Maximizing the Potential of Synthetic Data: Insights from Random Matrix Theory

We thank the reviewer for their thoughtful feedback.

Limitations of the Data Model: Our data model, based on Gaussian mixtures with symmetric clusters, is a simplification chosen for analytical tractability. It allows us to derive explicit theoretical results using random matrix theory, capturing key aspects of high-dimensional data behavior, such as covariance estimation inconsistency and its impact on distribution shifts. These insights provide a foundation for future extensions to more complex distributions.

Fitting One Gaussian per Class: Fitting one Gaussian per class with different means would introduce additional complexity. While possible, it would require careful handling of class-specific statistics and might lead to more intricate expressions in the theoretical results. We consider this an interesting direction for future work to better reflect scenarios with asymmetric class distributions.

Input-Dependent Label Noise: Modeling label noise that depends on the input introduces significant analytical challenges and may be intractable within our framework. We focus on input-independent noise for tractability, providing foundational insights that can inform more complex models in future studies. Empirical studies of input-dependent noise are valuable but beyond the scope of our theoretical analysis.

Correspondence of Label and Feature Noise to Real-World Errors: Label noise can arise from misannotations or errors in the labeling process, while feature noise can result from measurement errors or model approximations in synthetic data generation. Our model captures these through parameters $\varepsilon$ (label noise) and $\hat \eta$ (feature noise due to covariance estimation), helping understand their impact on performance.

Values of $\hat n$ in MNIST Experiments:

• Best Stats Estimation: $\hat n = 50{,}000$ (all training data).
• Weak Stats Estimation: $\hat n = 1{,}000$.
• Very Weak Stats Estimation: $\hat n = 100$.
• No Covariance Estimation: Covariance set to identity.

We will include these details in the revised paper.

LLM Safety Experiment and Presence of Feature Noise: While we focus on label noise, feature noise is inherently present because the synthetic data are generated by another language model. This reflects real-world scenarios where both types of noise affect performance.

Error Bars in Figure 7: Including error bars would enhance reliability. However, due to computational constraints, we prioritized adding more experiments, such as those involving LLMs. In future work, we aim to include error bars where feasible to improve the robustness of our experimental results.

Interpretation of $(\rho, \phi) = (0.5, 0.5)$ as Weak Supervision: With these values, there is a 50% chance of selecting any synthetic data point, reducing the amount of noisy data included compared to "No supervision" with $(\rho, \phi) = (1, 1)$. This mitigates the negative impact of incorrect labels, which is why we consider it weak supervision.

LLM Q&A Safety Generation Experiment Details: The accuracy is evaluated using Llama-Guard-3.1. We used two models to annotate labels to introduce variability and potential label noise, simulating realistic scenarios. Including a comparison of labeling models with ground truth is a valuable suggestion for future work. During supervised fine-tuning, the model is trained on data containing both safe and unsafe answers, learning to produce safe responses when given new prompts.

The reviewer thanks the author for the response.

1. Insights for Practitioners. My primary concern remains: could the author summarize the insights from this theoretical paper that could directly benefit practitioners? While I appreciate the theoretical contributions of this paper and understand that some results are derived under Gaussian data assumptions—which may approximate real high-dimensional data in certain cases—I feel the connection between theory and practical scenarios could be developed further. For example, the author states, "Our findings identify conditions where synthetic data could improve performance, focusing on the quality of the generative model and verification strategy." Could some of these conditions be verified in practice? Mixing real and synthetic data might lead to a U-shaped performance curve due to distribution shift. Could we predict whether such a U-shape would occur? Furthermore, with the random matrix analysis, what new insights are obtained compared to prior work by Feng et al. (2024)?

2. Input-Dependent Label Noise. From my understanding, the theoretical framework in Feng et al. (2024) encompasses input-dependent errors, whereas the current setting considers only input-independent noise. While the author provides stronger theoretical results, this seems to narrow the scope compared to previous work. As the author notes, "Label noise can arise from misannotations or errors in the labeling process", if the synthetic data are annotated using the same pipeline as the real data, the noise levels should be similar. However, if synthetic labels are generated by another model, input-dependent errors would arise. In the experiments, the authors write, "We used two models to annotate labels to introduce variability and potential label noise," implying that the introduced noise should also be input-dependent. This limitation should be explicitly discussed.

3. Figure 7 Clarifications. In Figure 7 (left), both feature noise and label noise are included. However, the subtitles of the two plots are somewhat misleading. The right plot is labeled "Feature Noise," while the left plot is labeled "Label noise $\epsilon=0.3$." This gives the impression that there is no feature noise in the left plot, which is incorrect. I suggest the author revise the subtitles for clarity.

4. Interpretation of $(\rho, \phi) = (0.5, 0.5)$ as Weak Supervision. Could the author clarify the meaning of the proportion of synthetic data, which is used as the x-axis in all figures? Is it the proportion of synthetic data in the entire training set before verification and selection? The paper assumes knowledge of which data are synthetic and which are real and the verification apply only to the synthetic portion then. If so, different lines with varying verification methods but the same proportion of synthetic data involve different total amounts of data being used for training. Why not control for the total number of training samples in each case? Initially, I assumed that the proportion of synthetic data referred to the training set after verification. Under that assumption, $(\rho, \phi) = (0.5, 0.5)$ would correspond to no supervision, i.e., $(\rho, \phi) = (1, 1)$. Even if verification applies only to synthetic data, the "weak supervision" here is actually randomly using half the synthetic data. Why is this termed supervision? Isn't it simply a way of altering the ratio between real and synthetic data, given the assumed knowledge of data sources?

5. In the caption of Figure 3, the authors write "The parameter $\epsilon$ is variable depending on the proportion of synthetic data by taking it equal to the misclassification error corresponding to training a classifier on synthetic data only". What does it mean?

The reviewer thanks the authors for their further responses and clarifications. I still have some follow-up questions:

1. Insights for Practitioners: I agree that including feature shift in analyzing synthetic data is important; however, I would like to see more actionable insights beyond simply stating that a distributional shift exists and affects performance. Since the authors provide a rigorous characterization, could some insights be included on how supervision may influence this shift and thereby affect performance?

Additionally, I do not fully agree with the statement: "the greater is this shift, the lower is the quality of synthetic data." Why should a shift directly correlate with lower quality? There could be beneficial distributional shifts. For example, in mathematical reasoning tasks, if the real distribution of problems spans different difficulty levels, it might be advantageous to augment more intermediate-level problems to help the model learn to solve harder problems. I hope this analogy can help the authors gain deeper insights into the distributional shifts and their potential impacts.

2. Input-Dependent Label Noise: In the Q&A experiments, why should the label noise in this experiment be independent of the input? Label noise is determined by the performance of the labeling models, which would naturally vary based on the input characteristics.

3. Interpretation of $(\phi, \psi) = (0.5, 0.5)$ as Weak Supervision: My concern here is that $(\phi, \psi) = (0.5, 0.5)$ does not actually involve supervision. It simply represents taking a random half of the synthetic data, as you already know which data is synthetic. To truly qualify as weak supervision, a setting such as $(\phi, \psi) = (0.6, 0.8)$ would involve some level of actual supervision.

We thank the reviewer for their feedback and for highlighting areas for improvement.

Comparison with Jain et al. (2024): Our work differs primarily in the statistical modeling of synthetic data. Jain et al. (2024) model differences between real and synthetic data through mean shifts. We consider both mean and covariance estimation errors, acknowledging that covariance estimation is inconsistent in high-dimensional settings. We also include label noise in our analysis, which is not considered in their work. This allows us to study the combined effect of feature and label inaccuracies on model performance.

U-Shaped Curve in Figure 6: The performance increase when the proportion of synthetic data exceeds 0.8 under "No supervision" is due to the large amount of synthetic data starting to outweigh the negative impact of label noise. With label noise $\varepsilon = 0.2$, 80% of the synthetic labels are correct. As more synthetic data is added, the correct labels provide sufficient signal for the model to improve. This effect diminishes when label noise is higher, as incorrect labels have a more detrimental impact.

Typos and Literature Inclusion: We will carefully revise the paper to correct any typos and ensure all relevant literature, including the papers you mentioned, is appropriately cited and discussed.

We thank the reviewer for their engagement and support to our work, and for all the guidelines to improve our manuscript.

We also want to point out that, in addition to correcting the typos (marked in blue), we also included the additional references in the related work section of the updated manuscript.

In the differences with Jain et al.: We apologize for our misunderstanding of the reviewer’s question. Beyond the differences in synthetic data modeling (as described in our initial rebuttal) and the types of experiments (outlined above), we would like to highlight another key distinction from Jain et al. (2024): our analysis relies on Random Matrix Theory, which provides a deterministic theoretical performance of the model (Theorem 4.2). This performance depends solely on the characteristics of the training data (such as the mean and covariance in Gaussian Mixture Models) rather than the random nature of the data itself, as in Jain et al. (2024). Additionally, our single analysis holds for both low and high-dimensional regimes, whereas in Jain et al. (2024), they provide a different analysis for each regime, making our method easier to exploit and more generalisable.

We hope our response was satisfactory to the reviewer and remain available for any further clarification.

We thank the reviewer for their valuable feedback.

Definition of Synthetic Data Quality: In our paper, the quality of synthetic data is characterized by:

1. Sample Size $\hat n$ for Estimating Statistics: The parameter $\hat n$ represents the number of real data samples used to estimate the mean and covariance in our generative model. Larger $\hat n$ leads to more accurate estimates and higher-quality synthetic data.

2. Label Noise Rate $\varepsilon$: Indicates the fraction of incorrect labels in the synthetic data. Lower $\varepsilon$ means fewer label errors, contributing to better data quality.

By considering both feature quality (through covariance estimation) and label accuracy, our model captures key aspects affecting synthetic data utility.

Use of Gaussian Distributions: While real-world data often deviate from Gaussianity, we chose Gaussian mixtures for analytical tractability, enabling explicit formulas and rigorous analysis. Our findings extend qualitatively to non-Gaussian settings, supported by related work showing that results hold under broader conditions like sub-Gaussian distributions or bounded moments. Experiments with MNIST and LLMs, which are non-Gaussian, validate the applicability of our results.

Focus on Label Verification and Feature Fidelity: Our analysis accounts for feature noise by incorporating distribution shifts due to covariance estimation errors. While we focus on label verification, including feature noise in our model provides insights into how feature inconsistency affects performance. Developing methods for feature verification is a promising direction for future work.

Typographical Error in Figure 1: We apologize for the error and confirm that the correct value is $\hat n = 5 \times 10^3$. We will correct this in the revised paper.

Performance Increase with High Synthetic Data Proportion: In Figure 3, under weak supervision, the performance increase when synthetic data proportion exceeds 0.8 occurs because the large volume of synthetic data begins to compensate for label noise and distribution shifts. The correct labels (even with some noise) provide sufficient signal for the model to improve as more data is added. This effect is more pronounced when label noise $\varepsilon$ is not excessively high. When $\varepsilon$ is high, adding more noisy data continues to degrade performance, as shown in Figure 8.

Interpretation of $\hat \eta$ as Distribution Shift: The parameter $\hat \eta$ represents the ratio $p / \hat n$, capturing the extent of distribution shift due to covariance estimation errors. A higher $\hat \eta$ indicates greater estimation error in the covariance matrix, leading to a more significant distribution shift between synthetic and real data. This affects model performance, highlighting the importance of accurate covariance estimation in generating high-quality synthetic data.

We thank the reviewer for their constructive remarks.

Model Relevance and Use of $L^2$ Loss: While using a regression model as a linear classifier is not standard in practice, we chose this setup for several reasons:

1. Analytical Tractability: The linear classifier with an $L^2$ loss allows us to derive closed-form solutions and perform rigorous theoretical analysis using random matrix theory. This helps us understand the fundamental effects of synthetic data, label noise, and distribution shifts in high-dimensional settings.

2. Isolating Effects: By focusing on a linear model, we can isolate and study the impact of synthetic data quality and verification strategies without the complexities introduced by non-linear models. This simplification aids in deriving insights applicable to more complex models.

3. Practical Scenarios: Linear models are still relevant, especially when used as classifiers on features extracted from neural networks, which is common in data-limited situations.

Regarding the connection to Liao and Couillet (2019), our work extends their analysis by incorporating a mixture of real and synthetic data, providing a framework to study the effect of synthetic data on model performance. While the $L^2$ loss is less common in classification, it provides analytical tractability. Our experiments demonstrate that our findings hold qualitatively when using other loss functions like cross-entropy.

Practical Relevance of Assumptions: Practitioners often know which data is real and synthetic, especially when augmenting datasets. Our analysis covers various proportions of real and synthetic data through the parameter $\pi$, allowing us to study scenarios where real data is limited. Although we cannot directly analyze the case with $\pi = 0$ (training only on synthetic data) in Theorem 4.2 due to undefined quantities, we address this separately in Subsection 4.2 (Corollary 4.3), generalizing previous results to include high-dimensional settings.

Mixing Strategy with Mixing Parameter $\alpha$: We appreciate the suggestion to consider a general mixing constant $\alpha$. We highlight that this can be easily incorporated into our current analysis using the exact same derivations. We will provide in the final version a general result incorporating the mixing parameter $\alpha$ as suggested by the reviewer and investigate how the optimal value of $\alpha$ depends on other constants like $\pi$, $\varepsilon$, and the verification quality.

Connection to Literature:

• Jain et al. (2024): Our work differs by considering both mean and covariance estimation errors in the synthetic data, acknowledging that covariance estimation is inconsistent in high-dimensional settings. We also incorporate label noise, which is not considered in their analysis.

• Dohmatob et al. (2024): We extend their study by including both label and feature noise, providing a more comprehensive analysis of the synthetic data's impact on model performance.

Scaling Laws and Concrete Setup: Our experiments illustrate the theoretical findings, showing how key parameters like the ratio $p/n$, label noise $\varepsilon$, and verification quality $(\rho, \phi)$ affect model performance. We would appreciate it if the reviewer could elaborate more on their question “Can you instantiate your results on a concrete setup and show the corresponding scaling laws?”.

Thanks for the response. I'm not satisfied with a couple of points made.

Practitioners often know which data is real and synthetic, especially when augmenting datasets.

This is simply not true, and will get worse over time, as more synthetic data pollution occurs; it's not possibly to reliable tell which part of the training corpus of a LLM is real and which part is synthetic (can't tell which images on the internet are real and which ones are synthetic). I'm not referring here to synthetic data generated and controlled by the practitioner; I mean synthetic data which finds its way in the training corpus (e.g because data was scrapped from the Internet, etc.).

Practical Scenarios: Linear models are still relevant, especially when used as classifiers on features extracted from neural networks, which is common in data-limited situations.

The question is not whether linear models are relevant or irrelevant. I'm fine with linear models for theoretical analysis of phenomena in relevant regimes. The issue here is that nobody in ML fits a linear model via a regression loss, to use it as a classifier at inference time. From a technical standpoint, I think you can get just as much "analytical tractability" by working with an actual classification loss (e.g logistic), and using classical tools around Gaussian comparison inequalities (Gordon, etc.). The justification for L^2 loss here seems superficial.

We thank the reviewer for their engagement and thoughtful feedback throughout the rebuttal process. We also sincerely appreciate the reviewer's interest and support for our work, as well as the increase in their score.