# learning_harmonized_representations_for_speculative_sampling__23727209.pdf

Published as a conference paper at ICLR 2025

LEARNING HARMONIZED REPRESENTATIONS FOR SPECULATIVE SAMPLING

Lefan Zhang, Xiaodan Wang, Yanhua Huang , Ruiwen Xu Xiaohongshu Inc. Shanghai, China {lefan,xiaodan2,yanhuahuang,ruiwenxu}@xiaohongshu.com

Speculative sampling is a promising approach to accelerate the decoding stage for Large Language Models (LLMs). Recent advancements that leverage target LLM s contextual information, such as hidden states and KV cache, have shown significant practical improvements. However, these approaches suffer from inconsistent context between training and decoding. We also observe another discrepancy between the training and decoding objectives in existing speculative sampling methods. In this work, we propose a solution named HArmonized Speculative Sampling (HASS) that learns harmonized representations to address these issues. HASS accelerates the decoding stage without adding inference overhead through harmonized objective distillation and harmonized context alignment. Experiments on four LLa MA models demonstrate that HASS achieves 2.81x-4.05x wall-clock time speedup ratio averaging across three datasets, surpassing EAGLE-2 by 8%-20%. The code is available at https://github. com/HArmonized SS/HASS.

1 INTRODUCTION

Generative Large Language Models (LLMs), such as GPT-4 (Achiam et al., 2023) and LLa MA (Touvron et al., 2023), have demonstrated remarkable capabilities across a wide range of tasks. Nevertheless, efficiently decoding from these models poses a significant challenge due to the inherent auto-regressive decoding mechanism, which restricts their applicability in time-sensitive scenarios. Speculative sampling (Chen et al., 2023; Leviathan et al., 2023) offers a solution by leveraging additional resources to increase concurrency. Specifically, it employs an efficient draft model to generate draft tokens auto-regressively, which are then concurrently verified by the target LLM. Based on the verification results, a subset of draft tokens that preserves the same distribution as the target LLM is accepted as the final output.

Leviathan et al. (2023) show that the practical performance of speculative sampling is highly related to two factors: the decoding cost of the draft model and its alignment with the target LLM. To develop efficient draft models that are well-aligned with the target LLM, previous works propose to leverage the target LLM s contextual information (Xiao et al., 2024; Li et al., 2024b;c; Du et al., 2024). For instance, EAGLE (Li et al., 2024b;c) employs previous hidden states of the target LLM as the draft model s input features. However, these approaches introduce inconsistent context between training and decoding, as illustrated in Figure 2. During training, the draft model always has access to the target LLM s hidden states in previous timesteps. However, during decoding, the draft model cannot access the target LLM s hidden states for unverified timesteps, resulting in a context misalignment between training and decoding. This issue can be viewed as a form of exposure bias (Bengio et al., 2015; Wang & Sennrich, 2020) at the feature level in speculative sampling.

Another discrepancy is also observed between the objectives of the training and decoding stages. During the decoding stage, the objective of the draft model is to propose tokens that the target LLM is likely to assign high probabilities to (Li et al., 2024c; Miao et al., 2024; Sun et al., 2024). In this scenario, the draft model should focus more on recalling the desired tokens, while the specific

Corresponding author.

Published as a conference paper at ICLR 2025

LLa MA2-Chat 7B T=0 LLa MA2-Chat 13B T=0 LLa MA3-Instruct 8B T=0 LLa MA3-Instruct 70B T=0 LLa MA2-Chat 7B T=1 LLa MA2-Chat 13B T=1 LLa MA3-Instruct 8B T=1 LLa MA3-Instruct 70B T=1 Models

HASS EAGLE-2 EAGLE

Figure 1: Speedup ratios of different methods on LLa MA2-Chat 7/13B and LLa MA3-Instruct 8/70B with temperature T {0, 1}, averaging over MT-bench, Human Eval, and GSM8K datasets.

order of these tokens can be somewhat de-emphasized. Moreover, most LLM applications perform nucleus sampling (Holtzman et al., 2020) or top-k sampling (Fan et al., 2018). For these decoding objectives, tokens with high probabilities play a more significant role in determining the output. Therefore, to develop efficient draft models, their training objectives should consider these properties encountered in the decoding stage. To the best of our knowledge, previous works on training draft models for speculative sampling have largely overlooked these decoding considerations.

One Auto-regressive Layer

Prediction Head & Sampling

Consistent forward

Training forward

Decoding forward

Unverified tokens

Figure 2: We use EAGLE (Li et al., 2024b) as an example to illustrate the context misalignment, where the speculation starts from timestep t. f (l) and f (s) represent hidden states from the target LLM and the draft model. When decoding draft token xt+2, the input context is inconsistent between training and decoding.

In this paper, we introduce HArmonized Speculative Sampling (HASS), a novel approach designed to address the aforementioned issues by learning harmonized representations. Specifically, to make draft models aware of the decoding strategy, HASS extends the idea of ranking distillation (Tang & Wang, 2018) from the recommender system to speculative sampling, resulting in a distillation loss focused on the most probable tokens within the target distribution. To mitigate the previously discussed context misalignment between training and decoding, HASS employs a contextaligned training strategy. Together, these two strategies of HASS improve the acceleration performance without any inference overhead and maintain training efficiency.

We conduct experiments across dialogue, code generation, and mathematical reasoning tasks using the MT-bench, Human Eval, and GSM8K datasets, respectively. Building with EAGLE-2 (Li et al., 2024c), HASS achieves 8%-16% acceptance length improvement over it on LLa MA2-Chat 7/13B and LLa MA3Instruct 8/70B, resulting in 2.81x-4.05x wallclock time acceleration compared with the vanilla inference on NVIDIA H800 GPU.

2 PRELIMINARY

Speculative sampling leverages the concept of speculative execution (Kung & Robinson, 1981; Hennessy & Patterson, 2011) to reduce wall-clock time from more concurrency. Specifically, given the target LLM M(l) that is the focus of acceleration, speculative sampling employs a draft model

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M(s) to speculatively and efficiently generate draft tokens. The conventional approach (Leviathan et al., 2023; Chen et al., 2023) decomposes the next step generation into three steps:

M(s) proposes an unverified draft sequence with length L by auto-regressive decoding.

M(l) evaluates posterior probabilities of L draft tokens in parallel. τ tokens that retain the target distribution are accepted by a modified rejection sampling schema based on the draft sequence and the distribution gap.

Leviathan et al. (2023) demonstrate that the wall-clock time improvement ratio is directly proportional to τ, while the arithmetic operation increment ratio is inversely proportional to τ. Consequently, τ, also known as the acceptance length, plays a crucial role in determining the performance of acceleration. This analysis also applies when using multiple draft sequences (Miao et al., 2024; Li et al., 2024c;b; Sun et al., 2024). Note that τ is closely related to the distribution gap between the target LLM and the draft model. With efficient decoding requirements, the draft model typically has limited capacity, resulting in a significant distribution gap compared to the target LLM. Fortunately, during inference, the acceptance rate is primarily influenced by the alignment of distributions on the desired tokens, i.e., the tokens to which the target LLM assigns high probabilities. However, previous speculative sampling works mainly focus on the entire vocabulary set w.r.t. knowledge distillation from the target LLM (Li et al., 2024b; Zhou et al., 2024), thereby disconnecting the training process from the practical decoding requirements.

EAGLE (Li et al., 2024b) is a lightweight draft model design, as shown in Figure 2. During decoding, it utilizes the LM Head of the target LLM to generate draft tokens. Specifically, we assume that the speculation starts from timestep t, meaning the first draft token is at timestep t + 1. To generate the draft token xt+1, the target LLM s hidden state f (l) t 1 in the second-to-top layer is concatenated with the embedding et to perform the input of the draft model. During training, EAGLE constructs a regression task between f (l)s and the predicted hidden states f (s)s of the draft model. However, due to the auto-regressive decoding, the draft model only accesses the target LLM s features at the beginning of the speculation. It uses the features produced by itself as input for subsequent steps. This context misalignment, stemming from feature inaccuracies, leads to error accumulation and hinders the performance of generating later draft tokens (Li et al., 2024b; Du et al., 2024). EAGLE-2 (Li et al., 2024c) employs the same model design but works on dynamic drafting structures instead of a static tree structure during the decoding stage, yet the aforementioned issue remains unresolved.

3 METHODOLOGY

As outlined before, previous speculative sampling methods suffer from disharmonies between training and decoding. This section introduces HArmonized Speculative Sampling (HASS) to tackle objective misalignment and context inconsistency through harmonized objective distillation and harmonized context alignment, respectively, as described below.

3.1 HARMONIZED OBJECTIVE DISTILLATION

HASS prioritizes the most decoding-desired tokens by leveraging the ranking distillation (Tang & Wang, 2018) idea from the recommender system. Specifically, ranking distillation aims to train a student model to assign higher ranks to the items that are top-ranked by the teacher model. In the context of speculative sampling, the draft model and the target LLM serve as the student and the teacher, respectively. Draft models with similar properties will perform at a higher acceptance rate in the decoding stage. Consider the set of K tokens with the highest probabilities from the target LLM s probability distribution as ˆΩ Ω, where Ωrepresents the entire vocabulary. HASS considers the following Top-K distillation loss:

x ˆΩ q(x) log p(x), (1)

where q and p are the next token probability distributions of the target LLM and the draft model, respectively. Note that, when integrated with EAGLE, the training stage can obtain ˆΩfrom hidden states of the target LLM. This implies that the proposed loss function benefits from the same efficient

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training cost as EAGLE. We evaluate the proposed Top-K distillation loss against six alternative losses, such as Bi LD (Li et al., 2024a) and Recall@k Surrogate loss (Patel et al., 2022), through ablation studies.

3.2 HARMONIZED CONTEXT ALIGNMENT

HASS Training Step 1 / EAGLE Training

HASS / EAGLE Decoding

HASS Training Step 2

!!"$ !!"# !! !!(#

"("), "($!)

HASS Training Step 3

"("), "($!), "($")

!!"$ !!"# !! !!(#

"("), "($!), "($"), "($#) !!"$ !!"# !! !!(#

HASS Training Step 4

Figure 3: Training with harmonized context alignment, where q and k refer to the query and key states in the transformer layer, respectively. Superscript (l) denotes tensors from the target LLM, and superscript (sj) denotes tensors from the j-th draft model forward. Note that during training (sj) refers to calling j times draft model in a batch, while during inference (sj) refers to j-th autoregressive decoding.

HASS follows a context alignment schema that aligns training and decoding on their contexts. The training procedure of HASS is divided into n steps, enabling the draft model to utilize contextual features consistent with those in the decoding stage and addressing the context inconsistency by adapting the inaccurate features generated in previous HASS training steps. Specifically, it is achieved by first taking the inaccurate feature from the last draft model as query, and then considering the inaccuracy accumulation in key-value part of the transformer block.

Formally, in the HASS training step j, given input token sequence x1, x2, .., x T , we optimize the draft model M(s) with the following objective function:

t=1 [Cross Entropy(P (l)(xt+1|x1, . . . , xt), P (s)(xt+1|x1, . . . , xt)) + Aux-loss], where

P (s)(xt+1|x1, . . . , xt) = Head(f (sj) t+1 )

= Head(M(s)(

From last draft z }| { f (sj 1) t| {z } query

From target LLM z }| { f (l) 1 f (l) t j+1

From previous draft models z }| { f (s1) t j+2 f (sj 1) t | {z } key & value

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Temperature = 0 Temperature = 1 Model Method MT-bench Human Eval GSM8K Mean MT-bench Human Eval GSM8K Mean

PLD 1.43 1.59 1.37 1.46 - - - - Lookahead 1.66 1.77 1.65 1.69 - - - - Sp S (V-68M) 2.02 2.03 2.04 2.03 1.72 1.50 1.65 1.62 Sp S (L-68M) 1.83 1.81 1.83 1.82 1.47 1.36 1.46 1.43 Medusa 2.34 2.48 2.37 2.40 2.35 2.56 2.40 2.44 EAGLE 3.68 3.90 3.77 3.78 3.45 3.67 3.62 3.58 EAGLE-2 4.44 4.78 4.60 4.61 4.23 4.47 4.50 4.40 HASS 4.99 5.29 5.17 5.15 4.84 4.91 5.01 4.92

PLD 1.46 1.70 1.44 1.53 - - - - Lookahead 1.64 1.85 1.69 1.73 - - - - Sp S (V-68M) 2.13 2.61 2.21 2.32 1.73 2.25 1.81 1.93 Sp S (L-68M) 1.83 1.67 1.70 1.73 1.50 1.34 1.45 1.43 Medusa 2.51 2.56 2.70 2.59 2.53 2.89 2.72 2.71 EAGLE 3.86 4.50 4.17 4.18 3.62 4.27 3.98 3.96 EAGLE-2 4.74 5.57 5.17 5.16 4.60 5.41 5.03 5.01 HASS 5.13 6.05 5.55 5.58 4.98 5.86 5.41 5.42

L3 8B EAGLE 2.91 3.66 3.57 3.38 2.67 3.35 3.30 3.11 EAGLE-2 4.21 4.93 4.42 4.52 3.90 4.73 4.30 4.31 HASS 4.68 5.54 5.02 5.08 4.26 5.30 4.85 4.80

L3 70B EAGLE 3.24 4.07 3.79 3.70 3.06 3.85 3.66 3.52 EAGLE-2 4.10 5.02 4.37 4.50 4.00 4.93 4.35 4.43 HASS 4.62 5.78 5.24 5.21 4.59 5.68 5.20 5.16

Table 1: Acceptance lengths τ of different methods on MT-bench, Human Eval, and GSM8K datasets with temperature T {0, 1}. L2 represents LLa MA2-Chat, while L3 represents LLa MA3-Instruct. Sp S stands for Vanilla Speculative Sampling, while V-68M and L-68M represent Vicuna-68M and LLa MA-68M, which are the draft models of Sp S.

P (l) is the auto-regressive probability distribution provided by the target LLM, the Aux-loss consists of the proposed Top-K loss and the feature regression loss (following EAGLE), Head and stand for the language modeling head and the concatenation operation respectively. When training tokens in the entire sequence in parallel, the above formulation adapts the inaccurate features from previous j 1 steps for all positions except the first j 1 positions. Note that compared to EAGLE, HASS takes additional training overhead due to the extra n 1 training steps for adapting inaccurate features, while maintaining the same decoding overhead. To accelerate the training procedure, we propose a modification to the attention mask mechanism, as outlined below:

The first step mirrors the training stage of EAGLE. At timestep t + 1, the draft model takes the target model s feature f (l) t as input and produces the draft feature f (s1) t+1 . In this step, the attention mask remains the same as the original causal mask without any modification.

In the second step, features from the first step are incorporated. For instance, in the selfattention mechanism at timestep t + 1, f (s1) t is used to calculate the current query. Keys and values are derived from f (l) :t f (s1) t , where f (l) :t includes features from timesteps earlier than t. The attention mask is adjusted to ensure that the previous feature seen by f (s1) i is always f (l) i 1, as shown in the HASS Training Step 2 part of Figure 3.

For step j 3, the feature from the previous step f (sj 1) t is utilized to calculate the query at timestep t+1, while keys and values are generated by f (l) :t j+2 f (s1) t j+2 . . . f (sj 1) t .

We empirically demonstrate that the acceleration effect converges with a small n so that the training of HASS is cost-efficient. The actual training overhead of HASS in terms of training speed, computational cost, and GPU memory is investigated in Appendix A.8.

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Temperature = 0 Temperature = 1 Model Method MT-bench Human Eval GSM8K Mean MT-bench Human Eval GSM8K Mean

Sp S (V-68M) 1.35x 1.38x 1.37x 1.37x 1.17x 1.02x 1.12x 1.10x Sp S (L-68M) 1.23x 1.24x 1.25x 1.24x 1.00x 0.94x 0.99x 0.98x Medusa 1.91x 1.96x 2.20x 2.02x 2.00x 2.25x 2.15x 2.13x EAGLE 1.90x 2.10x 2.04x 2.01x 1.50x 1.91x 1.87x 1.76x EAGLE-2 2.66x 3.06x 2.72x 2.81x 2.39x 2.87x 2.54x 2.60x HASS 2.99x 3.41x 3.32x 3.24x 2.70x 3.13x 2.87x 2.90x

Sp S (V-68M) 1.63x 1.98x 1.68x 1.76x 1.33x 1.72x 1.39x 1.48x Sp S (L-68M) 1.41x 1.29x 1.30x 1.33x 1.12x 1.04x 1.11x 1.09x Medusa 2.26x 2.25x 2.71x 2.41x 2.31x 2.47x 2.36x 2.38x EAGLE 1.80x 2.46x 2.41x 2.22x 1.84x 2.10x 2.21x 2.05x EAGLE-2 3.02x 3.64x 3.23x 3.30x 3.04x 3.45x 3.13x 3.21x HASS 3.23x 4.24x 3.48x 3.65x 3.28x 3.78x 3.37x 3.48x

L3 8B EAGLE 1.29x 2.00x 1.85x 1.71x 1.25x 1.41x 1.67x 1.44x EAGLE-2 2.64x 3.31x 2.54x 2.83x 2.39x 2.54x 2.48x 2.47x HASS 2.78x 3.43x 3.06x 3.09x 2.49x 3.05x 2.89x 2.81x

L3 70B EAGLE 2.14x 2.74x 2.42x 2.43x 1.80x 2.34x 2.12x 2.09x EAGLE-2 2.94x 3.98x 3.19x 3.37x 3.02x 3.61x 3.21x 3.28x HASS 3.40x 4.68x 4.08x 4.05x 3.43x 4.25x 3.87x 3.85x

Table 2: Speedup ratios of different methods on MT-bench, Human Eval, and GSM8K datasets with temperature T {0, 1}. L2 represents LLa MA2-Chat, while L3 represents LLa MA3-Instruct. Sp S stands for Vanilla Speculative Sampling, while V-68M and L-68M represent Vicuna-68M and LLa MA-68M, which are the draft models of Sp S.

4 EXPERIMENT

4.1 EXPERIMENTAL SETUP

Target LLMs. LLa MA2-Chat 7/13B and LLa MA3-Instruct 8/70B.

Tasks. We conduct evaluations on three generation tasks. For multi-turn conversation, code generation, and mathematical reasoning tasks, we choose the MT-bench (Zheng et al., 2024), Human Eval (Chen et al., 2021), and GSM8K (Cobbe et al., 2021) datasets, respectively. The batch size is set as 1 under all the experiments following Leviathan et al. (2023) and Zhou et al. (2024).

Metrics. HASS neither fine-tunes the target LLMs weights during training nor relaxes the acceptance conditions during decoding, making it a lossless acceleration method. Thus, the generation quality is promised with no need for evaluation. We use the following two metrics to measure the acceleration performance:

Speedup Ratio: The actual test speedup ratio relative to vanilla auto-regressive decoding.

Acceptance Length τ: The average number of tokens generated per drafting-verification cycle, indicating the number of tokens accepted by the target LLM from the draft model.

Note that the speedup ratio is sensitive to the hardware due to variations in computing power, and the acceptance length may be slightly affected by hardware due to numerical errors. Therefore, all inference processes are conducted on NVIDIA H800 GPU.

Comparisons. The vanilla auto-regressive decoding is taken as the baseline, which serves as the benchmark for speedup ratios (1.00x). We compare HASS with recent lossless speculative sampling methods, including PLD (Saxena, 2023), Lookahead (Fu et al., 2023), Vanilla Speculative Sampling (Chen et al., 2023), Medusa (Cai et al., 2024), EAGLE (Li et al., 2024b), and EAGLE-2 (Li et al., 2024c). PLD and Lookahead are free of traning, which respectively use string matched from the prompt and cached n-grams as draft tokens instead of generating draft tokens from a draft model s predicted probability distribution. Therefore, the results of PLD and Lookahead under temperature = 1 are not reported in Table 1.

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Implementation. Our code is built based on EAGLE-2 s open-source repository1. Experiments on EAGLE and EAGLE-2 reuse draft model weights trained by Li et al. (2024b). For harmonized objective distillation, K is set as 10, and the loss of harmonized objective distillation is added to EAGLE s original loss with a coefficient of w = 1.0. For harmonized context alignment, the draft model is aligned for 3 steps during training. For dynamic tree structure, we set the total number of draft tokens to 60 for all experiments with a draft tree depth of 6. We keep other settings, such as the fixed training dataset, i.e., the Share GPT2 dataset with 68,000 dialogues, and the optimizer, consistent with EAGLE-2.

4.2 EFFECTIVENESS & ABLATION STUDY

In this section, we first evaluate the effectiveness of HASS by comparing it with existing speculative sampling methods on acceptance length and speedup ratio. Then, we conduct ablation studies on harmonized objective distillation and harmonized context alignment. Inspired by Yi et al. (2024), we further conduct experiments by training on different proportions of the Share GPT dataset to investigate HASS s scalability in the face of data sparsity (see Appendix A.6), and by evaluating on the translation tasks to investigate HASS s robustness across different task types (see Appendix A.7). As shown from the results, HASS is more scalable than EAGLE-2 with fewer training data and achieves promising improvements over EAGLE-2 on translation tasks consistent with results on MT-bench, Human Eval, and GSM8K.

4.2.1 EFFECTIVENESS

We present different methods acceptance lengths and speedup ratios across three datasets in Tables 1 and 2, respectively. HASS performs the largest acceptance length and highest speedup ratio across all datasets and LLMs we tested. Most methods achieve their best performance on the Human Eval dataset, as the fixed templates in the code generation task are easier to draft and accelerate. Though PLD and Lookahead are free of training, they consistently show poorer performance than Medusa, EAGLE, EAGLE-2, and HASS.

4.2.2 ABLATION STUDY ON HARMONIZED OBJECTIVE DISTILLATION

0 0.1 0.2 0.5 1.0 2.0 Weight

Acceptance Length

1 5 10 50 100 K

T = 0 T = 1

Figure 4: Acceptance lengths τ of HASS with varied Ks and weights of the Top-K loss. The results are conducted on LLa MA2-Chat 7B and averaged over MT-bench, Human Eval, and GSM8K datasets with temperature T {0, 1}.

We first study the effects of different K and the weight w of the Top-K loss by varying these hyper-parameters and summarize the results in Figure 4. Training with the Top-K loss (w > 0) always improves performance compared to training without the Top-K loss (w = 0). HASS achieves the largest acceptance length when w = 0.5. A small value of K may result in performance degeneration, as the draft model only focuses on the token with the highest probability and consequently neglects other potential tokens. With a larger K, the Top-K loss generally brings better results, while the acceptance length is the largest when K = 5.

Since the harmonized objective distillation can be implemented with any loss function that focuses on the most probable tokens w.r.t. the target distribution, we further consider the following loss functions and compare them with the Top-K Loss:

Top-P Loss, where the ˆΩis formed by the most probable tokens whose cumulative probability is just larger than P.

Normed Top-K Loss, where the target and draft distributions are both normalized over ˆΩ. The normalization operation can be either linear or softmax.

1https://github.com/Safe AILab/EAGLE 2https://huggingface.co/datasets/Aeala/Share GPT Vicuna unfiltered

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Loss Function Temperature = 0 Temperature = 1 Mean

Top-K Loss 4.99 4.84 4.92 Top-P Loss 5.03 4.76 4.90 Normed Top-K Loss (Linear) 4.97 4.83 4.90 Normed Top-K Loss (Softmax) 4.98 4.72 4.85 Bi-directional Top-K Loss 4.99 4.72 4.86 Recall@k Surrogate Loss 4.97 4.76 4.87 Bi LD Loss 5.04 4.75 4.90

Table 3: Acceptance lengths τ of HASS with different kinds of loss functions for harmonized objective distillation. The results are conducted on LLa MA2-Chat 7B over the MT-bench dataset with temperature T {0, 1}.

Bi-directional Top-K Loss, where the distillation is conducted over the most probable tokens w.r.t. the target distribution as well as the draft distribution.

Recall@k Surrogate Loss (Patel et al., 2022), where a smooth approximation of the recall metric is obtained and is differentiable for direct optimization.

Bi LD Loss (Li et al., 2024a), where the internal logits ranking information is captured by constructing logits differences with long-tail noise filtered out.

After searching the optimal hyper-parameters for each of the compared loss functions, we summarize their best results in Table 3. Bi LD loss outperforms other loss functions under temperature T = 0, while Top-K loss outperforms others under temperature T = 1. Generally, Top-K loss shows the best performance. A better loss function may exist than Top-K loss to exploit the target LLM further. We leave this topic in future works.

We also conduct an experiment with LLa MA2-Chat 7B, where the fixed training dataset is replaced by the dataset generated by the target LLM (see Appendix A.4). We observe that when using nongreedy decoding, the acceptance length increases from 4.92 to 5.19 averaging over three datasets. Therefore, information obtained from harmonized objective distillation is not equivalent to direct distillation from target-model-generated data.

4.2.3 ABLATION STUDY ON HARMONIZED CONTEXT ALIGNMENT

Aligning Step MT-bench Human Eval GSM8K Mean

EAGLE-2 + Top-K 4.59 4.97 4.77 4.78 HASS Align-2 4.95 5.25 5.12 5.11 HASS Align-3 4.99 5.29 5.17 5.15 HASS Align-4 4.99 5.30 5.18 5.16 HASS Align-5 4.98 5.26 5.09 5.11

EAGLE-2 + Top-K 4.46 4.61 4.64 4.57 HASS Align-2 4.71 4.89 4.98 4.86 HASS Align-3 4.84 4.91 5.01 4.92 HASS Align-4 4.77 4.93 5.03 4.91 HASS Align-5 4.71 4.92 4.95 4.86

Table 4: Acceptance lengths τ of HASS with varied aligning steps in the harmonized context alignment. The results are conducted on LLa MA2-Chat 7B with temperature T {0, 1}.

We propose the harmonized context alignment, which eliminates the feature inconsistency of draft models between the training and decoding stages. To study the effect of increasing the aligning steps in the harmonized context alignment, we conduct experiments by varying the step number and summarize the results in Table 4.

As the first training step of HASS is the same as EAGLE-2, we continually train EAGLE-2 s draft model weights with the Top-K loss and consider it the baseline. Without harmonized context alignment (EAGLE-2 + Top-K), the draft model performs the worst across all datasets. Training with 3/4

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steps of harmonized context alignment generally obtains the most considerable acceptance length. When training with 5 steps of context alignment, the acceptance length decreases. We believe this is caused by the draft model s limited capacity, as it predicts less accurately on former steps tokens when paying too much attention to the latter ones. Figure 5 shows the acceptance rate α across speculation steps on the MT-bench dataset following Li et al. (2024c). In later speculation steps, HASS performs better acceptance rates than EAGLE-2, demonstrating the effectiveness of harmonized context alignment.

01- 23- 45- LLa MA2-Chat 7B T=0

01- 23- 45- LLa MA2-Chat 13B T=0

01- 23- 45- LLa MA3-Instruct 8B T=0

01- 23- 45- LLa MA3-Instruct 70B T=0

HASS EAGLE-2

01- 23- 45- LLa MA2-Chat 7B T=1

01- 23- 45- LLa MA2-Chat 13B T=1

01- 23- 45- LLa MA3-Instruct 8B T=1

01- 23- 45- LLa MA3-Instruct 70B T=1

Figure 5: Acceptance rates α (%) of HASS and EAGLE-2 across different speculation steps on the MT-bench dataset with temperature T {0, 1}.

As shown in Figure 5, the acceptance rates of HASS decrease compared to EAGLE-2 on LLa MA2Chat 13B and LLa MA3-Instruct 70B at the first step (0-α). The draft models degrade on the first speculation step with much attention paid to the latter speculation steps, while the first step s acceptance rates are crucial to larger acceptance lengths. We conduct experiments to emphasize the significance of former speculation steps by reweighting the training loss from each step with a factor β. In specific, the step j s training loss will be multiplied by βj 1. Table 5 and Figure 6 show the acceptance lengths and acceptance rates of HASS with different reweight factors on LLa MA3-Instruct 70B over the MT-bench dataset, respectively. With the factor β decreasing from 1.0 to 0.5, HASS achieves better acceptance lengths with different temperatures. Correspondingly, we perceive that the acceptance rate at the first speculation step is consistently higher with a smaller β, while the acceptance rates at the latter speculation steps generally decline. When the factor β decreases to 0.3, too much emphasis is assigned to the first speculation step, leading to deterioration in acceptance length. Since the further exploration of an appropriate trade-off between different speculation steps is out of this paper s scope, we leave it for future work.

Reweight Factor β T = 0 T = 1 Mean

1.0 (Default) 4.62 4.59 4.61 0.7 4.65 4.61 4.63 0.5 4.67 4.62 4.65 0.3 4.65 4.61 4.63

Table 5: Acceptance lengths τ of HASS with different reweight factors β for harmonized context alignment. The results are conducted on LLa MA3-Instruct 70B over the MT-bench dataset with temperature T {0, 1}.

01- 23- 45- LLa MA3-Instruct 70B T=0

01- 23- 45- LLa MA3-Instruct 70B T=1

= 1.0 = 0.7

= 0.5 = 0.3

Figure 6: Acceptance rates α (%) of HASS with different reweight factors β for harmonized context alignment. The results are conducted on LLa MA3-Instruct 70B over the MT-bench dataset with temperature T {0, 1}.

5 RELATED WORK

There have been a number of works on improving the acceptance rate of speculative sampling while maintaining the target distribution. Most of them fall into two categories. (1) The former category

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is aligned training that tries to obtain draft models aligned with the target LLM before the decoding stage. Zhou et al. (2024) propose a knowledge distillation approach and study several strategies to improve the alignment. Li et al. (2024b) demonstrate that hidden states of the target LLM as input of the draft model provide extra feature uncertainty information. Xiao et al. (2024) also utilize hidden states of the target LLM and introduce an RNN-based draft model design that achieves a comparable acceptance rate. GLIDE (Du et al., 2024) instead reuses the KV cache of the target LLM. It also notices the context misalignment when using information from the target LLM, but the proposed blockwise attention mask method can not solve the misalignment completely. (2) The latter category is efficient decoding, which designs sophisticated decoding strategies to utilize concurrency efficiently. Miao et al. (2024) propose to utilize multiple draft models and design a treebased attention mechanism to verify multiple draft sequences efficiently. Li et al. (2024c) introduce a dynamic structure to save computation by pruning inefficient paths in the draft tree. Sun et al. (2024) study improving the verification stage through optimal transportation. However, these works tend to only consider training or decoding, ignoring the linkage of these two stages. This work instead aims to link training and decoding, leading to harmonized speculative sampling.

6 CONCLUSION

This paper introduces HASS, a harmonized speculative sampling solution that addresses disharmonies between training and decoding on their objectives and contexts. Compared to its closest baseline, EAGLE-2, HASS improves the acceptance rate without any inference overhead. Experiments conducted on LLa MA2-Chat 7/13B and LLa MA3-Instruct 8/70B demonstrate the effectiveness and efficiency of HASS. Averaging on MT-bench, Human Eval, and GSM8K, HASS is 2.81x4.05x faster than vanilla auto-regressive decoding, 8%-20% faster than EAGLE-2.

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A.1 IMPLEMENTATION OF HARMONIZED CONTEXT ALIGNMENT

We present the pseudo code of harmonized context alignment, which is implemented without the customized attention mask, for better understanding. The actual implementation in our experiments is achieved by the customized attention mask as shown in Figure 3.

1 def train_batch( 2 draft_model, # draft model 3 lm_head, # language model head 4 optimizer, # optimizer 5 forward_num, # aligning steps in harmonized context alignment 6 hidden_states_target, # target LLM s feature 7 input_ids, # input tokens 8 ): 9 hidden_states_draft_list = [] 10 for forward_idx in range(forward_num): 11 optimizer.zero_grad() 12 predict = draft_model(hidden_states_target, input_ids, hidden_states_draft_list) 13 hidden_states_draft = torch.cat([hidden_states_target[:, :1], predict[:, :-1]], dim=1).detach() 14 hidden_states_draft_list.append(hidden_states_draft) 15 target_head, pred_head = lm_head(hidden_states_target), lm_head(predict) 16 loss = feature_loss(hidden_states_target, predict) + logit_loss(target_head, pred_head) 17 loss.backward() 18 optimizer.step()

1 def attention( 2 hidden_states_target, # target LLM s feature 3 attention_mask, # causal attention mask 4 hidden_states_draft_list, # list of draft model s features 5 ): 6 bs, seq_len = hidden_states_target.shape[0], hidden_states_target.shape[1] 7 query = q_proj(hidden_states_draft_list[-1]) if hidden_states_draft_list else q_proj(hidden_states_target) 8 key_t, value_t = k_proj(hidden_states_target), v_proj(hidden_states_target) 9 attn_weight = torch.matmul(query, key_t.transpose(2, 3)) / math.sqrt(query.shape[-1]) + attention_mask 10 indices = torch.arange(seq_len) 11 for i, hidden_states_draft in enumerate(hidden_states_draft_list[::-1]): 12 key_d, ind_q, ind_k = k_proj(hidden_states_draft), indices[i:], indices[:seq_len - i] 13 attn_weight_d = torch.matmul(query, key_d.transpose(2, 3)) / math.sqrt(query.shape[-1]) 14 attn_weight[:, :, ind_q, ind_k] = attn_weight_d[:, :, ind_q, ind_k] 15 attn_weight_normed = F.softmax(attn_weight, dim=-1) 16 attn_output = torch.matmul(attn_weight_normed, value_t) 17 for i, hidden_states_draft in enumerate(hidden_states_draft_list[::-1]): 18 value_d, ind_q, ind_k = v_proj(hidden_states_draft), indices[i:], indices[:seq_len - i] 19 attn_output[:, :, ind_q] += attn_weight[:, :, ind_q, ind_k][..., None] * (value_d[:, :, ind_k] - value_t[:, :, ind_k]) 20 attn_output = o_proj(attn_output.transpose(1, 2).reshape(bs, seq_len, -1)) 21 return attn_output

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A.2 HARMONIZED CONTEXT ALIGNMENT ON TOKENS

In this section, we attempt to verify whether applying token alignment as well as feature alignment brings better performance. In specific, we use the tokens generated by the draft model for training in harmonized context alignment instead of using the tokens from training data. We apply feature and token alignment to EAGLE-2 s draft model weights and summarize the results in Table 6 and Figure 7.

Temperature = 0 Temperature = 1 Mean

EAGLE-2 4.44 4.23 4.34 Feature Only 4.83 4.60 4.72 Feature + Token (0.1) 4.81 4.57 4.69 Feature + Token (0.2) 4.78 4.51 4.65 Feature + Token (1.0) 4.28 4.11 4.20

Table 6: Acceptance lengths τ of applying feature and token alignment to EAGLE-2 s draft model weights, where Token (x) denotes tokens from training data being replaced by draft-modelgenerated tokens with a probability of x. The results are conducted on LLa MA2-Chat 7B over the MT-bench dataset with temperature T {0, 1}.

01- 23- 45- LLa MA2-Chat 7B T=0

01- 23- 45- LLa MA2-Chat 7B T=1

Feature Only Feature + Token (0.1) Feature + Token (0.2)

Figure 7: Acceptance rates α (%) of applying feature and token alignment to EAGLE-2 s draft model weights, where Token (x) denotes tokens from training data being replaced by draft-modelgenerated tokens with a probability of x. The results are conducted on LLa MA2-Chat 7B over the MT-bench dataset with temperature T {0, 1}.

Feature only alignment brings the best performance, while adding token alignment leads to degeneration. With the probability of applying token alignment increasing from 0.1 to 1.0, the acceptance length decreases consistently. As shown in Figure 7, more token alignment generally causes lower acceptance rates. As a result, training with tokens generated by the draft model in harmonized context alignment hurts the acceleration performance.

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A.3 HYPER-PARAMETERS OF TOP-K LOSS

We conduct an ablation study on Top-K loss s hyper-parameters, i.e., K and w, in Section 4.2.2 and show the averaged acceptance lengths over three datasets in Figure 4. Here, we present the speedup ratios and acceptance lengths of HASS with varied Ks and ws in Table 7.

MT-bench Human Eval GSM8K Mean

K w Speedup τ Speedup τ Speedup τ Speedup τ

1 1.0 2.89x 4.94 3.24x 5.19 3.11x 5.10 3.08x 5.08 5 1.0 2.90x 5.00 3.44x 5.29 3.33x 5.18 3.22x 5.16 10 1.0 2.99x 4.99 3.41x 5.29 3.32x 5.17 3.24x 5.15 50 1.0 2.85x 5.01 3.46x 5.29 3.41x 5.17 3.24x 5.16 100 1.0 2.93x 5.00 3.45x 5.29 3.45x 5.18 3.28x 5.16 10 0.0 2.77x 4.93 3.38x 5.22 3.18x 5.11 3.11x 5.09 10 0.1 2.98x 4.96 3.40x 5.26 3.51x 5.16 3.30x 5.13 10 0.2 2.87x 4.98 3.41x 5.29 3.35x 5.16 3.21x 5.14 10 0.5 3.00x 5.02 3.32x 5.31 3.50x 5.18 3.27x 5.17 10 2.0 2.94x 4.98 3.37x 5.29 3.34x 5.17 3.22x 5.15

1 1.0 2.58x 4.70 2.79x 4.80 2.83x 4.95 2.73x 4.82 5 1.0 2.64x 4.81 3.13x 4.94 2.93x 5.02 2.90x 4.92 10 1.0 2.70x 4.84 3.13x 4.91 2.87x 5.01 2.90x 4.92 50 1.0 2.62x 4.77 3.01x 4.88 2.99x 5.03 2.87x 4.89 100 1.0 2.66x 4.74 3.14x 4.97 2.90x 5.03 2.90x 4.91 10 0.0 2.61x 4.71 2.76x 4.84 2.79x 4.96 2.72x 4.84 10 0.1 2.69x 4.75 3.05x 4.94 2.87x 5.00 2.87x 4.90 10 0.2 2.66x 4.75 3.16x 4.95 2.88x 5.01 2.90x 4.90 10 0.5 2.68x 4.80 3.15x 4.93 2.96x 5.03 2.93x 4.92 10 2.0 2.68x 4.75 3.11x 4.89 2.89x 5.03 2.89x 4.89

Table 7: Speedup ratios and acceptance lengths τ of HASS with varied Ks and ws of the Top-K loss on LLa MA2-Chat 7B over MT-bench, Human Eval, and GSM8K datasets with temperature T {0, 1}.

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A.4 SELF-DISTILLATION

In the main text, we use the fixed Share GPT dataset to train draft models for a fair comparison with EAGLE and EAGLE-2. Following existing speculative sampling methods (Zhou et al., 2024; Cai et al., 2024), we further use target-model-generated outputs to distill the draft model from the target model s real output distribution, dubbed as self-distillation. In specific, we feed the prompts from the Share GPT dataset into the target models recursively with temperature set to 0 and collect the responses as multi-turn conversations for self-distillation following Li et al. (2024b). To study the effect of self-distillation, we conduct experiments on HASS and EAGLE-2 by training the draft model with fixed data or model-generated data and summarize the results in Table 8.

MT-bench Human Eval GSM8K Mean

Model Method Data Speedup τ Speedup τ Speedup τ Speedup τ

L2 7B EAGLE-2 F 2.66x 4.44 3.06x 4.78 2.72x 4.60 2.81x 4.61 MG 2.86x 4.70 3.30x 5.12 3.03x 5.00 3.06x(+0.25) 4.94(+0.33)

HASS F 2.99x 4.99 3.41x 5.29 3.32x 5.17 3.24x 5.15 MG 3.13x 5.25 3.85x 5.70 3.40x 5.57 3.46x(+0.22) 5.51(+0.36)

L2 13B EAGLE-2 F 3.02x 4.74 3.64x 5.57 3.23x 5.17 3.30x 5.16 MG 3.04x 4.80 3.47x 5.46 3.19x 5.16 3.23x(-0.07) 5.14(-0.02)

HASS F 3.23x 5.13 4.24x 6.05 3.48x 5.55 3.65x 5.58 MG 3.34x 5.27 4.42x 6.00 3.63x 5.61 3.80x(+0.15) 5.63(+0.05)

L2 7B EAGLE-2 F 2.39x 4.23 2.87x 4.47 2.54x 4.50 2.60x 4.40 MG 2.49x 4.38 2.94x 4.73 2.69x 4.80 2.71x(+0.11) 4.64(+0.24)

HASS F 2.70x 4.84 3.13x 4.91 2.87x 5.01 2.90x 4.92 MG 2.75x 4.97 3.39x 5.24 3.13x 5.35 3.09x(+0.19) 5.19(+0.27)

L2 13B EAGLE-2 F 3.04x 4.60 3.45x 5.41 3.13x 5.03 3.21x 5.01 MG 3.08x 4.63 3.23x 5.25 3.04x 4.95 3.12x(-0.09) 4.94(-0.07)

HASS F 3.28x 4.98 3.78x 5.86 3.37x 5.41 3.48x 5.42 MG 3.33x 5.02 3.76x 5.80 3.60x 5.42 3.56x(+0.08) 5.41(-0.01)

Table 8: Speedup ratios and acceptance lengths τ of HASS and EAGLE-2 with fixed or targetmodel-generated training data. F and MG stand for Fixed and Model-Generated , respectively. L2 represents LLa MA2-Chat.

On LLa MA2-Chat 7B, self-distillation consistently brings improvements for HASS and EAGLE2. On LLa MA2-Chat 13B, self-distillation only achieves marginally better or comparable results, which is consistent with the observation from Li et al. (2024b) ( data from the target LLM marginally improves performance in its section 4.3.3). Especially, the acceptance lengths of the self-distilled HASS are lower than that of the vanilla HASS on the Human Eval dataset, while both the speedup ratios and acceptance lengths of the self-distilled EAGLE-2 are lower than that of the vanilla EAGLE-2 on Human Eval and GSM8K datasets. It may be due to the code generation dataset Human Eval and the mathematical reasoning dataset GSM8K being less similar to the training dataset Share GPT compared with MT-bench.

HASS outperforms EAGLE-2 on either fixed training data or model-generated training data. It is noted that HASS trained on the fixed dataset even achieves better performance than EAGLE-2 trained on the model-generated data consistently. With self-distillation, HASS consistently achieves more improvement or less degeneration in terms of the acceptance length compared with EAGLE-2.

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A.5 DRAFTING HYPER-PARAMETERS

Li et al. (2024c) find that the draft token s confidence score is strongly positively correlated with the acceptance rate, and accordingly propose the context-aligned dynamic draft tree, which can be dynamically adjusted with two hyper-parameters: depth and number of tokens . Depth decides the draft tree s depth during the expansion phase, while number of tokens decides how many draft tokens will be kept during the reranking phase. Increasing both these hyper-parameters surely leads to a larger acceptance length. Nevertheless, sending more draft tokens into the target model for verification causes a higher overhead in real applications. Therefore, we vary these hyper-parameters and report the speedup ratios in Table 9 to find a better trade-off.

Depth 5 6 7 8 9 # Tokens 40 60 80 100 40 60 80 100 40 60 80 100 40 60 80 100 40 60 80 100

T=0 L2 7B EAGLE-2 2.48x 2.78x 2.61x 2.69x 2.69x 2.66x 2.70x 2.79x 2.71x 2.86x 2.91x 2.95x 2.60x 2.60x 2.88x 2.89x 2.28x 2.54x 2.55x 2.65x HASS-MG 3.09x 3.02x 3.04x 3.22x 3.08x 3.13x 3.19x 3.22x 3.14x 3.11x 3.29x 3.27x 3.07x 3.16x 3.31x 3.32x 2.78x 2.75x 2.95x 3.03x

L2 13B EAGLE-2 2.63x 2.96x 3.04x 3.06x 3.01x 3.02x 3.18x 3.22x 2.78x 3.12x 3.14x 3.24x 2.98x 3.12x 3.19x 3.26x 2.49x 2.64x 2.69x 2.72x HASS-MG 3.25x 3.31x 3.24x 3.25x 3.33x 3.34x 3.49x 3.40x 3.19x 3.42x 3.36x 3.40x 3.15x 3.40x 3.40x 3.37x 2.70x 2.74x 3.09x 3.02x

T=1 L2 7B EAGLE-2 2.31x 2.37x 2.55x 2.36x 2.42x 2.39x 2.33x 2.40x 2.49x 2.66x 2.65x 2.44x 2.36x 2.48x 2.38x 2.64x 2.29x 2.22x 2.27x 2.42x HASS-MG 2.79x 2.89x 2.86x 2.88x 2.72x 2.75x 2.92x 2.82x 2.83x 2.76x 2.81x 2.75x 2.49x 2.68x 2.77x 2.77x 2.30x 2.35x 2.58x 2.50x

L2 13B EAGLE-2 2.92x 3.11x 2.88x 2.79x 3.06x 3.04x 3.16x 2.93x 3.05x 3.14x 3.14x 3.11x 3.00x 3.13x 3.15x 2.98x 2.61x 2.72x 2.65x 2.54x HASS-MG 3.24x 3.30x 3.27x 3.19x 3.33x 3.33x 3.40x 3.28x 3.19x 3.26x 3.24x 3.26x 3.15x 3.26x 3.19x 3.17x 2.62x 2.77x 2.74x 2.84x

Table 9: Speedup ratios of EAGLE-2 and HASS-MG with varied depths and numbers of tokens on the MT-bench dataset with temperature T {0, 1}, where HASS-MG denotes HASS trained with self-distillation. L2 represents LLa MA2-Chat.

When depth = 5, the acceptance length is relatively small. When depth = 9, the verification overhead is extremely high. Thus, neither of these settings achieves a promising speedup ratio. For both HASS-MG and EAGLE-2, the best performances are achieved when depth {6, 7, 8} and # tokens {60, 80, 100}. HASS-MG consistently obtains a superior performance compared with EAGLE-2 through hyper-parameter tuning across different LLMs and temperatures.

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A.6 NUMBER OF TRAINING TOKENS

Inspired by Yi et al. (2024), we randomly sample different proportions of the training dataset, i.e., the Share GPT dataset with 68,000 dialogues, to investigate the influences of training token numbers. In specific, we train the draft models of HASS and EAGLE-2 with 1/8, 1/4, 1/2 and the entire Share GPT dataset and summarize the results in Figure 8 and Table 10.

1/8 1/4 1/2 1/1 LLa MA2-Chat 7B

Acceptance Length

1/8 1/4 1/2 1/1 LLa MA2-Chat 13B

1/8 1/4 1/2 1/1 LLa MA3-Instruct 8B

1/8 1/4 1/2 1/1 LLa MA3-Instruct 70B

1/8 1/4 1/2 1/1 LLa MA2-Chat 7B

Speedup Ratio

1/8 1/4 1/2 1/1 LLa MA2-Chat 13B

1/8 1/4 1/2 1/1 LLa MA3-Instruct 8B

1/8 1/4 1/2 1/1 LLa MA3-Instruct 70B

HASS T=0 HASS T=1

EAGLE-2 T=0 EAGLE-2 T=1

Figure 8: Acceptance lengths τ and speedup ratios of HASS and EAGLE-2 averaging across MTbench, Human Eval, and GSM8K with different proportions of training dataset, i.e., the Share GPT dataset with 68,000 dialogues.

As shown from Figure 8, HASS consistently outperforms EAGLE-2 under different proportions of training dataset with temperature T {0, 1}. HASS with merely 1/4 training dataset achieves better or comparable performance compared to EAGLE-2 with the entire training dataset, which demonstrates HASS s superior data exploitation and scalability obtained through further aligning on objectives and contexts between training and decoding. The speedup ratio and acceptance length of HASS and EAGLE-2 are approximately logarithmically proportional to the scale of training data, which is consistent with the finding in Yi et al. (2024). As shown from Table 10, the decrease in training data contributes to more severe degradation on EAGLE-2 than that on HASS, reflecting HASS s robustness to data sparsity.

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MT-bench Human Eval GSM8K Mean

Model Method Proportion Speedup τ Speedup τ Speedup τ Speedup τ

1/8 1.74x 3.06 2.00x 3.39 2.12x 3.24 1.95x 3.23 1/4 2.08x 3.64 2.49x 3.93 2.21x 3.81 2.26x 3.79 1/2 2.36x 4.11 2.76x 4.46 2.64x 4.28 2.59x 4.28 1/1 2.66x 4.44 3.06x 4.78 2.72x 4.60 2.81x 4.61

1/8 2.32x 3.92 2.69x 4.30 2.56x 4.12 2.52x 4.11 1/4 2.64x 4.42 3.05x 4.76 2.70x 4.59 2.80x 4.59 1/2 2.85x 4.79 3.36x 5.05 3.18x 4.94 3.13x 4.93 1/1 2.99x 4.99 3.41x 5.29 3.32x 5.17 3.24x 5.15

1/8 1.88x 3.25 2.39x 3.79 2.27x 3.57 2.18x 3.54 1/4 2.38x 3.82 2.85x 4.53 2.79x 4.25 2.67x 4.20 1/2 2.74x 4.32 3.46x 5.18 3.08x 4.81 3.09x 4.77 1/1 3.02x 4.74 3.64x 5.57 3.23x 5.17 3.30x 5.16

1/8 2.59x 4.01 3.37x 4.79 3.01x 4.43 2.99x 4.41 1/4 2.89x 4.55 3.59x 5.51 3.18x 5.00 3.22x 5.02 1/2 3.16x 4.90 4.20x 5.86 3.41x 5.31 3.59x 5.36 1/1 3.23x 5.13 4.24x 6.05 3.48x 5.55 3.65x 5.58

1/8 1.54x 2.77 2.00x 3.12 1.61x 2.83 1.72x 2.91 1/4 1.89x 3.18 2.30x 3.59 2.08x 3.35 2.09x 3.37 1/2 2.24x 3.68 2.64x 4.16 2.42x 3.91 2.43x 3.92 1/1 2.64x 4.21 3.31x 4.93 2.54x 4.42 2.83x 4.52

1/8 2.14x 3.61 2.84x 4.22 2.30x 3.78 2.43x 3.87 1/4 2.46x 4.04 3.24x 4.83 2.48x 4.27 2.73x 4.38 1/2 2.72x 4.43 3.38x 5.28 2.99x 4.71 3.03x 4.81 1/1 2.78x 4.68 3.43x 5.54 3.06x 5.02 3.09x 5.08

1/8 2.09x 2.87 2.72x 3.45 2.29x 2.92 2.37x 3.08 1/4 2.47x 3.33 3.25x 4.01 2.71x 3.46 2.81x 3.60 1/2 2.76x 3.74 3.71x 4.57 3.11x 3.96 3.19x 4.09 1/1 2.94x 4.10 3.98x 5.02 3.19x 4.37 3.37x 4.50

1/8 2.73x 3.68 3.79x 4.61 3.10x 3.95 3.21x 4.08 1/4 3.05x 4.12 4.23x 5.18 3.56x 4.54 3.61x 4.61 1/2 3.27x 4.40 4.52x 5.57 3.87x 4.92 3.89x 4.96 1/1 3.40x 4.62 4.68x 5.78 4.08x 5.24 4.05x 5.21

1/8 1.60x 2.99 1.90x 3.21 1.86x 3.17 1.79x 3.12 1/4 1.89x 3.48 2.28x 3.71 2.27x 3.71 2.15x 3.63 1/2 2.26x 3.93 2.57x 4.14 2.46x 4.17 2.43x 4.08 1/1 2.39x 4.23 2.87x 4.47 2.54x 4.50 2.60x 4.40

1/8 2.19x 3.82 2.50x 4.06 2.41x 4.03 2.37x 3.97 1/4 2.50x 4.27 2.84x 4.46 2.61x 4.52 2.65x 4.42 1/2 2.63x 4.56 3.10x 4.75 2.81x 4.81 2.85x 4.71 1/1 2.70x 4.84 3.13x 4.91 2.87x 5.01 2.90x 4.92

1/8 1.91x 3.16 2.18x 3.71 2.23x 3.46 2.11x 3.44 1/4 2.31x 3.68 2.72x 4.40 2.77x 4.11 2.60x 4.06 1/2 2.78x 4.20 3.19x 5.00 3.08x 4.67 3.02x 4.62 1/1 3.04x 4.60 3.45x 5.41 3.13x 5.03 3.21x 5.01

1/8 2.49x 3.94 2.98x 4.70 2.99x 4.33 2.82x 4.32 1/4 2.87x 4.43 3.40x 5.35 3.11x 4.87 3.13x 4.88 1/2 3.22x 4.75 3.70x 5.69 3.26x 5.18 3.39x 5.21 1/1 3.28x 4.98 3.78x 5.86 3.37x 5.41 3.48x 5.42

1/8 1.51x 2.64 1.63x 3.01 1.64x 2.81 1.59x 2.82 1/4 1.77x 2.99 1.87x 3.51 1.92x 3.29 1.85x 3.26 1/2 1.90x 3.40 2.25x 4.03 2.38x 3.82 2.18x 3.75 1/1 2.39x 3.90 2.54x 4.73 2.48x 4.30 2.47x 4.31

1/8 1.96x 3.42 2.31x 4.10 2.25x 3.72 2.17x 3.75 1/4 2.22x 3.77 2.51x 4.63 2.45x 4.18 2.39x 4.19 1/2 2.43x 4.10 2.96x 5.07 2.82x 4.56 2.74x 4.58 1/1 2.49x 4.26 3.05x 5.30 2.89x 4.85 2.81x 4.80

1/8 2.16x 2.85 2.52x 3.35 2.19x 2.91 2.29x 3.04 1/4 2.26x 3.29 3.01x 3.94 2.57x 3.44 2.61x 3.56 1/2 2.70x 3.67 3.37x 4.47 3.00x 3.94 3.02x 4.03 1/1 3.02x 4.00 3.61x 4.93 3.21x 4.35 3.28x 4.43

1/8 2.80x 3.70 3.46x 4.52 2.98x 3.93 3.08x 4.05 1/4 3.10x 4.10 3.90x 5.11 3.41x 4.51 3.47x 4.57 1/2 3.28x 4.36 4.15x 5.46 3.72x 4.91 3.72x 4.91 1/1 3.43x 4.59 4.25x 5.68 3.87x 5.20 3.85x 5.16

Table 10: Speedup ratios and acceptance lengths τ of HASS and EAGLE-2 with different proportions of training dataset, i.e., the Share GPT dataset with 68,000 dialogues. L2 represents LLa MA2Chat, while L3 represents LLa MA3-Instruct.

Published as a conference paper at ICLR 2025

A.7 EVALUATION ON TRANSLATION TASKS

To investigate the robustness of HASS across different task types, we further evaluate HASS and EAGLE-2 on five translation tasks3 by following Yi et al. (2024). It is noted that both HASS and EAGLE-2 are trained on the fixed Share GPT dataset without adaptation for translation tasks. We conduct experiments on LLa MA2-Chat 7/13B and LLa MA3-Instruct 8/70B and summarize the results in Table 11.

De En Fr En Ja En Ru En Zh En Mean Model Method Speedup τ Speedup τ Speedup τ Speedup τ Speedup τ Speedup τ

L2 7B EAGLE-2 2.58x 4.06 2.47x 3.99 2.46x 3.79 2.23x 3.48 2.39x 3.68 2.43x 3.80 HASS 3.15x 4.55 2.99x 4.60 2.97x 4.26 2.64x 3.82 2.83x 4.10 2.92x 4.27

L2 13B EAGLE-2 2.95x 4.51 3.00x 4.41 2.67x 3.80 2.61x 3.65 2.60x 3.92 2.77x 4.06 HASS 3.63x 5.01 3.64x 4.94 3.05x 4.07 3.06x 4.03 3.02x 4.22 3.28x 4.45

L3 8B EAGLE-2 2.59x 3.89 2.34x 3.96 1.91x 2.97 1.90x 3.25 2.03x 3.17 2.15x 3.45 HASS 2.98x 4.30 2.79x 4.21 2.21x 3.19 2.28x 3.53 2.32x 3.38 2.52x 3.72

L3 70B EAGLE-2 3.10x 4.17 3.13x 4.07 2.35x 3.16 2.79x 3.76 2.52x 3.39 2.78x 3.71 HASS 3.75x 4.71 3.61x 4.47 2.67x 3.41 3.39x 4.25 2.84x 3.72 3.25x 4.12

L2 7B EAGLE-2 2.26x 3.86 2.41x 3.91 2.09x 3.58 1.98x 3.34 2.25x 3.61 2.20x 3.66 HASS 2.80x 4.44 2.99x 4.59 2.66x 4.11 2.53x 3.74 2.65x 4.05 2.73x 4.19

L2 13B EAGLE-2 2.97x 4.29 2.77x 4.31 2.45x 3.73 2.33x 3.51 2.47x 3.72 2.60x 3.91 HASS 3.45x 4.88 3.22x 4.84 3.02x 4.13 2.79x 3.97 2.83x 4.01 3.06x 4.37

L3 8B EAGLE-2 2.23x 3.67 2.21x 3.69 1.85x 2.79 1.94x 3.15 1.89x 3.03 2.02x 3.27 HASS 2.80x 4.13 2.73x 4.08 2.21x 3.18 2.35x 3.54 2.11x 3.34 2.44x 3.65

L3 70B EAGLE-2 2.97x 4.02 2.95x 3.89 2.37x 3.16 2.72x 3.65 2.44x 3.34 2.69x 3.61 HASS 3.71x 4.67 3.49x 4.38 2.83x 3.47 3.25x 4.11 2.75x 3.70 3.21x 4.07

Table 11: Speedup ratios and acceptance lengths τ of HASS and EAGLE-2 on five translation tasks, where draft models are trained with the fixed Share GPT dataset. De, Fr, Ja, Ru, Zh, and En stand for German, French, Japanese, Russian, Chinese, and English, respectively. L2 represents LLa MA2Chat, while L3 represents LLa MA3-Instruct.

As shown from Table 11, HASS consistently outperforms EAGLE-2 under all settings. HASS achieves 2.44x-3.28x wall-clock time speedup ratio averaging across five translation tasks, surpassing EAGLE-2 by 17%-24%. In terms of acceptance length, HASS achieves 8%-14% improvement over EAGLE-2. In consistent with results on dialogue (MT-bench), code generation (Human Eval), and mathematical reasoning (GSM8K) tasks, HASS shows promising improvements over EAGLE-2 on translation tasks, reflecting its robustness across different task types.

3https://github.com/Kthyeon/Multilingual-Spec Bench

Published as a conference paper at ICLR 2025

A.8 TRAINING OVERHEAD

As shown from Table 4, training with 3/4 steps of harmonized context alignment generally obtains the most considerable acceptance length, and so the aligning step of HASS is fixed to 3 (Standard) in this paper unless stated otherwise. To investigate the actual training overhead of HASS, we train draft models for LLa MA2-Chat 7/13B and LLa MA3-Instruct 8/70B on a single NVIDIA H800 GPU with batch size set to 2 and varied aligning steps, and summarize the results of training speed, computational cost, and GPU memory in Figures 9, 10, and 11, respectively. It is worth mentioning that the training overhead of HASS with 1 aligning step is the same as that of EAGLE-2.

Align-1 (EAGLE)

Align-2 Align-3 Align-4 Align-5 0

Training Speed (batch/s)

LLa MA2-Chat 7B

Align-1 (EAGLE)

Align-2 Align-3 Align-4 Align-5 0

LLa MA2-Chat 13B

Align-1 (EAGLE)

Align-2 Align-3 Align-4 Align-5 0

LLa MA3-Instruct 8B

Align-1 (EAGLE)

Align-2 Align-3 Align-4 Align-5 0

LLa MA3-Instruct 70B

Figure 9: Training speed (batch/s) of HASS with varied aligning steps, where the speed of Align-1 is the same as that of EAGLE/EAGLE-2.

The training speed is evaluated by how many batches can be processed in one second, i.e., batch/s, and the ratio between Align-1 and Align-j represents how much training time needed for executing the same amount of training data compared to EAGLE-2. As shown from Figure 9, the training speed decreases with more aligning steps. However, the actual training time of standard HASS (Align-3) is only 66.34% more than EAGLE-2 averaging over four target models, and the highest extra time cost compared to EAGLE-2 is just 91.47% (on LLa MA3-Instruct 8B). The training overhead of HASS is totally affordable, while HASS achieves superior performance and requires unchanged inference overhead.

0 5 10 15 20

Align-1 (EAGLE)

LLa MA2-Chat 7B

Align-1 (EAGLE)

LLa MA2-Chat 13B

0 10 20 30 40

Align-1 (EAGLE)

LLa MA3-Instruct 8B

0 20 40 60 80 100

Align-1 (EAGLE)

LLa MA3-Instruct 70B

Constant Part Attention Part Others

0 20 40 60 TFLOPs

Align-1 (EAGLE)

0 20 40 60 80 100 TFLOPs

Align-1 (EAGLE)

0 25 50 75 100 TFLOPs

Align-1 (EAGLE)

0 100 200 300 TFLOPs

Align-1 (EAGLE)

Forward Backward

Figure 10: Training FLOPs of HASS with varied aligning steps, where the computational cost of Align-1 is the same as that of EAGLE/EAGLE-2. The upper figures show the FLOPs of the forward pass, while the lower figures show the FLOPs in total (forward and backward passes).

The computational cost is evaluated by TFLOPs and can be divided into forward pass and backward pass, we depict the cost of forward and backward passes in upper and lower figures in Figure 10, respectively. The cost of forward pass is consisted of three parts:

Constant part is invariant to the number of aligning steps. Mapping target LLM s hidden state into q(x) (refer to section 3.1) with the LM head for distilling the draft model is included in constant part.

Published as a conference paper at ICLR 2025

Attention part is linearly proportional to the hidden state number fed into the draft model, which is accumulated across HASS training steps, i.e., Pj i=1 i for Align-j. Fusing token embeddings with hidden states, projecting hidden states into keys and values, and conducting attention operations between query and several kv-pairs sourced from different hidden states are included in attention part.

Others is linearly proportional to the number of aligning steps, i.e., j for Align-j. Computational costs except for constant and attention parts are included in others.

The cost of backward pass can be considered as (A + O) 2, where A and O represent attention part and others respectively, as the computation of constant part requires no gradient. Generally, the standard HASS (Align-3) requires approximately 3x computational cost of EAGLE-2.

Align-1 (EAGLE)

Align-2 Align-3 Align-4 Align-5 6

GPU Memory (GB)

LLa MA2-Chat 7B

Align-1 (EAGLE)

Align-2 Align-3 Align-4 Align-5

LLa MA2-Chat 13B

Align-1 (EAGLE)

Align-2 Align-3 Align-4 Align-5

LLa MA3-Instruct 8B

Align-1 (EAGLE)

Align-2 Align-3 Align-4 Align-5

LLa MA3-Instruct 70B

Figure 11: Training GPU memory of HASS with varied aligning steps, where the GPU memory of Align-1 is the same as that of EAGLE/EAGLE-2. Avg and Peak stand for the average and peak GPU memory across the training process, respectively.

The GPU memory is evaluated by GB and we report the average and peak GPU memory across the training process. Both the average and peak GPU memory increase with more aligning steps. The GPU memory requirement can be covered by a single NVIDIA H800 GPU even at Align-5 and batch size set to 2.