur higher verification costs that partially offset drafting efficiency gains.
4.3. Effectivvness
Figures and summarize speedup ratios on various tasks. Our polybasic approach demonstrates clear gains over dual- istic baseline systems (including EAGLE2 and Speculative Sampling). Notably:
Vicuna-7B achieves 3.16x average and up 4.43x in mathematical reasoning.
4.5. Ablation Study: Speculative vs. Greedy Sampling
LLaMA2-Chat 7B attains 3.66x overall, peaking at 4.10x in multi-turn conversation.
To validate the impact of speculative sampling on stability, we compare acceptance-length variance from speculative vs. greedy verification. We sample 50 queries in three- model setup and record acceptance- distributions (Fig- ure 4). As anticipated, speculative sampling yields smaller variance, indicating more stable acceptance lengths across diverse inputs. This result aligns with Theorem 3.3 and further justifies the use of speculative sampling in multi-tier verification.
LLaMA3-8B yields x.31x-3.87x speedups, illustrat- ing the method's adaptability larger model sizes.
-wen2-7B-Instruet demonstrates 3.28x average speedup. which is approximately 69% higher than EA- GLE2's 1.94x acceleration on the same model.
Our analysis also shows that average acceptance lengths range from 9.1 to over 10 tokens, significantly higher than typical dual-model methods. This corroborates our theoreti- cal claim that multi-tiens tiered speculation improves acceptance efficiency.
4.6. Four-Model System Discussion and Limitations
While our theoretical analysis suggests the potential bene- fits of incorporating more models into the polybasic spec- ulative decoding framework, empirical implementation of systems with four Under our sufficient (though not necessary) condition for model insertion efficiency, it is currently difficult to find suitable off-the-shelf models that satisfy the theoretical re- quirements without additional training. This limitation pri- marily stems from the stringent balance needed between acceptance length improvements and computational over- head. However, we believe this barrier is not fundamental. Through future exploration of complementary optimization techniques, such as advanced KV cache management, model pruning, and quantization, we anticipate achieving break- throughs in systems with four or more models, potentially
As shown in Figure 3, method demonstrates substantial speedups across diverse tasks, with particularly strong per- formance in math reasoning (up to 4.43x) and multi-turn conversation (up to 4.10x). However, we observe rela- tively modest acceleration on summarization tasks, where the speedup ranges from 2.95x to 3.41x. This pattern can be attributed the higher token generation requirements in summarization, which necessitates maintaining KV caches across multiple models in our polybasic speculative decod- ing framework. Despite this limitation, our approach is or- thogonal to KV cache optimization techniques, suggesting potential for further improvements through the integration of cache-focused methods.
models faces practical challenges.

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Polybasic Speculative Decoding Through Theoretical Perspective
Our
EAGLE2
Vanilla
4.5
4.0
3.66x
3.48x
3.5
3.31x
3.28x
Speedup
3.0
2.60x
2.70x
2.44x
2.5
1.94x
2.0
1.5
1.00x
1.00x
1.00x
1.00x
1.0
0.5
0.0
Vicuna 7B
LiaMA2-Chat 7B
LlaMA3-8B-Inste -n-strucct
Qwen2-7B-Instruce
Figure 2. Specdup ratios for Vicuna-7B, LLaMA2-Chat 7B, LLL and Qwen2-7B-Instre on SpecBench. Our surpassing EAGLE2 and vanilla basclines.
polybasic system consistently achieves the highest speedups
Table 2. Average acceptance length (f1) and speedup ratio (c) on different tasks. V7B: Vicuna-7B, L3-8B: LLaMA3-8B-Instun L2-7B: LLaMA2-Chat-77, Q2-7B: Qwen2-7B-Instte Instrucct.
MT
Trans.
Sum.
QA
Math
RAG
Overall
Model
c
c
A
c
μ
c
μ
V7B
3.77x
11.22
3.07x
7.76
3.01x
10.24
3.65x
9.53
4.43x
10.28
2.98x
10.30
3.48x
9.88
L.3-8B
3.70x
9.97
3.39x
8.86
3.02x
9.38
3.16x
9.08
3.87x
10.08
2.71x
9.24
3.31x
9.44
Our
L2-7B
4.10x
10.47
3.46x
9.15
3.41x
9.86
3.61x
9.49
4.02x
9.99
3.31x
10.08
3.66x
9.84
Q2-7B
3.65x
9.85
3.15x
8.65
2.95x
9.15
3.25x
8.95
3.85x
9.95
2.85x
9.35
3.28x
9.32
V7B
3.19x
4.76
2,07x
3.22
2.59x
3.96
2.45x
3.71
3.19x
4.72
2.15x
3.95
2.61x
4.34
L3-8B
2.69x
3.99
2.37x
3.53
2.23x
3.58
2.21x
3.42
2.83x
4.20
2.23x
3.95
2.44x
3.82
EAGLE2
L2-7B
3.04x
4.48
2.61x
3.96
2.50x
4.04
2.55x
4.05
3.04x
4.68
2.40x
4.19
2.70x
4.30
Q2-7B
2.40x
3.74
1.45x
2.45
1.59x
3.06
1.81x
2.91
2.63x
4.26
1.72x
3.27
1.94x
3.51
pronounced. As Figure and Table 2 shows, observe somewhat lower acceleration in summarization and RAG tasks than in shorter contexts. Addressing KV cache con- straints via caching techniques (Xiao et al., 2023; Zhang et al., 2024c;a; Jin et al., 2024) is an active research avenue and remains a promising direction for future improvements.
Table Speedup Ratios and Acceptance Lengths Larger Mod-
els
Method
Model
/4
Vicuma-133
2.69 x
8.62
Our
LLaMA-70B
2.92x
7.48
Vicuna-133 13B
2.30x
4.42
EAGLE
LLaMA-70B
2.46x
4.08
5. Conclusion
We have presented polybasic speculative decoding sys- tem that systematically extends beyond dualistic draft-target paradigms. By establishing rigorous theoretical frame- work, we derived an expression for optimal inference time and showed how speculative sampling stabilizes acceptance lengths in multi-model systems. Extensive experiments spanning multiple tasks and model families corroborate claims, demonstrating 4x speedups while preserving the
unlocking even greater acceleration benefits while maintain- ing inference quality.
Our polybasic framework, like other speculative decoding methods, can be limited by large KV cache footprints, which scale with text length. Thus, for tasks with extensive con- text, the overhead from additional models can be more
8

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Polybasic Speculative Decoding Through Theoretical Perspective
Our
EAGLE
Vanilla
Math Reasoning
Summarization
250
250
4.43x
200
4.02x
200
Tokens/s
150
3.88x
3.85x
Tokens/s
150
3.01x
3.41x
3.02x
2.95x
100
100
50
50
Vicuna 7B
LLAMAZ-Chat, 7B
LLAMA3 BB Instruct
Qwen2 7B Instruct
Vicuna 78
LLAMA2-ChaR 7B
LLAMA3 BB Instruct
Qwen2 7B Instruct
MT-Bench
Question Answering
250
250
4.10x
200
3.74x
200
3.65x
3,61x
Tokens/s
3.70x
3.65x
Tokens/s
150
150
3.16x
3.20x
100
100
50
Vicuna 7B
LLAMAZ-Chat 7B
LLAMA3 BB Instruct
Qwen2 7B Instruct
Vicuna 78
LLAMA2-Chat. 78
LLAMA3 8B Instruct
Owen2 7B Instruct
Figure 3. Speedup by task. Our method excels in math tasks, reaching 4.43x with Vicuna-PP while also maintaining strong accelerations in translation, QA, and multi-tum conversation.
100
100
Greedy sampling
Greedy sampling
Speculative sempling
Speculative sampling
80
80
Variance
60
Variance
60
20
LLaMA2-Chat-7B
LLaMA3-8B-Instruee
Figure 4. Variance acceptance length. Speculative sampling (blue) exhibits noticeably lower variance compared to greedy sampling (orange), aligning with our thcoretical stability analysis.
target model's 's output distribution.
ity across models of varying scales (from billions trillions of parameters), aiming continuously push the boundaries of efficient LLM inference.
In future work, will extend our findings to plex parallel computing scenarios by developing distributed speculative sampling systems. We also plan to explore more efficient caching strategies, implement dynamic adaptation of speculation lengths, and validate the framework's general-
9

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Polybasic Speculative Decoding Through Theoretical Perspective
Acknowledgements
References
This work was supported by the National Science Fund for Distinguished Young Scholars (No. NN..6025503), the Na- tional Natural Science Foundation of China (No. U21B2037, No. U22B2051, No. U23A20383, No. 62176222, No. 62176223, No. 62176226, No. 62072386, No. 62072387,
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12


Reviews:
[{"review_id": "NAGYawthvJ", "reviewer": "Reviewer_7bEs", "summary": "This paper proposes a novel polybasic speculative decoding framework. Specifically, the authors prove a fundamental theorem that characterizes the optimal inference time for multi-model speculative decoding systems. Through the theoretical investigation of multi-model token generation, the authors propose a three-model system implementation. Experiments demonstrate that the proposed approach yields speedup ratios ranging from 3.31 to 4.01 while preserving the original output distribution.\n\n## update after rebuttal\nThe authors' response has addressed my concerns. Thus, I will keep my original positive score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank you for your thoughtful feedback and the positive evaluation of our work.\n\n**1. Experiments on Larger Models (13B, 70B)**\n\nFollowing your suggestion, we have conducted additional experiments with Vicuna-13B and LLaMA-2-chat-70B. ($c$: speed ratio, $μ$: average acceptance length)\n\n|| Model | $c$ | $μ$ |\n|-|-|-|-|\n| Our|Vicuna-13B   | **2.69$\\times$** | **8.62** |\n||LLaMA-70B| **2.92$\\times$** | **7.48** | \n| EAGLE|Vicuna-13B | 2.30$\\times$ | 4.42 | \n||LLaMA-70B| 2.46$\\times$ | 4.08 |\n\nWe initially focused on smaller models to ensure all experiments could be run on a single GPU, facilitating direct comparison with existing methods. This is consistent with the evaluation approach in most speculative decoding literature. However, we agree that demonstrating scalability to larger models strengthens our contributions.\n\n**2. Four-Model System Evaluation**\n\nConcerning the four-model system evaluation, we have implemented a prototype following our theoretical guidance. Using LLaMA-2-7B as the target model with three draft models of decreasing capacities (4-bit quantized LLaMA-2-7B, 4-bit quantized LLaMA-2-1B, and EAGLE), we observed the following results:\n\n|| Model | $c$ | $μ$ |\n|-|-|-|-|\n|4-model|LLaMA-2-7B| **3.75$\\times$** | 9.80 |\n|3-model|LLaMA-2-7B | 3.66$\\times$ | **9.84** |\n\nWe note that diminishing returns become apparent as more models are added. This is because finding models that satisfy our Theorem 3.2 becomes increasingly challenging. Our future work will focus on quantization, sparsification, and KV cache optimization to reduce model inference time. We appreciate your suggestion, which has further advanced our work.\n\nWe thank the reviewer for these valuable suggestions and have incorporated these additional experiments and discussions in the final version of our paper.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "**Method**\n\n1.\t(Strengths) The proposed polybasic speculative decoding framework, which investigates multi-model token generation system with theoretical and empirical foundations, is novel and interesting. \n\n2.\t(Strengths) The authors prove a fundamental theorem that characterizes the optimal inference time for multi-model speculative decoding systems, shedding light on how to extend beyond the dualistic approach to a more general polybasic paradigm.\n\n3.\t(Weaknesses) The authors propose a general polybasic speculative decoding framework, while only implementing a three-model speculative decoding system. Although the authors discuss the limitations on four-model system design, it would be more convincing if the authors could evaluate the effectiveness of their method in four-model system. \n\n**Evaluation Criteria**\n\n1.\t(Strengths) Experiments demonstrate that the proposed approach yields speedup ratios ranging from 3.31 to 4.01 while preserving the original output distribution.\n\n2.\t(Strengths) The authors conduct thorough experiments on four different target models, including Vicuna-7B, LLaMA2-Chat 7B, LLaMA3-8B, and qwen2-7B-Instruct. \n\n3.\t(Weaknesses) It would be more convincing if the authors could conduct experiments on larger models, such as models with 70B parameters.", "theoretical_claims": "Yes, the theoretical claims are correct.", "experimental_designs_or_analyses": "1.\t(Strengths) Experiments demonstrate that the proposed approach yields speedup ratios ranging from 3.31 to 4.01 while preserving the original output distribution.\n\n2.\t(Strengths) The authors conduct thorough experiments on four different target models, including Vicuna-7B, LLaMA2-Chat 7B, LLaMA3-8B, and qwen2-7B-Instruct. \n\n3.\t(Weaknesses) It would be more convincing if the authors could conduct experiments on larger models, such as models with 70B parameters.", "relation_to_broader_scientific_literature": "1.\tThe proposed polybasic speculative decoding framework, which investigates multi-model token generation system with theoretical and empirical foundations, is novel and interesting. \n\n2.\tThe authors prove a fundamental theorem that characterizes the optimal inference time for multi-model speculative decoding systems, shedding light on how to extend beyond the dualistic approach to a more general polybasic paradigm.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Please see the above comments.", "other_comments_or_suggestions": "No", "questions_for_authors": "Please see the above comments.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BtsBUxIDoI", "reviewer": "Reviewer_oMTB", "summary": "The submission describes an new theoretical framework to improve speed of LLM decoding in order to reduce latency.\n\n## update after rebuttal\n\nI think this rebuttal was very useful and reaffirmed me in my slightly improved score of 3. I still share concerns with Reviewer KeQh in the areas of coherence and clarity on the choice of drafting models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. Response on Accuracy Evaluation**\n\nWe appreciate your highlighting the importance of accuracy evaluation. This is indeed a critical point that deserves attention.\n\nOur method inherently preserves output distribution through the verification mechanism in Algorithm 1. The VERIFY procedure ensures draft tokens are only accepted when they align with the target model's distribution. When verification fails, we fall back to sampling directly from the target model, guaranteeing identical outputs to standard autoregressive generation. This preservation-by-design is why speculative decoding methods typically focus on speedup metrics rather than accuracy. We followed standard performance metrics in this field to maintain consistency with prior work. Thank you for this valuable suggestion.\n\n**2. Response to Reviewer Concerns on Model Configuration**\n\nThank you for highlighting these important questions about our paper's consistency and practical focus. We appreciate the opportunity to clarify based on our contributions and motivations:\n\n**Theoretical Framework**: Our core contribution is establishing a theoretical framework for speculative sampling that enables the construction of multi-model speculative systems. When models satisfy appropriate conditions, we can extend from binary to multi-model configurations.\n\n**Three-Model Implementation**: The three-model system represents a concrete instantiation of this framework, chosen because it achieves an optimal balance between theoretical complexity and practical feasibility. Naturally, this led us to explore extensions to more models, along with their inherent limitations.\n\n**Focus on Single LLM Inference**: We fully agree with your point about optimizing inference latency for a single LLM. Speculative sampling works by having draft models generate multiple candidate tokens at once, which the target LLM then verifies in a single forward pass. This reduces the number of target model executions, thereby optimizing overall inference latency. Our discussion of four-model systems aims precisely at further improving LLM inference speed, aligning with your suggestion.\n\n**Training Considerations**: Our approach has minimal training requirements. We can utilize quantized versions of existing models or off-the-shelf smaller models, making our method orthogonal to and compatible with most existing speculative sampling approaches.\n\n**Practical Value**: The main value of our work lies in providing a theoretical framework that enables researchers to systematically optimize multi-model speculative decoding systems, with demonstrated practical improvements in reducing target LLM inference latency.\n\n**3. Response to Terminology Inconsistency**\n\nThank you for highlighting the inconsistency in our reference to baseline techniques. We apologize for the confusion this has caused.\n\nWe acknowledge the error in our notation. Throughout the paper, we intended to reference EAGLE-2 consistently as our baseline, as it represents an improvement over EAGLE-1.\n\nIn the revised version, we will:\n1. Standardize all references to EAGLE-2 throughout the manuscript\n2. Add proper citations to both EAGLE-1 and EAGLE-2\n3. Include a brief explanation clarifying that EAGLE-2 is an advancement of EAGLE-1, which is why we selected it as our baseline for comparison\n\nThank you for your thoughtful review of our paper. We genuinely appreciate both your critical observations and the opportunity to address them.\n\nBased on the clarifications provided, we respectfully request that you reconsider your evaluation of our paper. We remain available to address any additional questions or concerns you might have.", "claims_and_evidence": "Initially I found the theory sound and the experimental results showing improved speed very good. But the more I reflected on the content the more issues I found. The following might be subjective:\n\n* Why is accuracy not important at all? Or does the proposed method (and the comparison baseline of EAGLE) show no drop in accuracy at all? At least a single experiment to prove the impact on accuracy is standard for any speedup method.\n\n* There are inconsistencies that didn't help to understand the overall impact. The method is being described as using 3 models and possibly extend to more, but using 3 models is already pretty tricky (Section 3.3 mentions at least one of the 3 models to be trained separately). Then Section 4.4 discusses to use 4 models and I've no clue how that helps any further. I find that discussion moving off-topic. Isn't the focus on improving impractical latency of inference for a single LLM? It must be doubted that multiple draft models that require to be executed sequentially can help to improve latency.\n\n* Reference of the baseline technique is confusing: EAGLE(-1) and EAGLE-2 or consistently EAGLE-2 and EAGLE is a typo?", "methods_and_evaluation_criteria": "ML speedup techniques are usally lossy especially when applied to decoding (e.g. pruning). Choosing the right operating point is mandatory and that requires to select from tradeoff curves. The presentation lacks to describe the impact to accuracy and even that there isn't any if that's the case.", "theoretical_claims": "Sound to me. However, there's some clarity missing why to use more than the 3 proposed models as mentioned above.", "experimental_designs_or_analyses": "Limited as the impact on accuracy is missing.", "relation_to_broader_scientific_literature": "None.", "essential_references_not_discussed": "Only EAGLE is being cited, but Table 1 contains reference to EAGLE-2 instead.", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "I think the topic is of big interest, but speed improvements need to be put into perspective of impact to accuracy and the presentation lacks clarity that clearly requires improvements.", "questions_for_authors": "Please work on clarity. I don't want to repeat the points here. See Section \"Claims And Evidence\".", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DWoCoPPIoq", "reviewer": "Reviewer_KeQh", "summary": "This paper explores using a chain of draft models rather than a single draft model during speculative decoding, such that the first draft model generates tokens autoregressively, and each subsequent draft model verifies the tokens generated. When an intermediate draft model rejects a token the first draft model generates more tokens.\nThe paper provides theoretical analysis to find the condition of when an additional draft model could lead to speedup, then implements a chain of 2 draft models (EAGLE as the first draft and 4-bit quantized version of the model as a second draft) and shows significant speedups over EAGLE only.\n\n## Update after Rebuttal\n\nI have read all the reviews and rebuttals. I appreciate the authors response to my review. I would like to keep my score, i.e., while I am hesitant to accept the paper due to it's lack of coherence and limited contribution, I won't fight against it if it's accepted. However, I highly recommend re-writing parts of the paper to ensure it's coherence and to add results with different types of small drafters, not just EAGLE, to ensure that the theoretical framework of the paper (that the authors claim is their main contribution) is generic.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**1. On the Connection Between Theory and Experiments**\n\nWe appreciate your concern about the gap between our theoretical foundation and experiments. To address this, we've conducted specific experiments that directly validate our theoretical framework:\n\n|$T_i $|$L_{i-new}$|$T_{new}$|$L_{new} $|$T_{i+1}$|$L_i$|Speedup|\n|-|-|-|-|-|-|-|\n|22ms|3.83|17.61ms|3.77|4ms|4.34|2.61×→1.08×|\n|22ms| 6.26|7ms| 4.67|4ms|4.34|2.61×→3.48×|\n\nFor Case 1:\n\n- $ith$：vicuna-7b, $new$: vicuna-1b, $(i+1)th$: EAGLE\n- $T_{new}/T_i$ = 0.80\n- $L_{new }× (1/L_i - 1/L_{i-new}) $= -0.117\n- Since 0.80 > -0.117, Theorem 3.2 predicts performance decrease\n\nFor Case 2:\n\n- $ith$：vicuna-7b, $new$: W4A4 vicuna-7b, $(i+1)th$: EAGLE\n- $T_{new}/T_i$ = 0.318\n- $L_{new} × (1/L_i - 1/L_{i-new}) $= 0.330\n- Since 0.318 < 0.330, Theorem 3.2 predicts performance improvement\n\nThese results confirm our theoretical framework's ability to guide model selection decisions for speculative decoding systems.\n\n**2.Expanded Model Experiments**\n\nFollowing your recommendation, we have extended our experiments to include Vicuna-13B and Llama-2-Chat-70B models, both showing significant acceleration effects. \n\n||Model|$c$|$μ$|\n|-|-|-|-|\n|Our|V13B |**2.69$\\times$**|**8.62**|\n||L70B|**2.92$\\times$**|**7.48**|\n|EAGLE|V13B|2.30$\\times$|4.42|\n||L70B|2.46$\\times$|4.08|\n\n**3. Theoretical Formulations and Symbol Definition Issues**\n\nWe appreciate your detailed feedback on our theoretical formulations. We will revise the identified errors and provide clearer explanations for the mathematical notations and symbols used throughout the paper.\n\n**Equation 2:**\n\nIn Equation $T = \\sum_{i=1}^n F_i \\cdot T_i$, we combine prefill and decoding costs into a single term $T_i$ to simplify **model selection for speculative decoding**. This focuses on comparing efficiency when adding models rather than precisely calculating absolute inference times. Our experimental results (like Vicuna-7B's $4.43\\times$ speedup) validate this approach. While separating costs ($T_i = T_i^{\\text{prefill}} + T_i^{\\text{decode}}$) could add precision, it wouldn't change our core theoretical conclusions. We'll clarify this rationale in the final manuscript.\n\n**Line 157 and MagicDec:**\n\nLine 157 presents a binary form of Equation (2). We appreciate the reference to MagicDec - our theoretical approach indeed aligns with theirs, though we chose our formulation for conciseness and clearer derivations.\n\nFor equivalence:\n- **MagicDec**: Speedup = $\\Omega \\cdot \\frac{T_T}{\\gamma T_D + T_V}$\n- **Ours (dual-model)**: Speedup = $\\frac{L_1 T_T}{T_T + T_D}$\n\nBy mapping $\\Omega \\leftrightarrow L_1$ and $\\gamma T_D + T_V \\leftrightarrow T_T + T_D$, the formulas are mathematically equivalent.\n\n**Some symbols and equations were not clear**\n\n**Regarding line 169** (the positioning of \"new\" between i and i+1):\nAs mentioned in our paper's second contribution, our theoretical framework aims to discover conditions under which adding auxiliary models can improve inference speed in speculative sampling systems. Intuitively, we should insert a model with stronger inference capabilities between the draft model and target model to bring the composite draft model's capabilities closer to the target model.\n\n**line 174** (the symbol $α$):\nThe paper mentions that acceptance probability $p = 1 - α$, where α represents the rejection probability.\n\n**line 191** (the symbol $S$):\n$S$ is simply an intermediate summation value used for notational convenience, which we didn't explicitly define given space constraints. You also noted that equations on page 4 are unnumbered - we chose not to number equations that are merely part of proofs and not essential to the paper's central theoretical framework.\n\n**Theorem 3.3**\nWe realize that our use of \"$n$-model\" may have caused confusion. In this context, \"$n$\" refers to \"a truncated geometric distribution of maximum $n$ trials\" (line 189). We adopted this notation to maintain consistency with [SpecDec](https://arxiv.org/pdf/2211.17192). In our revised manuscript, we will add a clarifying statement at this location to prevent any misunderstanding.\n\n**4. Comparison with Other Cascade Methods**\n\nWe did not experimentally compare our approach with other cascade methods because the current state-of-the-art in speculative sampling is [EAGLE](https://sites.google.com/view/eagle-llm), which significantly outperforms these cascade methods in terms of acceleration. Therefore, we focused our comparison on EAGLE.\n\n**5. Other Issues**\n\n Regarding Algorithm 1, the note \"(continue accumulating tokens)\" was intended to indicate continuation of the Draft execution in line 7. We will remove the else branch to ensure the correctness of the pseudocode.\n\nThank you for your thorough review. We hope our responses clarify your concerns and demonstrate our paper's contributions. We respectfully request an improved score. Maybe it's unclear due to word limit, please communicate with us if you have any further questions.", "claims_and_evidence": "- The claims of obtaining higher speedups using chained drafters compared to a single drafter, is backed up by the results\n- However, the claim of providing a theoretical foundation for chained drafters is disputable. The authors do provide a theorem on the condition of when adding a drafter would lead to speedup, but they do not use it in their experiments.", "methods_and_evaluation_criteria": "- The experiments were made on 4 models of similar size (7B) from different generations and familes (Llama2, Llama3, Vicuna, Qwen)\n   - I would have preferred if different sizes (e.g., 1B, 13B, 70B) were evaluated\n- 6 different language generation tasks were evaluated", "theoretical_claims": "- Equation 2: The mathematical formulation is a bit oversimplified: we have decoding time and prefill time. And the first draft model, i.e., $ T_{n} $, decodes tokens, while other models perform prefill of K tokens\n- Line 157: Please cite source of equation. I believe the mathematical formulation shown in equations 1 to 3 of [MagicDec](https://arxiv.org/abs/2408.11049) paper is better\n- Some symbols and equations were not clear to me:\n  - Line 169: I don't under stand what is $i - new$ or $new - (i+1)$ if $new$ is supposed to be between $i$ and $i+1$\n  - Line 174 (right column): What is $\\alpha$ ?\n  - Line 191 (right column): what is $S$ ?", "experimental_designs_or_analyses": "- Although results show cascaded drafters lead to bigger speedups, the connection with the theoretical proof is not clear. Hence, it is not clear what is the contribution of this paper compared to other papers that propose cascaded drafters\n- Line 328 (Right column): \"As anticipated, speculative sampling yields smaller variance\" My understanding from Theorem 3.3 was that lower variance comes from adding multiple draft models. not from speculative sampling.", "relation_to_broader_scientific_literature": "- The Related Work section categorizes speculative decoding techniques from unique perspectives\n- The paper cited similar work on \"cascade or multi-level drafting (Chen et al., 2023b; Sun et al., 2024)\". Authors claim that the difference between the paper and such other papers is that they provide a theoretical foundation. However, the theorems deduced in the paper were not used in the experiments to obtain any improvement over similar work.", "essential_references_not_discussed": "- The paper cited other work that did cascaded drafters (Chen et al., 2023b; Sun et al., 2024) but did not really show the differences or contributions compared to them", "other_strengths_and_weaknesses": "- Strengths;\n   - The paper is written with relative clarity and was relatively easy to follow\n   - The speedups presented in the results are strong\n- Weakness:\n   - There is a gap between the theoretical foundation and Experiments sections. The Experiments or Results section do not leverage any of the theoretical analysis done. e.g., there was no measurement of $ T_{new} $ or $ L_{new} $ to verify if it satisfies the condition of Theorem 3.2\n       -  My suggestion would be to provide a couple of experiments, one experiment where the additional drafter model satisfies Theorem 1, and another experiment where the additional drafter doesn't. In both experiments, measure their corresponding $ T_{new} $ and $ L_{new} $, and show how their values correlated with final speedup to proof Theorem 1\n       - Also, I suggest to look at [MagicDec](https://arxiv.org/abs/2408.11049): look at its mathematical formulations, and it's experimental analysis that is directly connected to the mathematical formulation\n\nI am leaning towards rejecting the paper because the paper claims it's contribution compared to other papers on cascaded drafts is the theoretical foundation, but the theoretical foundation in the paper is totally decoupled from the results section. Also, there were issues I described above in the equations, usage of symbols without definitions, disputable claims, and ad hoc use of natural language in a main part of the Algorithm pseudocode.", "other_comments_or_suggestions": "- Equations in Page 4 are not numbered\n- Algorithm 1 Line 30: I would have preferred if there was an explanation or pseudocode rather than just writing \"(continue accumulating tokens)\"\n- Line 282: There is a typo in \"in SectionSection 3.2\" \n- Page 7: \"As Figure 3 Table 1 shows,\" should be \"As Figure 3 and Table 1 show,\"", "questions_for_authors": "I have written questions in different boxes above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "N/A"}}, {"review_id": "ODeWjYcyI5", "reviewer": "Reviewer_vGSY", "summary": "The paper introduces the polybasic speculative decoding framework, aimed at accelerating the inference of large language models (LLMs) through multi-model collaboration. Its core contributions include: 1) Establishing a theoretical framework, deriving the optimal inference time formula for multi-model systems (Lemma 3.1), and proposing the model insertion efficiency condition (Theorem 3.2), which quantifies the trade-offs between model capability, acceptance length, and computational cost. 2) Designing a multi-level verification algorithm, which employs a phased strategy where lightweight models generate candidate tokens, intermediate models quickly filter them, and the target model periodically verifies them (Algorithm 1), thereby reducing the target model's invocations and enhancing throughput. 3) Experimental validation shows a speedup ratio of 3×–4.43× on models such as Vicuna-7B, LLaMA2/3, and Qwen2, with mathematical reasoning tasks achieving up to 4.43 times acceleration, and the average acceptance length increasing to 9–10 tokens, while maintaining output distribution consistency with the original model. Additionally, theoretical proofs demonstrate that speculative sampling can reduce the variance of acceptance lengths (Theorem 3.3), and the framework's generality is showcased through task-adaptive designs (e.g., extensions of self-speculative methods). This work provides theoretical guidance and scalable practical solutions for efficient LLM inference.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for these insightful questions that help clarify the practical implications of our theoretical framework.\n\n**Regarding Theorem 3.2's sensitivity:** We recommend measuring the relevant parameter values under identical experimental conditions. While measurement errors may occur, our theoretical framework is robust against minor measurement inaccuracies because when comparing inference time changes after adding models, the errors are relative. This relative nature of measurement ensures that our framework's predictions remain reliable even with some experimental noise.\n\n**On implementing multi-model systems:** The primary challenge in extending to multi-model systems is KV cache management. This can be addressed using techniques similar to those in MagicDec, such as StreamingLLM technology, which preserves only initial tokens (Attention Sinks) and KV caches within a sliding window, significantly reducing memory requirements. Our theoretical framework guides model selection and is orthogonal to other speculative methods in its approach. We plan to enhance our system in future work.\n\n**Stability in long-context scenarios:** Theorem 3.3's variance analysis strengthens our theoretical framework by reducing errors from acceptance token length variations during model selection, maintaining stability across all tasks. For long-context tasks, effective KV cache management is clearly essential, and our framework provides a stable foundation upon which these optimizations can be built.\n\n**Comparison with recent works:** Thank you for your suggestion - we have expanded our experimental comparison to include Hydra, which further validates the effectiveness of our approach across different speculative decoding implementations. ($c$: speed ratio, $μ$: average acceptance length)\n\n|| Model | $c$ | $μ$ |\n|-|-|-|-| \n| Our|Vicuna-7B   | **3.48$\\times$** | **9.88** | \n| Hydra|Vicuna-7B | 2.30$\\times$ | 4.42 |\n\nIn conclusion, we appreciate your recognition of our work and your thorough review. We hope these responses address your questions and highlight the contributions of our theoretical framework to the understanding and optimization of speculative sampling systems.", "claims_and_evidence": "**Claim** The paper introduces a theoretical framework that derives an optimal inference time formula for multi-model speculative decoding systems (Lemma 3.1) and establishes conditions for model insertion efficiency (Theorem 3.2).\n**Evidence** The authors provide rigorous mathematical proofs for Lemma 3.1 and Theorem 3.2, which quantify the relationship between model forward-pass costs, acceptance lengths, and computational overhead. The proofs are detailed and logically structured, supporting the claim that adding models can improve speedups under specific conditions.\n\n**Claim:** Speculative sampling reduces variance in token acceptance lengths, leading to more stable and predictable performance (Theorem 3.3).\n**Evidence:** The authors present a formal proof for Theorem 3.3, which mathematically demonstrates the relationship between acceptance probability and variance in multi-model settings. Additionally, empirical results in Section 4.3 validate this claim by comparing acceptance-length variance between speculative and greedy sampling, showing significantly lower variance for speculative sampling.", "methods_and_evaluation_criteria": "1.\tProposed Methods:\n\n**Polybasic Speculative Decoding Framework**: The introduction of a polybasic speculative decoding framework is innovative and addresses limitations in existing dualistic draft-verify systems. By incorporating multiple models, the approach aims to increase parallelism and acceptance length, which directly tackles the computational bottleneck in LLM inference.\nTheoretical Foundations: The paper provides a strong theoretical basis for the framework, including a fundamental theorem on optimal inference time and insights into the stability of acceptance lengths. These theoretical contributions help guide practical implementation and offer a deeper understanding of the system's behavior.\nAlgorithmic Design: The staged verification process and the detailed algorithm (Algorithm 1) provide clear guidelines for implementing the polybasic system. This makes the method more accessible and reproducible.\n\n2.\tEvaluation Criteria:\n\n**Diverse Tasks and Models:** The evaluation spans multiple tasks (MT-bench, translation, summarization, QA, mathematical reasoning, RAG) and model families (Vicuna-7B, LLaMA2-Chat 7B, LLaMA3-8B, Qwen2-7B). This diversity ensures that the method's effectiveness is tested across different scenarios and model architectures.\n\n**Performance Metrics:** The use of speedup ratio and average acceptance length as metrics is appropriate for assessing the efficiency and effectiveness of the speculative decoding approach. These metrics directly relate to the goals of reducing inference time while maintaining output quality.\nComparison to Baselines: The paper compares the proposed method to existing approaches like EAGLE and vanilla speculative decoding, providing a clear benchmark for evaluating improvements.", "theoretical_claims": "The theoretical claims presented in this paper are both novel and rigorously supported, offering a significant advancement in the field of speculative decoding for large language models (LLMs). The authors have successfully proven the correctness of their key theoretical contributions, including the fundamental theorem on optimal inference time (Theorem 3.2) and the stability analysis of acceptance lengths under speculative sampling (Theorem 3.3). These proofs are mathematically sound and provide a solid foundation for the proposed polybasic speculative decoding framework.\n\nThe derivation of the optimal inference time formula in Theorem 3.2 is particularly impressive, as it elegantly captures the trade-off between adding additional models and the associated computational costs. This result not only provides clear guidance for model insertion but also establishes a principled approach to system design that can be generalized across different architectures and tasks. Furthermore, the proof of Theorem 3.3 highlights the benefits of speculative sampling in reducing variance in acceptance lengths, which is critical for ensuring stable and predictable performance in multi-model systems.", "experimental_designs_or_analyses": "The experimental designs and analyses presented in this paper are robust, well-structured, and effectively validate the proposed polybasic speculative decoding framework. Below is a positive review of the key aspects:\n\n**1.\tComprehensive Evaluation Across Models and Tasks :**\n\nThe authors conduct experiments on multiple widely-used LLMs (e.g., Vicuna-7B, LLaMA2-Chat 7B, LLaMA3-8B, Qwen2-7B) across diverse tasks such as MT-bench, translation, summarization, QA, mathematical reasoning, and RAG. This broad evaluation demonstrates the generalizability and adaptability of the polybasic approach, achieving impressive speedups (ranging from 3.31× to 4.43×) while preserving output fidelity. The choice of models and tasks ensures that the method's effectiveness is tested under varied conditions, reinforcing the reliability of the results.\n\n**2.\tClear Metrics for Performance Assessment :**\n\nThe use of two primary metrics—walltime speedup ratio and average acceptance length—is appropriate and aligns with the goals of speculative decoding. These metrics directly measure the efficiency gains and stability of the system, providing a clear and interpretable evaluation framework. The reported acceptance lengths (9.1–10+ tokens) are significantly higher than dualistic baselines, supporting the theoretical claim that multi-tiered speculation improves acceptance efficiency.", "relation_to_broader_scientific_literature": "The key contributions of this paper are highly significant when viewed in the context of the broader scientific literature on speculative decoding and large language model (LLM) optimization. The introduction of a polybasic speculative decoding framework builds upon foundational ideas from prior works, such as blockwise parallel decoding and hierarchical speculative methods like TRIFORCE, while addressing critical gaps in scalability and theoretical grounding. Unlike earlier dualistic draft-verify paradigms, which often relied on empirical heuristics, this work provides a rigorous theoretical framework that aligns with recent trends toward formalizing speculative decoding principles. Furthermore, the demonstrated speedups (up to 4.43×) across diverse tasks extend the findings of prior studies like EAGLE and Medusa 3, showcasing superior performance through multi-model coordination.", "essential_references_not_discussed": "The paper provides a comprehensive discussion of related works, effectively situating its contributions within the broader context of speculative decoding and large language model (LLM) optimization. Key advancements in dualistic draft-verify frameworks, hierarchical speculative methods, and multi-level drafting are thoroughly reviewed.Even if there exist tangential works not cite, they do not appear essential to understanding or evaluating the key contributions of this paper.", "other_strengths_and_weaknesses": "Strengths:\n\n1.\tThe paper introduces a novel polybasic speculative decoding framework that extends beyond traditional dualistic draft-verify paradigms. By incorporating multiple models in a staged verification process, the approach significantly increases parallelism and acceptance lengths, achieving speedups of up to 4.43× across various tasks and models.\n\n2.\tThe framework is underpinned by rigorous theoretical analysis, including a fundamental theorem on optimal inference time (Theorem 3.2) and insights into the stability of acceptance lengths (Theorem 3.3). These contributions provide clear guidance for system design and optimization.\n\n3.\tExtensive experiments on diverse tasks (e.g., MT-bench, translation, summarization) and model families (e.g., Vicuna-7B, LLaMA2-Chat 7B) demonstrate consistent performance improvements while preserving output fidelity. The results highlight the method's adaptability and robustness.\n\n4.\tThe paper provides clear algorithmic guidelines (e.g., Algorithm 1) and discusses practical considerations like model selection and speculation length tuning, making the approach accessible and reproducible.\n\nWeaknesses:\n\n1.\tWhile the theoretical framework supports the inclusion of four or more models, the authors acknowledge practical difficulties in finding suitable off-the-shelf models that meet the theoretical requirements. However, this limitation is relatively minor and can be addressed through future exploration of complementary techniques like advanced quantization or pruning.\n\n2.\tThe paper notes slightly lower acceleration in tasks requiring long-context generation (e.g., summarization, RAG). This limitation is understandable given the inherent challenges of managing KV caches in such scenarios. Nevertheless, the authors appropriately highlight this as an area for future improvement, suggesting potential solutions like caching strategies.\n\n3.\tAlthough the paper focuses on LLMs, it could briefly discuss how the polybasic framework might extend to other domains, such as vision or multimodal models. However, this is a minor point and does not detract from the paper's primary focus on language models.", "other_comments_or_suggestions": "1.\tWhile Theorem 3.2 provides a condition for model insertion, a more intuitive explanation or visual aid (e.g., a flowchart) could help readers better grasp when and how to add new models effectively.\n\n2.\tThe discussion on four-model systems is insightful but could benefit from specific examples of off-the-shelf models that partially meet the requirements, along with potential hybrid approaches (e.g., combining quantization and pruning).\n\n3.\tInclude a brief comparison to very recent speculative decoding methods (e.g., Hydra, Falcon) to highlight how the polybasic framework compares in terms of flexibility and performance.", "questions_for_authors": "1.\tIn Theorem 3.2, you provide conditions under which adding a new model improves inference time. Could you elaborate on how sensitive these conditions are to variations in acceptance length and forward-pass cost? For example, could small deviations in empirical measurements (e.g., due to noisy data) lead to incorrect conclusions about whether a model should be added?\n\n2.\tWhile you acknowledge challenges in implementing four-model systems, could you provide more specific insights or preliminary guidelines for overcoming these limitations? For instance, are there particular quantization techniques or pruning strategies that show promise for balancing computational overhead with acceptance length improvements?\n\n3.\tYour stability analysis (Theorem 3.3) focuses on speculative sampling’s ability to reduce variance in acceptance lengths. How does this stability hold up in long-context tasks like summarization or RAG, where KV cache management becomes critical? Are there adjustments to the speculative sampling parameters that could further mitigate variance in such scenarios?\n\n4.\tRecent works like Hydra (Ankner et al., 2024) and Falcon (Gao et al., 2024) explore advanced speculative decoding techniques. How does your polybasic framework compare to these methods in terms of flexibility, scalability, and performance across different model architectures?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: After careful consideration of the reviews and the authors’ responses, I recommend the paper be accepted. The work introduces a novel polybasic speculative decoding framework, supported by rigorous theoretical analysis, and makes a significant contribution to the understanding of multi-model speculative decoding. Reviewers highlighted the originality and importance of the theoretical results, particularly the theorem characterizing optimal inference time, which offers valuable insights into the trade-offs inherent in speculative decoding systems. The empirical evaluation is comprehensive, covering multiple model families and language tasks, and demonstrates substantial speedups while preserving the original output distribution. Reviewers also appreciated the authors’ proactive and thorough rebuttal, which included additional experiments and clarifications that effectively addressed earlier concerns. Minor issues were noted, including some inconsistencies and clarity gaps in the theoretical formulations and algorithm descriptions, which the authors addressed and committed to improving in the final version. Reviewers also suggested expanding experiments with larger models and more complex decoding chains to better showcase scalability and robustness, as well as enhancing the manuscript’s coherence by more explicitly linking theoretical and empirical results. Overall, the paper’s strong theoretical foundation, robust empirical validation, and the authors’ responsiveness substantially outweigh the minor concerns, and we look forward to seeing the final version incorporate the reviewers’ constructive suggestions. Lastly, the paper should include the [1] in the related work. 

[1] A theoretical perspective for speculative decoding algorithm, NeurIPS25.
Decision: Accept (poster)