Smaller, Weaker, Yet Better: Training LLM Reasoners via Compute-Optimal Sampling

## Abstract

Abstract Training on high-quality synthetic data
from strong language models (LMs) is a common strategy to improve the reasoning performance of LMs.
In this work, we revisit whether this strategy is compute-optimal under a fixed inference budget (e.g., FLOPs). To do so, we investigate the trade-offs between generating synthetic data using a stronger but more expensive (SE) model versus a weaker but cheaper (WC) model. We evaluate the generated data across three key metrics: coverage, diversity, and false positive rate, and show that the data from WC models may have higher coverage and diversity, but also exhibit higher false positive rates. We then finetune LMs on data from SE and WC models in different settings: knowledge distillation, self-improvement, and a novel weak-to-strong improvement setup where a weaker LM teaches reasoning to a stronger LM. Our findings reveal that models finetuned on WC-generated data consistently outperform those trained on SE-generated data across multiple benchmarks and multiple choices of WC and SE models.
These results challenge the prevailing practice of relying on SE models for synthetic data generation, suggesting that WC may be the compute-optimal approach for training advanced LM reasoners.

## 1 Introduction

Language models (LMs) have demonstrated impressive reasoning capabilities, but their success heavily relies on being trained on vast amounts of (problem, solution) pairs. Collecting this data from humans is costly and time-consuming. Recent studies have demonstrated the feasibility of synthetically generating this data using LMs themselves, offering a more scalable and efficient approach to training data acquisition.
One widely-adopted approach is to sample multiple candidate solutions for a problem from an LM, filters them for final answer correctness, and finetune models on the correct solutions (Zelikman et al., ). Several works show that LMs trained with such synthetic solutions outperform those trained with human-written solutions (Yuan et al., ; Yu et al., ; Yue et al., ; Singh et al., ; Pang et al., ).
Practitioners often sample solutions from strong LMs to ensure high quality (Teknium, ; Roziere et al., ; Mukherjee et al., ; Xu et al., ). However, sampling from strong LMs is expensive and resource-intensive, and limits the number of solutions that can be generated for practical sampling budgets.

Figure: Figure 2: Illustration of the approach. Given a fixed sampling budget, one can either generate fewer samples from a stronger but more expensive (SE) model or more samples from a weaker but cheaper (WC) model. The latter may lead to solving a wider range of problems and also more correct solutions per question. We compare the utility of these two synthetically generated datasets for training LM reasoners in various supervised finetuning setups and show that training with the data from WC consistently outperforms training on data from SE.
Refer to caption: x3.png

In this paper, we explore an alternative sampling approach. Given a fixed compute budget, we investigate sampling from a weaker but cheaper (WC) model as opposed to the commonly-used approach of sampling from a stronger but more expensive (SE) model. We start by comparing data from WC vs SE across three axes that play crucial roles in the utility of such synthetic data: 1- *coverage*, the number of unique problems that are solved, 2- *diversity*, the average number of unique solutions we obtain per problem, and 3- *false positive rate (FPR)*, the percentage of problems that arrive at the correct final answer but with a wrong reasoning. We find that since we can generate more samples from the WC model compared to the SE model under a fixed budget, the data from WC may exhibit higher coverage and diversity. However, due to the lower quality of the WC model, it may also have a higher FPR. As a particular example for the Gemma2 family (Team et al., , ) on the MATH dataset (Hendrycks et al., ), Gemma2-9B achieves $11\%$ higher coverage and $86\%$ higher diversity, but also with $7\%$ higher FPR compared to Gemma2-27B.

We then fine-tune models on data from SE and WC (see Figure 2) across diverse setups corresponding to three paradigms: 1) knowledge distillation, where a student LM learns from a teacher LM (Hinton et al., ); 2) self-improvement, where an LM learns from self-generated data (Huang et al., ); and 3) a new paradigm we introduce called Weak-to-Strong Improvement, where a strong student LM improves using synthetic data from a weaker teacher LM.
Using two (WC, SE) model pairs, one from the Gemma2 family and another from the Gemini 1.5 family (Reid et al., ), we show on multiple benchmarks that training on WC-generated data consistently outperforms training on SE-generated data under the three setups, with relative gains of up to $31.6\%$ percent (see Figure 1 for a summary of the results).
Our results indicate that it is more compute-optimal to sample from a WC model as opposed to the common-practice of sampling from a SE model.
With the performance gap between small and large LMs getting narrower over time (especially at larger scales), our results establish a solid foundation for training the next generation of LM reasoners.

## 2 Preliminaries

Let $\mathcal{D}=\{q_{i},a_{i}\}_{i=1}^{i=n}$ be a training dataset of size $n$ with reasoning questions $q_{i}$ and final answers (aka labels) $a_{i}$. A successful approach to leverage such data to improve models for reasoning is as follows. We sample multiple solutions for each $q_{i}$ at a non-zero temperature and create the synthetic data $\mathcal{D}_{G}=\{q_{i},\{(\hat{r}_{ij},\hat{a}_{ij})_{j=1}^{j=k}\}\}$, where $k$ is the number of samples, $\hat{r}_{ij}$ is the $j$-th reasoning chain (i.e. solution) generated by the model for $q_{i}$, and $\hat{a}_{ij}$ is the model’s final answer for $q_{i}$ in the $j$-th sample. Then, we filter the incorrect solutions by comparing $\hat{a}_{ij}$ to $a_{i}$ and removing the solutions whose final answer do not match that of the gold answer(^1^11While it is possible to use more sophisticated approaches for filtering (e.g., process-based or outcome-based reward model (Uesato et al., )), in this work we focus on final answer correctness for filtering as it has shown to be strong.). Finally, we supervise finetune a model on the remaining data $\tilde{D}_{G}$ to maximize $J(\theta)=\mathbb{E}_{(q,r,a)\sim\tilde{D}_{G}}\left[\log(p_{\theta}(r,a|q))\right]$, i.e. the probability of generating the reasoning $r$ and final answer $a$ given the question $q$. This approach was first proposed in (Zelikman et al., ) and was then extended in multiple works including (Zelikman et al., ; Singh et al., ).

For a dataset $\mathcal{D}_{G}$, we compute $coverage@k$ (aka $pass@k$) (Chen et al., ) as $\mathbb{E}_{\mathcal{D}_{G}}\left[1-\binom{M-c}{k}/\binom{M}{k}\right]$
where $c$ is the number of solutions, out of $M$, with correct answers and $\mathbb{E}_{\mathcal{D}_{G}}[.]$ denotes the expectation over the problems and solutions in the generated dataset.
Conceptually, $coverage@k$ measures the fraction of unique questions that have at least one correct solution, assuming that we sample $k$ solutions per question from the model. We also define $diversity@k$ as the average number of unique correct solutions we obtain per question when we sample $k$ solutions per question. Finally, we define *false positive rate (FPR)* as the percentage of solutions in $\tilde{D}_{G}$ where the reasoning is incorrect, despite the final answer being correct.

Different choices of the LM to sample solutions from and the LM to finetune lead to different setups. *Knowledge Distillation* (Hinton et al., ) corresponds to training a student LM on the synthetic data sampled from a stronger and larger LM. *Self-Improvement* (Huang et al., ) corresponds to training an LM on samples generated from itself.

## 3 Compute-Matched Sampling and Training

To generate a dataset $\mathcal{D}_{G}$ with synthetic solutions from $\mathcal{D}$, one can leverage different models for generating solutions. Specifically, at a fixed sampling budget (FLOPs), one can generate more samples from a weaker but cheaper (WC) model or fewer samples from a stronger but more expensive (SE) model. Given a WC model with $P_{WC}$ parameters and SE with $P_{SE}$ parameters, we compute the sampling ratio at a fix budget for the two models, focusing on decoder-only transformer models (Vaswani, ). Following (Kaplan et al., ), we note that the FLOPs per inference token is $2P$, for a model with $P$ parameters. As a result, the FLOPs for $T$ inference tokens is $2PT$. Further, we assume that generating each solution requires an average of $W$ inference tokens for both models(^2^22This is a reasonable assumption given that the solution to a question is expected to be model-agnostic. We note, however, that it is possible for some questions that one model solves a question using a more optimal way compared to the other model thus producing a smaller solution.).
Let $S_{WC}$ and $S_{SE}$ represent the number of samples we generate per question for the two models. The total cost of generating samples for the dataset $\mathcal{D}$ will then be $Cost_{WC}=n\times S_{WC}\times W\times(2P_{WC})$ and $Cost_{SE}=n\times S_{SE}\times W\times(2P_{SE})$ for the cheap and expensive models, respectively. At a fixed sampling budget, we have:

$$ $n\times S_{WC}\times W\times(2P_{WC})=n\times S_{SE}\times W\times(2P_{SE}) \quad\Rightarrow\quad\boxed{S_{WC}=\frac{P_{SE}}{P_{WC}}S_{SE}}$ (1) $$

Equation 1 indicates that at a fixed sampling budget, for each question we can generate $P_{SE}/P_{WC}$ more samples from WC; the ratio scales linearly with the model parameters ratio(^3^33Note that this may also depend on the available hardware, which we ignore in this work.). Sampling more solutions from WC may increase the likelihood of correctly solving a larger subset of the problems (high coverage) and obtaining more correct solutions per question (high diversity).

Given a fixed budget, we can either generate fewer samples from a SE model or more samples from a WC model, and then finetune models for a fixed number of steps on the data from each of these models to measure and compare the utility of the data from each model. Specifically, we generate $P_{SE}/P_{WC}$ more samples from the WC model compared to the SE model. We consider three finetuning setups that consists of diverse finetuning paradigms. The paradigms include the widely used knowledge distillation, the emerging framework of self-improvement, and a novel weak-to-strong improvement paradigm we introduce in this work. We define weak-to-strong improvement (W2S-I) as enhancing the reasoning capabilities of a strong model using samples generated from a weaker model. The three setups are as follows (a summary of the three setups and the finetuning paradigms that each case corresponds to can be found in Table 1).

Student-LM finetuning: Conventionally, the supervised finetuning data for training student LM is acquired from SE models to ensure high-quality (Teknium, ). However, we aim to understand whether WC models can replace SE models for distillation at the fixed sampling budget. To do so, we finetune a student LM separate from the WC and SE models on the WC and SE data, which corresponds to distillation in both the cases.

WC-LM finetuning: Prior work (Singh et al., ) has shown that finetuning a WC model through self-generated data lags behind distillation from SE data. However, their setup spends a higher sampling budget on collecting data from SE than WC. In this work, we revisit this finetuning setup under the fixed sampling budget and finetune the WC model on the WC and SE data at a fixed budget for both. Note that training the WC model on its own data corresponds to self-improvement whereas training WC on the data from SE corresponds to distillation. Hence, this setup compares self-improvement on WC data with distillation from SE data.

SE-LM finetuning: It is commonly believed that to improve a SE model, we either need synthetic data from the SE model itself or from an even stronger (and perhaps more expensive) model. Here, we test an alternative approach to understand whether the synthetic data from the WC model can improve the SE model. To this end, we finetune the SE model on the WC and SE data. Training SE on data from WC corresponds to W2S-I and training SE on data from SE corresponds to self-improvement. Overall, this setup compares W2S-I by WC data with self-improvement by SE data.

**Table 1: Summary of the supervised finetuning setups. We finetuned the language models under three setups: (a) Student LM, (b) Weak-Cheap (WC) LM, and (c) Strong-Expensive (SE) LM. For each setup, we employed different finetuning paradigms based on the source of the synthetic data. For example, training a separate student LM with data from both WC and SE models falls under the knowledge distillation paradigm. In contrast, training a WC model with its own samples is self-improvement. Finally, we also introduce a new paradigm, weak-to-strong improvement, where the samples from the WC model is used to improve the reasoning capabilities of the SE model at the fixed compute budget.**
| Data ($\downarrow$) / Finetuning setup ($\rightarrow$) | Student-LM | WC-LM | SE-LM |
| --- | --- | --- | --- |
| WC (Compute-matched) | Knowledge distillation | Self-improvement | Weak-to-strong improvement |
| SE | Knowledge distillation | Knowledge distillation | Self-improvement |

## 4 Experimental Setup

We briefly explain our setup here and provide more detail in Appendix G.

Datasets: We mainly experiment with MATH (Hendrycks et al., ) and GSM-8K (Cobbe et al., ) datasets, which are widely adopted in the literature.

Data Generation: We use Gemma2 models for synthetic data generation, with pretrained Gemma2-9B and Gemma2-27B acting as the WC and SE models respectively. Since the 9B model is roughly 3 times smaller than the 27B model, at a fixed sampling compute budget we can sample $3\times$ more sample solutions per problem for Gemma2-9B. For our experiments, we consider two sampling budgets: a low budget, where we generate 1 and 3 candidate solutions per problem from Gemma2-27B and Gemma2-9B, respectively, and a high budget, where we generate 10 and 30 candidate solutions per problem. Further, we study the transfer of the reasoning capabilities for the models trained on MATH at the high sampling budget on the Functional MATH dataset.

Model Finetuning: We summarize the details for our finetuning setups in the Table 1. In the Student-LM finetuning setup, we finetune the Gemma-7B model (Team et al., ), for WC-LM we finetune Gemma2-9B, and for SE-LM we finetune Gemma2-27B. Further, we train the LMs across different setups with the human-written solutions as a ground-truth baseline.

Synthetic Data Evaluation: To assess the quality of the synthetic data from the SE and WC models, we measure the coverage, diversity and *fpr* at a fixed cost. From Equation 1, we know that sampling one solution from SE takes the same FLOPs as sampling $P_{SE}/P_{WC}$ solutions from WC. Therefore, we compare $coverage@k$ for SE to $coverage@(\frac{P_{SE}}{P_{WC}}k)$ for WC to allow a similar budget to both models. Specifically, we compare $coverage@k$ and $coverage@3k$ for our SE and WC models. Similarly we compare $diversity@k$ and $diversity@3k$ for our SE and WC models.
Since FPR cannot be computed automatically, we compute it using two proxies: 1- a human evaluation on a subset of the data, where $50$ solutions from each model were selected randomly and rated for reasoning correctness by the authors, and 2- automatic evaluation where we sampled $500$ solutions and prompted Gemini-Pro-1.5 (Reid et al., ) to rate the correctness of the reasoning paths. To sample solutions, for the MATH dataset we selected uniformly from each diversity level. In our experiments, we find that the FPR estimates are close to each other for the human and automatic evaluation. We provide a few qualitative examples for the false positive instances in Appendix C.

Evaluating Finetuned Models: We use pass@1 accuracy to evaluate the performance of the finetuned LMs. Specifically, we generate a single solution for the problem (zero-shot) from the test split, using a sampling temperature of $0.0$ (greedy decoding) for the fine-tuned LM and measure the percentage of problems that where the final answer matches the golden final answer. We also report maj@k ($k=1,4,8,16$) for part of our experiments, where we generate $k$ solutions per problem at a sampling temperature of 0.7 and select the final answer that appears most among the $k$ samples.

## 5 Experiments and Results

We compare data from WC and SE models along several axes. First, we analyze the data along various quality metrics (§5.1). Subsequently, we present the supervised finetuning results for the different setups (§5.2). Finally, we perform ablation studies to study the impact of dataset size, sampling strategy, and the role of quality dimensions in the model performance (§6).

Figure: (a) Coverage on MATH.
Refer to caption: x4.png

### 5.1 Synthetic Data Analysis

We compare WC and SE data across three key quality metrics (coverage, diversity, and FPR) at a fixed sampling budget. We present the results for MATH at the low and high sampling budgets in Figure 3 and for GSM-8K in the Appendix – Figure 21.

Coverage: We find that the data from Gemma2-9B (WC) outperforms Gemma2-27B (SE) by $11\%$ and $6\%$ at the low and high sampling budgets, respectively, for the MATH dataset, and $8\%$ and $1\%$ for GSM-8K. This highlights that the higher number of samples for the WC model aids in solving more unique problems for both the reasoning datasets. We provide the coverage trends for diverse sampling budgets in Appendix D. In addition, we observe that the coverage of the WC model increases across various difficulty levels in the MATH dataset for the high sampling budget (see Appendix – Figure 22). This highlights that synthetic data from the WC model can solve more unique questions at various difficulty levels compare to the SE model, at a fixed sampling budget (Tong et al., ). Further, we provide a qualitative example that gets solved by repeated sampling from Gemma2-9B but remains unsolved by Gemma2-27B at the fixed high sampling budget (Table 5).

Diversity: The diversity for the data from Gemma2-9B is higher than Gemma2-27B by $86\%$ and $125\%$ at the low and high sampling budgets for the MATH dataset, and $134\%$ and $158\%$ at for the GSM-8K dataset. This implies that many unique reasoning chains in the synthetic data from the WC model lead to the correct solutions. We also observe that the absolute diversity scores are lower for MATH compared to GSM-8K at high sampling budget, indicating that models generate fewer correct solutions for the more challenging datasets when using repeated sampling.

FPR: Since we utilize the final answer correctness for filtering the synthetic data, it does not remove the solutions with incorrect intermediate reasoning steps. Our human evaluations suggest that the FPR for the WC-generated solutions is $7\%$ and $2\%$ higher than SE-generated solutions on the MATH and GSM-8K, respectively. The trends from the automatic evaluation are similar to that of human evaluation.
Due to the differences in the difficulty of the problems, we note that the absolute FPRs are much lower for GSM-8K compared to MATH. We also note that the automatic verification of the reasoning steps can also have errors and is still an open problem (Lightman et al., ).

Given the mixed signals of high coverage and diversity coupled with a high FPR, it remains unclear whether it is compute-optimal to sample from the WC model or the SE model for training strong reasoners. We study this in the next section.

### 5.2 Compute-Optimality Results for Training

We compare the utility of the synthetic data generated from the Gemma2-9B (WC) and Gemma2-27B (SE) model for the MATH and GSM-8K dataset across the diverse finetuning paradigms in Figure 4 and Figure 5, respectively. In addition, we present the results for training with human-written chain-of-thoughts from the original training sets as a baseline.

Figure: (a) Finetuning Gemma-7B.
Refer to caption: x7.png

Figure: (a) Finetuning Gemma-7B.
Refer to caption: x10.png

Student-LM Finetuning. The Gemma-7B finetuned with the synthetic data from WC consistently outperforms the one finetuned on data from SC with a relative gain of $6\%$ and $5.8\%$ at the low and high sampling budgets, respectively, for the MATH dataset and $4.2\%$ and $1.3\%$ for GSM-8K. Contrary to the common belief of stronger models being better for knowledge distillation, our results indicate that finetuning on data from WC is more compute-optimal than data from SE.

Figure: (a) Gemma-7B evaluation.
Refer to caption: x13.png

Figure: Figure 7: False positive rates of finetuned models. The false positive rates (FPR) of finetuned models on MATH assessed by Gemini-1.5-Pro, for (Left) models finetuned with Gemma2-27B and Gemma2-9B data (compute-matched) and (right) models finetuned with Gemini-Pro and Gemini-Flash data (price-matched).
Refer to caption: x16.png

Figure: (a) Finetuning Gemma-7B.
Refer to caption: x18.png

WC-LM Finetuning. We compare the performance of Gemma2-9B finetuned with the WC data (i.e. self-generated data) and SE data (i.e. data from Gemma2-27B). The results for MATH and GSM-8K are reported in Figures 4(b) and 5(b). We observe that the self-generated data (WC data) improves over knowledge distillation from a strong model (SE data), achieving relative gains of $3.8\%$ and $2\%$ at the low and high sampling budgets, respectively, for the MATH dataset, and $1.5\%$ at the low sampling budget for the GSM-8K dataset. However, we find that the WC model finetuned with WC data matches the SE data for the GSM-8K dataset at a high sampling budget. This is mainly due to the lower difficulty of the GSM-8k dataset, where it becomes saturated at higher sampling budgets (see Figure 21(a)). Interestingly, our empirical findings suggest that training a WC model on synthetic data from its own is more compute-optimal than distillation from a stronger model.

SE-LM finetuning. We present the results for finetuning Gemma2-27B with the Gemma2-9B generated data and self-generated data. The results for MATH and GSM-8K are reported in Figure 4(c) and 5(c). Surprisingly, we observe that the model finetuned with the WC data outperforms the SE data, achieving relative gains of $5.8\%$ and $4.3\%$ at the low and high sampling budget, respectively, for the MATH dataset and $1.2\%$ and $1.5\%$ for the GSM-8K dataset. This result is even more surprising given that the Gemma2-27B data is expected to be more in-distribution than the Gemma2-9B data. Contrary to the common belief of self-generated data or data from a stronger model being better, our empirical findings show that training a model in a W2S-I setup from a WC data may be more compute-optimal than training it in a self-improvement setup on its own data. This result also establishes a new paradigm for improving frontier models in a compute-efficient way, by generating synthetic data from much smaller models.

Figure: Figure 9: Understanding the role of coverage and diversity for training strong reasoners with WC model. We compare the performance of training the LMs with synthetic data acquired by collecting (a) 1 solution per problem (low diversity, low coverage), (b) 30 solutions per problem (high diversity, high coverage), and (c) 30 solutions per problem but keeping just one correct solution (high coverage, low diversity). We find that both high diversity and coverage are helpful for training strong reasoners.
Refer to caption: x21.png

FPR of Finetuned Models: We showed that models finetuned on WC data achieve higher final answer accuracy. However, since WC data had a higher FPR compared to SE data, a question that may arise is whether the WC finetuned models mainly learn to arrive at the correct final answer but with wrong reasoning chains. To study this, similar to the experiment in Figure 3(c), we use Gemini-1.5-Pro as a judge to estimate the FPR of the finetuned models. To reduce noise, we do this three times and average the results. We report the results for finetuned models with (Gemma-27B, Gemma-9B) and (Gemini-Pro, Gemini-Flash) as the (SE, WC) data in Figure 7. Despite the larger FPR of the WC data, we observe that the FPR of the WC finetuned models is as good as the FPR of the SE finetuned models across different finetuning setups and choices of SE/WC data.

Generalization. Here, we aim to study the transfer capabilities of the models trained with the WC and SE data. Specifically, we evaluate the models finetuned with the synthetic solutions for the MATH datasets at the high sampling budget on the Functional MATH dataset. The results in Figure 6 show that the Gemma-7B finetuned with the WC data consistently outperforms the SE data, where the relative gains range from $5.8\%-6.5\%$ at different values of $k$. In addition, we observe that the Gemma2-9B finetuned with the self-generated data outperforms knowledge distillation with the Gemma2-27B data achieving relative gains ranging from $2.5\%-4.5\%$ at different values of $k$. Moreover, finetuning Gemma2-27B with WC data matches closely with the SE data, except for $k=8$ where the gap is a relative gain of $2\%$. Our results highlight that finetuning the LMs with the WC data enhances the generalization capabilities over the SE data at the fixed sampling budget. So far, we have presented results on math datasets. In Appendix B, we extend our results to coding where we observe that the benefits from the WC can be context-dependent.

## 6 Ablation Studies

Impact of Dataset Size: We study whether the benefits of the synthetic data from the WC model hold at different dataset sizes. We repeat our experiments for the MATH dataset at the high budget, but when only having access to $500$ training data (selected randomly from the training set).
We present the results for the finetuned models in Figure 8. We observe that models trained with the WC data outperform those trained with the SE data, achieving relative gains of $12.93\%$, $11.4\%$, and $5.1\%$ for the three paradigms, respectively. This highlights the utility of generating more data from the WC model instead of the SE model in the low-problem regimes at the fixed sampling budget.

Figure: Figure 10: Finetuning models with mixing strong and weak (compute-matched) data. The results present the performance of the models finetuned with mixing the data from Gemma2-27B (SE) with the data from Gemma2-9B (WC) for the fixed sampling budget. Specifically, we mix $5$ solutions (5S) per problem from SE model with the $15$ solutions (15S) per problem from WC model.
Refer to caption: x22.png

Coverage and Diversity:
We aim to understand the role of coverage and diversity in enhancing the performance of models trained with WC-generated synthetic data. To this end, for the MATH dataset, we consider the original high-sampling (30 solutions per problem) WC dataset as a *(high coverage, high diversity)* dataset. We then construct a *(high coverage, low diversity)* version by only selecting one correct solution per question from our samples. This reduces the diversity of the original WC dataset from 11 to 1, while maintaining the coverage. We also create a *(low coverage, low diversity)* dataset where we generate just one solution per problem from the WC model and filter it for the correctness of the final answer. The coverage of this dataset ($27\%$) is lower than that of the WC dataset with 30 solutions per problem ($43\%$). We train models across the three finetuning setups on these sets and present the results in Figure 9. Our results indicate that across all setups, the high coverage and high diversity data is better than high coverage and low diversity, and high coverage and low diversity is better than low coverage and low diversity. This reveals that both the coverage and diversity play a critical role in training strong reasoners from the smaller LMs.

Mixing Strong and Weak-matched Data: Here, we aim to study the impact of distributing our fixed budget on sampling candidate solutions from both the SE and WC models. To do so, we sample $5$ solutions per problem from the Gemma-27B (SE) and $15$ solutions per problem from the Gemma-9B (WC) data. We compare this data with two non-mixture settings: 1- $10$ solutions per problem from SE model and no solutions from the WC model, and 2- $30$ solutions per problem from WC model and no solutions from the SE model. We observe the mixed data has a coverage of $68.8\%$ in comparison to the $70.7\%$ from WC data. This indicates that the compute-matched sampling from WC model solves more unique problems than mixing SE and WC data at the same sampling budget. We then finetune models on the mixed data and present the results for Student-LM, WC-LM, and SE-LM finetuning in Figure 10. We observe that in the student-LM and SE-LM setups, mixed data underperforms whereas in the WC-LM setup it slightly outperforms the non-mixed setups. This could be due to the fact that mixing two datasets results in two data distributions that might be harder for models to learn. Overall, our results highlight that the usefulness of data mixing might be context-dependent. We leave a rigorous study of SE and WC data mixing for optimal performance as a future work.

Default vs Compute-Optimal Sampling from Cheap LMs: We anticipate that the reason why data from SE models has been previously preferred over data from WC is because they have been tested in a setup where an equal number of samples have been generated from the two models (e.g., see (Singh et al., )), as opposed to a compute-matched setup.
To verify this, we generated $1$ solution per problem (number-matched) from the WC model for the MATH and GSM-8K datasets and trained the models under the three fine-tuning setups on this generated data, after filtering for final answer correctness. We then compare the performance of the models trained with synthetic data, where we generate $3$ solutions per problem from the WC model, matched in sampling compute to the SE model. We present the results in Figure 11. We see that the models trained with the number-matched WC data are sub-optimal in comparison to the models trained with the compute-matched WC data, and lead to worse models compared to training with the SE data. This highlights that the future comparisons between synthetic data from weak and strong models should be made in the sampling compute-matched regime.

Figure: (a) Finetuning LMs on MATH data.
Refer to caption: x23.png

## 7 Scaling to State-of-The-Art Language Models

Figure: Figure 12: We finetune Gemma models (7B/9B/27B) on synthetic data generated by the state-of-the-art LMs Gemini-1.5-Pro and Gemini-1.5-Flash. We find that finetuning with Flash-generated data consistently outperforms Pro-generated data not only at the same sampling monetary cost as Gemini-1.5-Pro, but also at $\approx 0.15\times$ of the cost.
Refer to caption: x25.png

In the prior experiments, we focused on the synthetic data acquisition from open LMs. Here, we aim to show that data from the weaker SoTA LM can train better reasoners than stronger SoTA LM at a fixed sampling budget. To this end, we scale our method to sampling data from Gemini-1.5-Pro and Gemini-1.5-Flash. As the model sizes are not publicly available, we utilize the ratio between their pricing per output token as a proxy to perform compute-matched sampling. As of August 2024, we note that the price per million output tokens is $\$10.5$ and $\$0.3$ for Gemini-1.5-Pro and Gemini-1.5-Flash, respectively. Hence, we sample $1$ and $35$ solutions per problem from 1.5-Pro and 1.5-Flash, respectively. We conduct our experiments on the MATH dataset.

We perform knowledge distillation on the Gemma-7B, Gemma2-9B, and Gemma2-27B LMs with the synthetic data from Pro (SE) and Flash (WC). We present the results in Figure 12. Interestingly, we find that finetuning with the WC data outperforms the SE data, achieving relative gains of $31.6\%$, $14.4\%$, and $10.9\%$ for Gemma-7B, Gemma2-9B, and Gemma2-27B, respectively. This can be attributed to the difference in the coverage of the models at the fixed sampling budget, which is $61.1\%$ and $81\%$ for 1.5-Pro and 1.5-Flash, respectively.

Reducing the cost of data sampling. Further, we investigate training the LMs with the WC data that is less expensive than collecting $1$ solution per problem from the SE model. Specifically, we create a dataset by sampling $5$ solutions per problem from the Flash (WC) model, which is $7\times$ more economical than generating $1$ solution from the Pro (SE) model, in terms of the price ($). Upon training the LMs on the $0.15\times$ cost data regime (Figure 12), we find that training on this data can also outperform training with SC data, achieving relative gains of $19.1\%$, $9.8\%$, and $5.7\%$ for finetuning Gemma-7B, Gemma2-9B, and Gemma2-27B, respectively. This can be attributed to higher coverage of the weaker model ($69\%$), even in the more economical scenario, in comparison to the stronger model ($61.1\%$).

## 8 Extending Results to Scenarios Lacking Ground-truth Labels

In the prior experiments, we assumed having access to final gold answers which allowed us to filter the synthetically generated solutions through final answer correctness. Here, we extend our approach to scenarios where ground-truth labels are unavailable. In particular, we consider two scenarios: 1- the MATH dataset while assuming we do not have the ground-truth labels (§8.1), and 2- single-turn chat (instruction-following) data which lacks the concept of ground-truth labels (§8.2).

Figure: (a) Analyzing Gemma2-9B and 27B data.
Refer to caption: x26.png

Figure: (a) Finetuning w/ Gemma data without filtering.
Refer to caption: x28.png

Figure: (a) Finetuning w/ Gemini data without filtering.
Refer to caption: x30.png

### 8.1 Performance on Reasoning

We study the impact of two settings on the performance of the finetuned models using SE and WC data at a fixed sampling budget. In the first setting, we perform no verification of the candidate solutions; that is, we include all the synthetic solutions in the finetuning mix. In the second setting, we perform verification for the candidate solutions using a model-based verifier. Specifically, we use an language model (LM) as a judge (Zheng et al., ) setting for verification where, akin to prior work (Yuan et al., ), an LM is prompted to verify if a solution is correct or not. Note, however, that in practice one can use any other type of verifier, including a verifier that has been previously trained to judge the quality of the solutions. Due to the lack of ground-truth data, LM as judge is expected to be better than no verification but worse than oracle verifier in filtering incorrect solutions from the data.

Setup: We experiment with the same (WC, SE) model pairs as in the previous experiments, i.e. (Gemma-9B, Gemma-27B) and (Gemini-1.5-Flash, Gemini-1.5-Pro). Following the compute-matched setup, we generate $10$ and $30$ solutions per problem from Gemma-27B and Gemma-9B; following the price-matched setup, we generate $1$ and $35$ solutions per problem from Pro and Flash. We also consider a cheaper version where we collect $5$ solutions per problem from Flash, as done in the previous experiments. Post-generation, we use the Flash model to verify the final answers for the Gemma-9B and Flash data, and the Pro model to verify the final answers for Gemma-27B and Pro data. This is to ensure that we do not spend more compute (or cost) for the WC setup. Subsequently, we perform supervised finetuning of Gemma-7B/9B/27B with the (un-)filtered synthetic data.

Data Analysis: We start by analyzing the data in the no-verification and LM as a judge setups and present the percentage of synthetic data that leads to incorrect final answer for the two strategies in Figure 13. We find that the majority of the synthetic solutions from Gemma-9B and Gemma-27B, $65\%+$, lead to incorrect final answer without any verification. However, we observe that LM as a judge verification significantly reduces the amount of bad solutions from Gemma-9B and Gemma-27B (down to $\sim 25\%$). On the other hand, we observe that the percentage of bad solutions is between $40\%-48\%$ for Gemini-Pro and Gemini-Flash without any verification. Similar to Gemma models, the amount of bad data reduces to $23\%$ after LM as judge verification. Now, we will study the impact of finetuning LMs on this data.

Results: The results for finetuning LMs on the Gemma-9B (WC) and Gemma-27B (SE) data with no verification and LM as a judge is presented in Figures 14(a) and 14(b). We observe that finetuning models with the SE data slightly outperforms WC data across the two strategies. Further, we present the results for finetuning LMs on the Gemini-Flash (WC) and Gemini-Pro (SE) data in Figure 15(a) and 15(b). We observe that the finetuned models with the WC data consistently outperform the SE data across both strategies. Interestingly, we observe that when we use 5 solutions per problem for Flash, we obtain better performance than when we use 35 solutions per problem, for training Gemma-7B and Gemma-9B without any verification (Figure 15(a)). This can be attributed to the presence of a larger number of bad solutions among $35$ solutions in comparison to $5$ solutions in the finetuning mix. Overall, the trends suggest that whether WC data is superior to SE data or not in the case of lacking ground truth data depends on the quality of the models from which we sample as well as the finetuning setup.

Figure: Figure 16: Performance of finetuned models on IFEval. The instruction-level accuracy (%) on IFEval for models finetuned with Gemini-Pro and Gemini-Flash (price-matched) data.
Refer to caption: x32.png

### 8.2 Performance on Instruction-following Task

Apart from reasoning, synthetic data from SE models is also used for instilling instruction-following (chat) capabilities (Taori et al., ; Teknium, ). Due to the subjectivity of the chat data, the notion of final answer correctness may be ill-defined. For instance, there is no ground-truth for the instruction ‘poem on strawberries and beaches’. Here, we study the usefulness of synthetic responses from WC and SE data at a fixed sampling budget, for training instruction-following LMs.

Setup: We use Gemini-1.5-Pro and Gemini-1.5-Flash as the SE and WC models, respectively, as they have the capability to follow user instructions. In particular, we prompt the generators with $5000$ random instructions from the OpenAssistant1 dataset (Köpf et al., ). We generate $1$ and $35$ responses per instruction for Pro and Flash respectively, following a price-matched setup. Subsequently, we perform supervised finetuning of for Gemma-7B, 9B and 27B with the synthetic instruction-following data. Finally, we evaluate the finetuned models on the IFEval data (Zhou et al., ) and report the instruction-level accuracy.

Results: We present the results in Figure 16. Interestingly, we observe that finetuned models with WC data significantly outperform the SE data across different model sizes. In particular, the instruction-level accuracy of Gemma-9B trained with Flash data outperforms Pro data by achieving a relative gain of $12.8\%$. Our results highlight the usefulness of WC data over SE data for training capable instruction-following models at a fixed sampling budget.

## 9 A Future Perspective

We showed that for the current WC and SE models, training reasoners through sampling from WC models may be more compute-optimal.
Here, we aim to discuss the relevance of these results for the future set of WC and SE models. To do so, we surveyed $17$ LMs that pass the following criteria: 1- the model size is known and falls within [1B, 9B] or [20B, 80B] range, 2- the model is released in the past one year, 2- the technical report of the model reports results on the MATH dataset and the model is capable on it ($>20\%$), 4- ranks high on the OpenLLM leaderboard under the pretrained models category (HF, ). This resulted in models from seven families including Gemma-2 (Team et al., ), LLaMA-3 (Dubey et al., ), Mixtral (Jiang et al., ), Qwen (Team, ; Yang et al., ), Grok-1 (xAI, ), DeepSeek-v2 (Shao et al., ), and Yi (Young et al., ). We grouped these models into small LM (1B to 9B) and large LMs (20B to 80B). We then plotted in Figure 17 the model performances on the MATH dataset against their date of the publication release on arxiv and fitted trendlines to the data points representing the small and large LMs using the least squares method(^4^44We consider the number of active model parameters for mixture-of-experts LMs.).

Our analysis reveals that, despite the variance, the trendline for the smaller LMs is steeper than that of the larger LMs. This indicates that the reasoning performance of the small LMs may be improving more rapidly over time compared to the larger LMs. The rapid rise in the performance of the small LMs can be attributed to factors such as the enhanced quality and scale of the pretraining data (e.g., LLaMA-3 employs 15T tokens), pruning and knowledge distillation (Muralidharan et al., ). With the performance gap between small and large LMs narrowing over time, we anticipate that our results will become even more relevant in the future.

Figure: Figure 17: Variation in the performance of open LMs on the MATH dataset over time. The fitted trendlines suggest that the quality of smaller LMs is improving more rapidly than that of larger LMs over time. This highlights that our findings on utilizing smaller LMs for training strong reasoners will become increasingly relevant in the future.
Refer to caption: x33.png

## 10 Related Work

LMs for reasoning. The ability to solve reasoning tasks has been a long standing goal of artificial intelligence (Reid et al., ; Achiam et al., ; Dubey et al., ; Team, ; Anthropic, ; AI, ). In this regard, LMs trained on the internet-scale data have achieved great success for math, code, and other reasoning tasks (Lewkowycz et al., ; Azerbayev et al., ; Kazemi et al., ). There have been several works that aim to enhance the reasoning capabilities of the LMs either via prompting (Kojima et al., ; Wang et al., ; Zheng et al., ; Kazemi et al., ) or finetuning (Yue et al., ; Yu et al., ). In this work, we focus on finetuning the LMs with task-specific datasets to build strong reasoners. Specifically, our method closely aligns with the widely adopted STaR (Zelikman et al., ) where the synthetic data from the LMs are used to elicit strong reasoning capabilities.

Finetuning LMs. Within the finetuning paradigm, there have been several works that improve reasoning with synthetic data. Broadly, these works focus on knowledge distillation from a strong but expensive LM (Wu et al., ; Yue et al., ) or self-improvement (Gulcehre et al., ; Singh et al., ). While it is common to filter the synthetic data for the final answer correctness (akin to Zelikman et al. ()), there are several works that aim to build task-specific verifiers to train strong reasoners (Lightman et al., ; Wu et al., ; Hosseini et al., ; Yuan et al., ). In this work, we explore the utility of the synthetic data from the weak but cheap LMs for training strong reasoners. We do not explore using model-based verifiers with the synthetic data for enhanced reasoning, and leave it as a future work. Our weak-to-strong improvement paradigm, where a strong model is trained with the generations from the weak model, is related to several prior work (Bowman et al., ; Burns et al., ; Yang et al., ) which study the ability of a strong LM to learn from the data generated by a weaker LM. However, the aim of these works is to recover the full capabilities of the strong model from weaker data, whereas we aim to enhance the strong model capabilities further. Our work also studies compute-optimal sampling from weak and strong models, which is absent in previous work.

Large and small LMs. While training large LMs has led to significant advancements across various tasks, there has recently been a growing interest in developing capable small LMs (HF, ; Javaheripi et al., ). Specifically, a capable small LM is faster to run, and easier to serve to millions of users on the edge devices (Gunter et al., ). As a result, several recent works aim to understand the utility of the weak but cheaper LMs in comparison to the strong but expensive LMs for reasoning. Specifically, Brown et al. (); Song et al. (); Snell et al. () show that the solve rate of the small LMs can increase significantly with repeated sampling. In addition, Hassid et al. () demonstrate that repeated generations from smaller LMs can outperform the data generated by larger LMs at a fixed sampling computational budget during inference for coding tasks. In this work, we go beyond these works and show the utility of the synthetic data from the small LMs for training strong reasoners across a diverse set of supervised finetuning setups.

## 11 Conclusion

In this work, we provide a framework for compute-optimal sampling from weak but cheap LM for reasoning tasks. Specifically, we show that at a fixed sampling compute budget, repeated sampling from a smaller model can achieve higher coverage and diversity than from a strong but more expensive model. Furthermore, our empirical findings highlight that fine-tuning LMs with data from the small LM can consistently outperform data from the large LM under the same compute budget.
Our results can serve as a foundation for training LM reasoners, especially as the performance gap between small and large LMs continues to narrow over time.

## Acknowledgements

This work was done during HB and AH’s internship at Google. We thank Tania Bedrax-Weiss, Hugo Larochelle and Hamidreza Alvari for feedback on this paper. We thank Chirag Nagpal, Katrin Tomanek, and Benjamin Estermann for support in setting up infra, which was crucial for our experiments.

## Appendix A Discussion

In this work, we introduce compute-matched sampling in the context of data generation from a weak and cheap (WC) model and a strong and expensive (SE) model. We demonstrate that WC data can train stronger language models (LM) for reasoning tasks than SE data when constrained by a fixed compute budget. A relevant area for future work, and a current limitation of this study, is to explore the conditions under which WC data consistently outperforms SE data in model finetuning (e.g., based on relative gains/losses in terms of coverage, diversity, and false positive rate). Additionally, we focus on establishing the utility of WC data through sequence-based supervised finetuning, given its widespread use. However, it would also be valuable to examine the behaviors of WC and SE data in iterative finetuning (Singh et al., ), as well as supervised finetuning through logit matching. Finally, an essential aspect of training reasoning models involves verification (Cobbe et al., ), and it would be appropriate to investigate the impact of WC and SE data on training LM verifiers for reasoning tasks.

## Appendix B Extending our results to coding tasks

Figure: (a) Coverage on MBPP.
Refer to caption: x34.png

Figure: (a) Finetuning Gemma-7B.
Refer to caption: x36.png

Here, we aim to understand the utility of the synthetic data from the Gemma2-9B (WC) and Gemma2-27B (SE) model on coding tasks. To this end, we generate candidate solutions for the MBPP (Austin et al., ) dataset from WC and SE models at the low and high sampling budgets and finetune models in three setups on these data. We use the santizied version of MBPP(^5^55[https://huggingface.co/datasets/google-research-datasets/mbpp/viewer/sanitized](https://huggingface.co/datasets/google-research-datasets/mbpp/viewer/sanitized)) containing $427$ problems overall; we used $3$ problems for fewshot prompting (used for sampling from the models), $324$ problems for synthetic training data generation, and $100$ problems for validation. The candidate solutions are filtered by the unit tests that accompany each instance of the dataset. After finetuning, we evaluate the LMs on $164$ problems from the HumanEval dataset (Chen et al., ).

We compare the coverage and diversity of the synthetic datasets in Figure 18 and observe that the coverage of the WC model is higher than SE at low data regime while it is similar to SE in the high sampling budget regime. In addition, we find that the diversity of the WC model is more than that of the SE model for the low and high sampling budgets. Subsequently, we finetune Gemma-7B, Gemma2-9B, and Gemma2-27B models with the ground-truth and synthetic datasets and evaluate on HumanEval (Figure 19). Our empirical findings indicate that finetuning with WC data outperforms SE data for the student-LM and WC-LM finetuning setups, while the performances are similar for SE-LM finetuning setup at the low sampling budget. At the high sampling budget, where the models have similar coverage, we find that training with the SE data is better for student-LM finetuning while WC-data is better for WC-LM finetuning. This might be attributed to the limited dataset size of MBPP and similar coverage by WC and SE models at the high sampling budget.

## Appendix C Qualitative Examples

We present a few qualitative examples for model-generated solutions that lead to the correct final answer with incorrect (or correct) reasoning for the MATH dataset. We provide two bad reasoning examples in Table 2 and Table 3. The existence of such examples contributes to the false positive rates in the synthetic dataset. In addition, we provide a good reasoning example in Table 4.

**Table 2: Qualitative example for incorrect reasoning the model generated data for MATH.**
| Problem: |
| --- |
| When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc=180$. |
| Ground-truth: |
| We first factor 180 into three positive integers from the set $\{1,2,3,4,5,6\}$. Since $180>5^{3}=125,$ at least one of the integers must be 6. Since $180>5^{2}\cdot 6=150$, at least two integers must equal 6. Indeed, $180=5\cdot 6\cdot 6$ is the only such way to factor 180. Therefore, $(a,b,c)=(5,6,6),(6,5,6),(6,6,5)$ are the only possibilities for $a,b,c$. Each occurs with probability $\left(\frac{1}{6}\right)^{3}=\frac{1}{216}$, so the probability that $abc=180$ is $3\cdot\frac{1}{216}=\boxed{\frac{1}{72}}$. |
| Generated solution (Incorrect reasoning): |
| There are $6^{3}=216$ possible outcomes. The numbers $1,3,6$ are the only possible values for $a,b,c$. If $a=1$, then $b=3$ and $c=6$, so the probability of this outcome is $\frac{1}{216}$. If $a=3$, then $b=1$ and $c=6$, so the probability of this outcome is also $\frac{1}{216}$. If $a=6$, then $b=3$ and $c=1$, so the probability of this outcome is also $\frac{1}{216}$. Therefore, the probability that $abc=180$ is $\frac{1+1+1}{216}=\boxed{\frac{1}{72}}.$<br>Final Answer: The final answer is $\frac{1}{72}$. I hope it is correct. |

**Table 3: Qualitative example for incorrect reasoning the model generated data for MATH.**
| Problem: |
| --- |
| A point is selected at random from the portion of the number line shown here. What is the probability that the point is closer to 4 than to 0? Express your answer as a decimal to the nearest tenth.<br>[asy]unitsize(0.2inch);<br>draw((0,0)--(5,0));<br>draw((0,-0.1)--(0,0.1));<br>draw((1,-0.1)--(1,0.1));<br>draw((2,-0.1)--(2,0.1));<br>draw((3,-0.1)--(3,0.1));<br>draw((4,-0.1)--(4,0.1));<br>draw((5,-0.1)--(5,0.1));<br>label(""0"",(0,-0.1),S);<br>label(""1"",(1,-0.1),S);<br>label(""2"",(2,-0.1),S);<br>label(""3"",(3,-0.1),S);<br>label(""4"",(4,-0.1),S);<br>label(""5"",(5,-0.1),S);<br>[/asy]. |
| Ground-truth: |
| We find the point which is equidistant from 0 and 4. Clearly, this occurs at 2. So, for all $x>2$, $x$ is closer to 4 than 0. So, the probability is equal to the length of this region $\frac{5-2}{5}=\boxed{.6}$. |
| Generated solution (Incorrect reasoning): |
| The total length of the number line is $5-0=5$. The total length of the portion of the number line from 1 to 4 is $4-1=3$. Therefore, the probability is $\frac{3}{5}=0.6=0.60=\boxed{0.6}$.<br>Final Answer: The final answer is $0.6$. I hope it is correct. |

**Table 4: Qualitative example for correct reasoning the model generated data for MATH.**
| Problem: |
| --- |
| How many ways are there to put 4 distinguishable balls into 2 indistinguishable boxes? |
| Ground-truth: |
| In this problem we don’t care which box is which, we only care which balls are together and which ones aren’t. For each ball, there are 2 choices of which box to place it in. Since this choice is independent for each of the 4 balls, we multiply the number of choices together. Hence there are $2^{4}=16$ ways to place 4 distinguishable balls into 2 distinguishable boxes. We then divide by the number of ways to arrange the boxes. There are $2!=2$ ways to arrange the 2 boxes, so there are $\frac{16}{2}=\boxed{8}$ ways to arrange 4 distinguishable balls into 2 indistinguishable boxes. Note: This method does not generalize if there are more than 2 boxes. |
| Generated solution (Correct reasoning): |
| If we label the boxes as 1 and 2, then there are $2^{4}=16$ ways to put 4 balls into two boxes. However, since the boxes are indistinguishable, we must divide by $2!$, so the answer is $\frac{16}{2}=\boxed{8}$.<br>Final Answer: The final answer is $8$. I hope it is correct. |

## Appendix D Coverage Trends

We present the coverage trends for diverse sampling budgets on the MATH and GSM-8K dataset in Figure 20(a) and Figure 20(b), respectively.

Figure: (a) MATH dataset.
Refer to caption: x39.png

## Appendix E Data analysis: GSM-8K

We presented the coverage, diversity, and false positive rate of the synthetic data from Gemma2-27B and Gemma2-9B on the MATH dataset in the main text. In Figure 21, we present these metrics for the GSM-8K dataset.

Figure: (a) Coverage on GSM-8K.
Refer to caption: x41.png

## Appendix F Solving problems across levels for MATH

Figure: Figure 22: Coverage across different levels of the MATH dataset. With compute-matched sampling, the coverage of the Gemma2-9B model is more than the Gemma2-27B model across all levels of the MATH dataset.
Refer to caption: x44.png

We present the effect of repeated sampling from the weak but cheaper LM and stronger but expensive LM on solving the problems across different levels for the MATH dataset in Figure 22.

**Table 5: Qualitative example from Level 5 of MATH dataset that gets solved by repeated sampling from Gemma2-9B (30 solutions) but remains unsolved by Gemma2-27B (10 solutions) at fixed sampling budget.**
| Problem: |
| --- |
| Solve for $n$: $\frac{2-n}{n+1}+\frac{2n-4}{2-n}=1$. |
| 9B-generated solution: |
| Let $x=\frac{2-n}{n+1}$. Then we have x+2n-42-n=1⇒x+2(n-2)2-n=1⇒x+-2(2-n)2-n=1⇒x-2=1⇒x=3.Therefore, $\frac{2-n}{n+1}=3$, so $2-n=3n+3$ and $4n=-1$ and $n=\boxed{-\frac{1}{4}}$.<br>Final Answer: The final answer is $-\frac{1}{4}$. I hope it is correct. |

## Appendix G Experimental Setup Details

As mentioned in the main text, we mainly experimented with MATH (Hendrycks et al., ) and GSM-8K (Cobbe et al., ) datasets, which are widely adopted for evaluating reasoning and mathematical problem solving. MATH consists of competition level problems with various levels of difficulty (Level 1-5) and GSM-8K comprises of grade school level math problems. Each dataset contains $7500$ math problems in their training split.
We evaluate the models on $500$ problems from the MATH test split (Lightman et al., ) and $1319$ problems from the GSM-8K test split. Further, we use $500$ problems from the MATH test split and $500$ problems from GSM-8K as the validation dataset.

We generate the solutions for the problems in the MATH using a 4-shot prompt and for GSM-8K using an 8-shot prompt. We generated the candidate solutions in the synthetic dataset using TopK (K$=3$) strategy with a temperature of $0.7$. We finetuned the Gemma2-9B and Gemma2-27B models with a batch size of $32$ for $600$ and $6000$ steps under the low and high sampling budget, respectively. During the fine-tuning process, we save 10 equally-spaced checkpoints and choose the one that yields the highest validation accuracy. In addition, we train the Gemma1-7B model with a batch size of $8$ for $2400$ and $24000$ step under the low and high sampling budget, respectively. We perform a hyperparameter search for the learning rates $\{1e-7,5e-7,1e-6\}$ based on the model performance on the validation datasets.

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: I will try and cluster my questions in sensible groups.

1. _Theoretical Understanding_:
   - Can you provide theoretical insights into when WC sampling should outperform SE sampling?
   - How does the optimal sampling ratio change with model size and task complexity?
   - What are the key factors that determine the success of weak-to-strong improvement?

2. _Methodology_:
   - How would the results change with more sophisticated filtering strategies?
   - Could you provide more details about the specific prompting strategies used?
   - How sensitive are the results to the choice of temperature and sampling parameters?

3. _Generalisation_:
   - What characteristics of a task make it more/less suitable for WC sampling?
   - How would the results scale to even larger model sizes?
   - What is the relationship between FPR and final model performance?

4. _Practical Implementation_:
   - How would you recommend implementing this in scenarios without ground truth?
   - What modifications would be needed for different domains or tasks?
   - Could you provide more detailed guidance on optimal sampling strategies for different scenarios?
soundness: 3
strengths: 1. _Originality_:
   - Introduces a novel compute-matched sampling framework with clear mathematical foundations
   - Proposes a new "weak-to-strong improvement" training paradigm that challenges conventional wisdom
   - Provides a fresh perspective on the compute-quality trade-off in synthetic data generation

2. _Experimental Rigour_:
   - Comprehensive evaluation across multiple dimensions:
     * Multiple model pairs (both open and closed models)
     * Various compute budgets and training paradigms
     * Different dataset sizes and difficulty levels
   - Thorough ablation studies that isolate the impact of coverage and diversity
   - Both human and automated evaluation of false positive rates
   - Clear validation of results through transfer learning (Functional MATH)

3. _Practical Impact_:
   - Demonstrates significant cost savings potential (0.15x cost for comparable or better performance)
   - Shows consistent improvements across model sizes (7B to 27B)
   - Provides actionable insights for practitioners
   - Results particularly relevant given the trend of improving smaller models

4. _Technical Depth_:
   - Rigorous mathematical formulation of compute-matching
   - Analysis of traed-offs between coverage, diversity, and error rates
   - Ablation studies support main claims
   - Clear empirical validation of theoretical framework
summary: This paper challenges the common practice of using strong but expensive (SE) language models to generate synthetic training data, proposing instead that using weaker but cheaper (WC) models may be more compute-optimal. The authors introduce a "compute-matched sampling" framework that enables fair comparison between WC and SE models by accounting for their relative compute costs. At a fixed compute budget, this framework shows that one can generate P_SE/P_WC more samples from a WC model than an SE model. The authors evaluate this approach across multiple model pairs (Gemma2 9B/27B and Gemini Flash/Pro), tasks (primarily mathematical reasoning), and training paradigms (knowledge distillation, self-improvement, and a novel "weak-to-strong improvement"). They assess the generated data along three key dimensions: coverage (problems solved), diversity (unique solutions per problem), and false positive rate (correct answers with incorrect reasoning). The results consistently show that training with WC-generated data outperforms SE-generated data when properly compute-matched.
weaknesses: 1. _Theoretical Foundation_:
   - Lacks formal analysis of when WC sampling should outperform SE sampling
   - No theoretical bounds on the optimal sampling ratio
   - Missing analysis of the relationship between model size and optimal sampling strategy
   - Limited exploration of failure modes and their characteristics

2. _Methodology Limitations_:
   - Heavy reliance on ground truth for filtering solutions
   - Limited exploration of alternative filtering strategies
   - FPR evaluation methodology could be more robust (50 human samples probably insufficient)
   - Some key implementation details relegated to appendices

3. _Generalisation Concerns_:
   - Primary focus on mathematical reasoning tasks
   - Limited exploration of other domains (coding results show context-dependency)
   - Unclear scalability to larger model sizes
   - Performance on more complex reasoning tasks not fully explored

4. _Practical Considerations_:
   - Deployment challenges in scenarios without ground truth not fully addressed
   - Resource optimisation strategies could be explored more
   - Limited discussion of integration with existing training pipelines
   - Cost-benefit analysis could be more comprehensive across different scenarios

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. Although both low and high budgets are studied, could you please provide the results of an extremely high budget where the cost is not an important factor? This should be indicative of diverse data scales.
2. Despite the train-test splits of MBPP, this paper only trains models on MBPP and tests them on HumanEval. The testing results on MBPP are expected to be provided for a more comprehensive understanding.
3. Writing：
- 1. All the ref links are invalid.
  2. l101, l104: "i.e." should be "i.e.".
  3. l103: grammar error for "we supervise finetune".
  4. l109: use \citet{} for "(Zelikman et al., 2024;Singh et al., 2023).".
soundness: 3
strengths: 1. The research question is significant, focusing on performance comparison of data sampled from WC and SE models, respectively.
2. The findings are impressive, challenging the traditional belief that data from a strong model is better for finetuning models.
3. The evaluation settings are diverse, demonstrating the effectiveness and robustness of this method, despite only the Gemma series models.
summary: This paper revisits the trade-offs between generating synthetic data using a stronger but more expensive (SE) model versus a weaker but cheaper (WC) model, and finds that at a fixed sampling compute budget, finetuning LMs with data from a WC model can consistently outperform data from a SE model in multiple settings.
weaknesses: 1. This paper centers exclusively on the Gemma series, and it is essential to extend the analysis to the Llama series to demonstrate the robustness of the conclusions.
2. While the paper aims to highlight the lower computational cost of the WC model for data synthesis (particularly important for large-scale data generation), all the experiments are conducted on relatively small datasets. This discrepancy undermines the overall contribution of the paper.
3. Compared to the SE model, the WC model can be regarded as a more diverse yet lower-quality variant. Therefore, it is crucial to compare it with techniques designed to enhance output diversity. Specifically, if adjusting the sampling temperature of the SE model consistently results in performance degradation relative to the WC model, this suggests that the WC model provides a superior quality-diversity trade-off compared to merely increasing the sampling temperature.

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confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: It seems that the difference in data quality between the WC and SE models becomes larger at lower budgets. Is it possible that the WC and SE models generate data of similar quality when the budget is very high?
soundness: 3
strengths: The paper is well-structured and clearly written, making the methodology and results easy to follow.

The experiments are well-executed and provide convincing evidence of the benefits of the proposed approach.

It addresses a critical issue in synthetic data generation, offering a valuable contribution to this area of research.
summary: This paper presents the novel observation that generating synthetic data using a weaker but cheaper (WC) model is more effective than using a stronger but more expensive (SE) model. The authors demonstrate that, under the same budget, data generated by WC models tend to have higher coverage and diversity, though with a corresponding increase in false positive rates. Additionally, they show that models fine-tuned on WC-generated data consistently outperform those trained on SE-generated data.
weaknesses: The conclusion may not hold when using models from different companies. Based on my experience, under the same budget, data generated by a larger model like Qwen2.5 7B could outperform that of a smaller one like Gemma2 2B.

The paper could benefit from experimenting with more complex reasoning tasks, such as tree search algorithms, and using a reward model to evaluate the quality of the generated data.

---

confidence: 3
contribution: 2
initial_rating: 6
presentation: 4
questions: * Your distillation setup is limited to finetuning. One setup where it would be more realistic to not have enough budget is pretraining. Do you have any results on this? I of course do not expect to pretrain a network until convergence during the rebuttal, but it would already be helpful if you could show the first couple of iterations just to make sure the worse data (higher FPR) does not seem to converge to a much worse model.
* I'd be interested in sample-matched figures. The figures where I'd be most interested in a sample-matched comparison are Figures 4c and 5c. This would allow finding out if a small model can successfully improve a larger model, which would challenge beliefs in the field.
* Just to go sure: In the self-improvement setups, you keep training a model iteratively on its own generations from the current parameters? Or do you mean that you finetune a "fresh" 7B model using an already converged 7B model?
soundness: 4
strengths: * The paper investigates not only knowledge distillation, but also a self-improvement setup
* In Figures 4 and 5, it is interesting that training on model-generated solution paths (one per question; at least in the "27B low compute" setup) gives a better performance than training on human-provided solution paths (also one per question)
* Figure 7 carries an interesting finding: Despite training on the data generated by the small model which has more errors, the ultimate trained model does not have more errors in its reasoning. This implies that the additional data mitigates its lower quality, which might add evidence to the discourse beyond the setup studied in this paper.
* The writing and flow of experiments is mostly clear
summary: The paper investigates whether it is better to (self-)distill from a Gemma-27B LLM or to distill three times more finetuning data from a three times smaller Gemma-9B model. It finds that the three times more data of the smaller model, despite including more errors, leads to a higher performance of the finetuned student model.
weaknesses: * In 21 of the 22 Figures, the paper hinges on "matching the compute", i.e., being allowed to generate 3 times more data when using the 3 times smaller LM. This confounds two factors of variation, making it hard to interpret the findings. This is the main weakness of the paper. One idea to improve on this weakness would be to test out distilling 1, 3, and 9 samples from the large LLM and 3, 9, and 27 samples from the small LLM (instead of the current 1+10 vs 3+30), so that there are both overlapping settings with a matched number of samples and with a matched compute.
* In the only figure where the small LLM is compared to the large LLM without this advantage (Figure 20 in the appendix), the large LLM produces better training data. It can be expected that if we use the large LLM to generate enough data until the student model converges, it will make a better distilled model. Thus, the only real application of the proposed method is when we do not have enough budget to produce enough data to converge. For the finetuning setup of the paper, that would amount to not being able to generate data for 8k-12.5k questions. This is a setup with limited applicability in practice. It would increase the contribution (score) of the paper to investigate problems where budget limits are hit more frequently in practice, like pretraining, see also my question below.
* Relative and absolute increases are reported inconsistently. E.g., in Figure 3b the fact that the proposed small model finds 11 instead of 5 solution paths per question (when it is allowed to generate 3 times more paths in total) is reported as a 125% increase (line 268), whereas the fact that 24% of its solutions paths are wrong compared to 17% of the large model is reported as a 7% increase (line 310). This inconsistency becomes problematic when reporting the increase on percentage numbers (e.g., line 258), where it is unclear whether this is a relative or absolute increase. Keeping the reporting consistent would increase both the presentation and the scientific soundness scores.
* The paper only evaluates Gemma (/Gemini) models. It would help judge the generalization of the claims (and increase the contribution score) to test it out on at least one other LLM, like a Llama model.
* The datasets are very limited to two math datasets, limiting the contribution. As above, more datasets would help judge the range of applicability, especially whether it also works on non-math and non-reasoning datasets.
* The paper does not compare to baselines, despite citing multiple closely related approaches
* The method still requires annotated data, because the LLM-generated data needs to be filtered out if it does not match the GT. It would increase the applicability of the score (and thus the contribution score) if there would be an ablation without filtering, i.e., answering whether the unfiltered erroneous data from the smaller model can still train a better model.

Small notes that did not influence my score and don't need to be rebuttled, I just note them to make the camera-ready better:
* The first paragraph of Section 3 could be shortened; it's message (in Equation 1) is just "if a model has x times more parameters, it takes x times longer to generate".
* typo in line 54, "filters"
* typo in line 103 "we supervise finetune"
* typo in line 151, "consists"
* typo in line 157, "for training student LM"
* typo in line 241, "that where"
* The references exclusively list arxiv versions of the papers, not their actual published versions
* The reference .bib file should best use double brackets for "{{OpenAI}}", "{{The Llama Team}}", to prevent the ill formatting in line 483 ("Team, 2024; Anthropic, 2024; AI, 2024")

Meta-review: The authors study scaling for post-training data and whether it's cost (either in $ or compute) efficient to use more data from weaker models vs less data from stronger ones. The authors show in extensive experiments that weaker models can give gains over stronger models. 

The paper studies a timely and important area (synthetic data and scaling) and has an interesting take on the problem (weak models are better than strong models). As a drawback, however, the paper's focus on post-training synthetic data limits it given the comparatively low cost of this type of approach compared to human-generated post-training data or pre-training scaling. As reviewer gC6v notes, "I see the paper as a scientifically thorough study of a niche problem." Reviewer rLZh's comment "While the paper aims to highlight the lower computational cost of the WC model for data synthesis (particularly important for large-scale data generation), all the experiments are conducted on relatively small datasets. This discrepancy undermines the overall contribution of the paper." this comes from a similar place of arguing that the generally low overall cost of these settings limits is broader applicability.

Decision: Accept (Poster)