Multi-Draft Speculative Sampling: Canonical Decomposition and Theoretical Limits

**Weakness** > I think the contributions of this work is a little weak. Given that speculative decoding shows poor performance with batch size larger than 1, the multi-draft sampling is usually not an appropriate option for speculative decoding. I think the authors should emphasize the contributions to real-world speculative decoding applications. We observed that whenever there is a moderate gap between the draft and target model, the improvement in the acceptance rate in the multi-draft setting can be worth the additional compute over the single-draft scheme. This is already demonstrated in our experiments in Fig 3 and 4 where we show improved token rate over baseline schemes. The multi-draft setting also naturally lends itself to parallelism as each draft model can be allocated a different processor in a multi-node GPU system. This could alleviate the compute overhead while preserving the improved acceptance rate, leading to further speedups. Finally, the use of multiple drafts opens the possibility of training them using a variety of ensemble learning methods. This can lead to a diverse set of draft models that better approximate the target model and lead to higher acceptance rates. > While I appreciate that the authors add some examples for better explanation, I think the authors should add a clear notation section with distinct description. We have included a new section in the Appendix (Section K) summarizing the notation in the paper. > For the truncated LP draft selection algorithm, there exists an operation of sorting, which takes a complexity of $O(n\log n)$ and may influence the final speedup. The authors should report its real-world time consuming and compare to vanilla multi-draft selection, especially the empirical results on models with large vocab size, e.g., Llama 3.1 with vocab size 128k. The sorting operation mentioned by the reviewer is on the set of tokens selected after top-p sampling. As the distributions tend to be concentrated on a small number of tokens, we did not find the sorting operation to be a bottleneck. In any case it should be noted that top-p sampling, a commonly used technique in many real world LLM tasks also has the $O(n \log n)$ complexity We believe that the fact that we get speedups in the actual token rate in the experiments already demonstrates that the sorting operation is not resulting in slowdowns. Motivated by your suggestion, we have added a section on the computational complexity in Section G in the Appendix to illustrate the cost from the sorting operation and the LP program. In fact in our experiments we observed that in many cases the computational burden can be dominated by the LP program. Finally if needed, the sorting operation can also be entirely avoided in the truncated LP. We have proposed a variant in Appendix (Section G.1) that replaces the $O(n\log n)$ cost with $O(n \cdot s)$ where $s$ is our truncation threshold. This method also has the same theoretical guarantees as our original proposal. > Besides, for multi-draft scenarios with $K>2$ , the authors propose a multi-stage manner, which may affect the final speedup as well. We present a new experimental result in Section J.2 involving three drafts. We observe gains in block efficiency and token rates similar to our other experiments. > I think the authors should conduct experiments on more benchmarks (e.g. HumanEval [1], GSM8K [2] and MT-bench [3]) for a fair comparison. Besides, the authors should report the real-world speedup to demonstrate the effectiveness of proposed method. We appreciate the suggestion to conduct experiments on the additional tasks. While we already have results on three diverse tasks: WMT, XSUM and Dolly in the paper, we will to report experiments on some of these should the paper be accepted.

> By and large the biggest complaint I have with this draft is the background exposition. Prior to reading this I was not familiar with speculative decoding. While I understand space is tight, many key concepts such as the role and form of accepting probability in the speculative decoder were not clearly explained. This is a key aspect of the work, and the draft would strongly benefit for giving it more treatment. To be honest, I couldn't understand anything the first time I read the paper and had to go back to the original speculative sampling paper [1] before I could piece things together. Nevertheless, after reading, I now understand that this paper is targeting the acceptance criteria as it determines the number of candidate tokens used via a Bernoulli like-process. This plus the strong theoretical analysis are enough for me to lean towards acceptance. We have included a review of speculative sampling in Appendix A in the revision and in particular highlighted the role of the acceptance probability. > Could you give some intuition about the conjecture in remark 2. It says that the exponent over the sum of the p-weights should match K (the number of tokens to generate per round). It's not at all clear to me why these should be so tightly coupled and why other lower order terms wouldn't appear. Our first discovery of such a relation was in the context of Theorem $2$ where we provide a necessary and sufficient condition for the acceptance probability to equal $1$. As explained in Appendix D, we noticed numerically that for the case of $K$ drafts, a natural counterpart of Theorem $2$ exists with the power of $2$ simply replaced by $K$. To provide further intuition, we have included a new subsection (Appendix D.1) where we consider the case of $K=3$ drafts and an alphabet of size $3$ in detail. Our key observation is that when we establish the augmented system of inequalities, only terms of the form $(\sum_{s\in {\mathcal S}}p_i)^3$ survive for the draft probabilities and hence lower order terms do not appear. This observation then motivated the expression in Remark 2, which was of course verified numerically in several cases.

> In Figure 3, experiments are conducted for draft temperatures $> 1.0$. However typically, target model temperature is often set to 1.0 or less than 1.0. Given this, I was expecting draft temperatures also to be $<= 1.0$. I was wondering how these draft temperatures were chosen? Would it be possible to say how the gap between the proposed approach and previous approaches vary as the target model temperature varies from 0.0 to 1.0? Our proposed IS scheme outperforms the baselines in terms of block efficiency for most temperature parameters. However it only provides improvements in the token rates when the block efficiency improvement is large enough. Indeed, when there is a "big enough" gap between the target and the draft model we can better demonstrate how the various methods are able to close it. This motivated the choice of the temperature parameters in our experiments. We have also added Fig. 4 in Section 5 in the main paper based on your suggestion. In this new experiment we vary the target model temperature from 0.2 to 1.0 and the draft model temperatures are set to 1.0. We did experiment setting temperatures of the draft model below 1, but the gains in acceptance rate were not big enough (c.f. Table 2), so there wasn't enough "room" to demonstrate gains with better schemes. > Theorems 2 and 3 provide a nice characterization of acceptance probability. However, given two distributions $p$, $q$, evaluating the condition seems to require $2^\text{vocabulary size}$ number of computations. Is there a faster way to evaluate these quantities? We did not focus on actual computation of the acceptance probability, as our proposed truncated LP scheme does not require this computation for implementation. We do compute the optimal acceptance probability in Table 2. Here we first apply top-k sampling with $k=5$, so the resulting vocabulary is small and an exhaustive search can be applied. >There are some approximate ways to solve the optimal transport by Sinkhorn methods e.g., https://amsword.medium.com/a-simple-introduction-on-sinkhorn-distances-d01a4ef4f085. I was wondering if these ideas are applicable here? We believe that a direct application of the Sinkhorn method to the optimal transport formulation in the K-draft setting of Sun et al [1] may still be slow in practice. In particular, the cost matrix associated with the optimal transport formulation will have $\Omega(n^K)$ non-zero entries, where $n$ denotes the alphabet size. As a result each iteration of the Sinkhorn algorithm could have a computational cost $\Omega(n^K)$, which may not result in improvements to the token rate. [1] Ziteng Sun et al., Spectr: Fast speculative decoding via optimal transport. Advances in Neural Information Processing Systems, 36, 2024. >Architectures in ML, typically refer to neural architectures. Given several works which propose different drafting methods for speculative decoding, I was thinking ``canonical architectures" refers to some new drafters. It might be good to clarify this distinction early on in the paper. We have replaced the term "canonical architecture" with "canonical decomposition" in most of the paper. >Can you please expand on what token rate means? Is it the number of tokens / second? Yes we define it as follows: $$ \mathrm{Token~Rate} = \frac{\text{Number of Decoded Tokens}}{\text{Total Time Elapsed (seconds)}}. $$ Instead of reporting the absolute value of token rate, in our experiments we report the percentage improvement in the token-rate with respect to the single draft baseline, to make it easier to note the improvement from having multiple drafts.

> I would have liked the presentation in Section 4 to be a little clearer - request you to write out the full linear program, and then explain the truncation a bit more - what is the truncated program computing? We have rewritten Section 4 to make the presentation clearer. We first present the full linear program and introduce the variables $w_{i,j}$ in the optimization program. Thereafter we explain the truncation procedure and how we reduce the number of variables involved in the optimization. We hope that the revised exposition makes the presentation clearer. > In the experiment section, please include results for Temperature $< 1$, which is much more common in practice. We have added Figure 4 in Section 5. In this experiment, we vary the target model temperature from 0.2 to 1.0 and the draft model temperatures are set to 1.0. While our proposed IS scheme outperforms the baselines in terms of block efficiency for most temperature parameters, it can provide improvements in the token rates when the block efficiency improvement is "large enough". When the draft temperature is set to less than $1$ we observed that all methods appear to have acceptance probability close to each other (c.f. Table 2). Thus there is not enough "room" for improvement with better schemes. > I would like to see a comparison to the results in https://openreview.net/forum?id=9KxnxWOBA5 Thank you for bringing this paper to our attention. We explain the differences with our paper below. - Theorem 1 in our paper, which presents the canonical decomposition result (see Fig. 1) is not present in the other paper. Our result shows how importance weighted sampling and speculative decoding can be elegantly combined for optimal token selection in the multi-draft setting. Building upon this theoretical result, we propose a practical token selection scheme that mimics the proposed decomposition and provides a a way to trade-off the computational speed with the acceptance probability. We present experimental results that demonstrate that our proposed scheme can achieve improvements over baselines in standard metrics such as token-rate and block-efficiency. - The other paper focuses on computing the optimal acceptance probability and provides a generalization of Theorem 3 in our work, which only considers the case of two (identical) drafts. However in Appendix D and Section 3 in our paper, we also provide a procedure for numerically verifying the generalization (of Theorem $2$) beyond two drafts. This numerical procedure is based on computing the dual representation of polyhedral cones using a technique known as the double description method. We believe that such an approach could also be of significant interest to the research community, in addition to the analytical results. - The experiments reported in the other paper appear to be limited to a specific draft sampling scheme and primarily use the average token acceptance probability as a performance metric. In contrast, our paper presents a new token level selection scheme and evaluates its performance using standard metrics (token rate and block efficiency) and demonstrates improvements over baselines in a number of different tasks. - We noticed that in the "author response" the authors of the other paper have raised a question about how we computed the acceptance probability in Table 2 in our paper and noted that a brute-force search may not be practical. We would like to clarify that we had explicitly mentioned (in the paragraph below table 2) that for this table top-k sampling with $k=5$ was used in this experiment. This made it feasible to compute the optimal acceptance probability using Theorem 3. We also note that in many of our other experiments in the paper, we use the more commonly deployed top-$p$ sampling with a default value of $p=0.95$. To clarify our paper, we have put experimental results involving top-k and top-p sampling in separate sub-sections in Section 5.