Towards a Theoretical Understanding of Synthetic Data in LLM Post-Training: A Reverse-Bottleneck Perspective

## Abstract

Abstract Synthetic data has become a pivotal resource in post-training tasks for large language models (LLMs) due to the scarcity of high-quality, specific data. While various methods have been developed to generate synthetic data, there remains a discernible gap between the practical effects of synthetic data and our theoretical comprehension.
To address this challenge, we commence by presenting a detailed modeling of the prevalent synthetic data generation process.
Building upon this modeling, we demonstrate that the generalization capability of the post-trained model is critically determined by the information gain derived from the generative model, as analyzed from a novel reverse-bottleneck perspective.
Moreover, we introduce the concept of Generalization Gain via Mutual Information (GGMI) and elucidate the relationship between generalization gain and information gain.
This analysis serves as a theoretical foundation for synthetic data generation and further highlights its connection with the generalization capability of post-trained models, offering an understanding about the design of synthetic data generation techniques and the optimization of the post-training process.
We open source our code at https://github.com/ZyGan1999/Towards-a-Theoretical-Understanding-of-Synthetic-Data-in-LLM-Post-Training .

## 1 Introduction

The efficacy of large language models (LLMs) is extensively influenced by both the volume and quality of the training data, as established by the widely acknowledged scaling laws (Kaplan et al., ).
Given the inherent sparsity of data available during the post-training phases of LLMs, synthetic data plays a critical role, particularly during fine-tuning and alignment processes. Over the past decades, the LLM community has increasingly employed synthetic data to augment training in scenarios where real data is scarce.
As of September 2024, there are over 1,000 datasets labeled as “synthetic” on the Hugging Face platform(^1^11https://huggingface.co/). Several leading-edge large language models, including LLaMA (Dubey et al., ), Falcon (Almazrouei et al., ), Qwen (Bai et al., ),
and GPT-4 (OpenAI et al., ), have also reported utilizing synthetic data during their post-training stages. These instances underscore the pivotal role of synthetic data in enhancing the post-training of LLMs.

Numerous methodologies for synthetic data generation have been advanced (Patel et al., ; Møller et al., ; Park et al., ), yet the most prevalent and efficacious approach within the community involves generating synthetic data through sampling from a proficiently trained generative model, often another LLM tailored for specific domain tasks. To delineate this process more precisely, Long et al. () describe the generation of synthetic data as follows: a well-trained generative model $M$
is utilized, and synthetic data $S_{\text{gen}}$ is produced by sampling from $M$, conditioned on a set of prompts $p$, just as illustrated in the lower part of Figure 1 (a).

Synthetic data with such a generation manner is widely recognized and has been verified to be effective in LLM post-training practice.
However, several challenges persist that compromise its potential benefits. First, the quality and diversity of synthetic data can vary significantly depending on the generation method and the underlying model parameters (Koo et al., ). This variability can lead to inconsistent training outcomes and may not fully address the sparsity in real data.
Additionally, while synthetic data offers a promising solution to enrich the limited real data, ensuring that it sufficiently mimics real-world complexities without carrying over biases or errors from the original data is still a daunting task (Villalobos et al., ).
Addressing these challenges requires a nuanced understanding of both the generation processes and their interaction with model training dynamics.

Unfortunately, there remains a significant gap in the rigorous modeling of synthetic data, which in turn limits a deeper understanding of its inherent mechanisms (Liang et al., ). This lack of a comprehensive theoretical framework hinders our ability to predict the effectiveness of synthetic data across different LLM applications and constrains the optimization of generative models for more targeted data synthesis (Giles et al., ). Consequently, advancing our knowledge on how synthetic data interacts with LLMs during training phases is crucial for enhancing model performance and reliability,
and can enable the development of tailored synthetic datasets that more effectively address specific gaps in training data, thereby enhancing the overall performance and generalization capabilities of large language models.

In this paper, we endeavor to examine the influence of synthetic data on the post-training phases of large language models (LLMs) through an analytical lens focused on data distribution and information content.
Our investigation seeks to address the following theoretical questions:

- •
What underlies the effectiveness of synthetic data? How can we model the data generation process and connect it with the generalization capabilities of post-trained models?
- •
What is the reason for the effectiveness of synthetic data in LLM post-training?

In response to these inquiries, we introduce a theoretical framework designed to dissect the impacts of synthetic data on LLM post-training. The principal contributions of our study are outlined as follows:

- 1.
We develop a modeling of synthetic data generation from a distributional perspective, providing a theoretical foundation for understanding the generation process and its implications on LLM post-training.
- 2.
Drawing on this modeling, we propose a reverse-bottleneck framework that elucidates the mechanisms through which synthetic data influences LLM post-training.
- 3.
We perform a theoretical analysis from an information-theoretic standpoint, delivering several upper bounds that quantifies the expected generalization capabilities of LLMs when trained with synthetic data.

Figure: Figure 1: An overview of the synthetic data generation modeling and the relationships between the distributions. (a) The synthetic data generation process and the corresponding distribution compression process. (b) The relationships between the distributions in the generation process.
Refer to caption: x1.png

The remainder of this paper is structured as follows.
In Section 2, we provide a comprehensive review of literature pertinent to our research.
In Section 3, we first delineate the symbols and foundational concepts critical to our analysis, then introduce the modeling for synthetic data generation and bridge its connection with generalization capability of post-trained models.
Section 4 introduces our novel reverse-bottleneck framework, designed to assess the effects of synthetic data on post-training stages of LLMs, and to establish generalization error upper bounds.
The paper concludes with Section 5, summarizing our findings and discussing potential avenues for future research.

## 2 Related Work

### 2.1 Generative Data Augmentation

Generative models constitute a category of machine learning models that are specifically trained to create new data points mimicking the original data distribution. Various types of generative models have been developed, each suited to particular data types and model architectures. Notable among these are Variational Autoencoders (Kingma, ), Generative Adversarial Networks (Goodfellow et al., ), Normalizing Flows (Rezende & Mohamed, ), and, more recently, diffusion models (Rombach et al., ).
Building on this premise, generative data augmentation has emerged as a promising approach to bolster machine learning model performance (Yamaguchi et al., ). This technique involves scaling up the available training dataset by generating new data points from a limited pool of labeled data using generative models. Empirical evidence suggests that generative data augmentation is particularly effective across various tasks, including knowledge graph reasoning (Maharana & Bansal, ), text-to-image generation (Yin et al., ), and relation extraction from natural language texts (Hu et al., ).
Theoretical investigations have also been conducted to elucidate the underlying mechanisms through which generative data augmentation delivers these benefits (Zheng et al., ). Collectively, these advancements highlight generative data augmentation as a highly promising avenue for improving machine learning model performance, especially in scenarios characterized by a scarcity of labeled data.

### 2.2 Synthetic Data in LLMs

Large language models (LLMs), a specialized subset of generative models tailored for the text domain, have demonstrated remarkable capabilities in generating high-quality text data. Similar to traditional generative data augmentation, synthetic data produced by these models is increasingly utilized to enhance LLMs, particularly during post-training phases. Given the scarcity of labeled data in specific domains, synthetic data plays a crucial role in boosting the performance of LLMs across a variety of downstream tasks, including text classification (Li et al., ), clinical text mining (Tang et al., ), and code generation (Tsai et al., ).
However, unlike classic generative data augmentation, synthetic data within LLMs is typically generated by the language models themselves and often predominates the training data in post-training stages. This predominance stems from the high-quality demands of synthetic data in LLM contexts, which necessitates alignment with human intent. Efforts to enhance the quality of synthetic data in LLMs have included integrating methodologies such as active learning (Wagner et al., ) and reinforcement learning (Setlur et al., ).
Despite these advancements, the theoretical understanding of how synthetic data influences the learning process in LLMs remains limited. Key questions persist regarding the mechanisms through which synthetic data impacts LLM training and the optimal strategies for designing synthetic data to maximize LLM performance (Long et al., ). Addressing these questions is essential for furthering our comprehension and utilization of synthetic data in enhancing large language model efficacy.

### 2.3 Information Bottleneck Theory & Generalization Capability

The information bottleneck (IB) theory, as introduced by (Tishby et al., ), serves as a theoretical construct designed to elucidate the learning processes within neural networks. In essence, for a given Markov chain $X\rightarrow Z\rightarrow Y$,
the IB theory aims to optimize the learning process by maximizing the mutual information between $Y$ and $Z$ while minimizing the mutual information between $X$ and $Z$ (Hu et al., ).
IB theory has been widely adopted across various deep learning fields, such as text classification (Slonim et al., ), sentence summarization (West et al., ), and image clustering (Hu et al., ).
Expanding upon these foundations, further research has explored generalization error upper bounds that incorporate mutual information (Russo & Zou, ; Xu & Raginsky, ). These studies have established a connection between the generalization capabilities of deep neural networks (DNNs) and IB theory (Alquier et al., ).
More recent advancements have also highlighted the links between mutual information bounds and the PAC-Bayes framework (Banerjee & Montúfar, ).
This type of bound suggests that the generalization error is intrinsically limited by the relevance between the training data and the learned model parameters.

## 3 Preliminaries

### 3.1 Notations & Experimental Setup

Let $S_{\text{anchor}}$ represent the real data utilized for generation, and $S_{\text{gen}}$ denote the synthetically generated data.
The LLM employed in the generation process is designated as $M$, with the input prompt labeled as $p$. The distribution of the post-training target task $\mathcal{T}$ is referred to as $\mathcal{D}$,
while the output distribution of the LLM is denoted by $\mathcal{D}_{M}$. Additionally, the distribution corresponding to the synthetic data is represented as $\mathcal{D}_{\text{gen}}$.
The generalization error associated with the under-aligned LLM $\pi$ on the synthetic data $S_{\text{gen}}$ is expressed as $\operatorname{Err}(\pi^{S_{\text{gen}}})$, and the generalization error related to the anchor data is indicated by $\operatorname{Err}(\pi^{S_{\text{anchor}}})$.
We define $H(\cdot)$ as the entropy of a random variable, $I(\cdot,\cdot)$ as the mutual information between two random variables, $D_{\text{KL}}$ as the Kullback-Leibler divergence, and $D_{\text{TV}}$ as the total variation distance.
The detailed definitions are listed in Appendix A.

To provide a more intuitive demonstration, we use an example in the Gaussian mixture model (GMM) setting during the explanation.
In simple terms, we assume that the target of the post-training task contains $K+J$ Gaussian distribution components, and set up a corresponding ground-truth GMM (gt-GMM, $G$) to represent the target of the post-training task.
After that, we randomly sample from the first $K$ components of the gt-GMM as anchor data.
To simulate the generative model $M$, we added $L$ random components to the gt-GMM, which may include extra distributions, making $M$ a GMM with total $K+J+L$ components. Finally, we randomly sampled data from $M$ to obtain the simulated synthetic data.
The detailed experimental setup is listed in Appendix B.

### 3.2 Modeling Synthetic Data Generation

Long et al. () provided a brief summary for the synthetic data generation, the overall process of synthetic data generation can be modeled as
$S_{\text{gen }}\leftarrow M_{p}\left(\mathcal{T},S_{\text{anchor }}\right)$,
where $S_{\text{gen}}$ is the generated synthetic data, $M$ is the generation model (usually a well-trained LLM), $p$ is the prompt for generation, $\mathcal{T}$ is the downstream task, and $S_{\text{anchor}}$ is the anchor data (real data).
More specifically, the prompt $p$ is derived from the generation task $\mathcal{T}$ and the anchor data $S_{\text{anchor}}$, and consists of three crucial elements:
$p(\mathcal{T},S_{\text{anchor}})\leftarrow E\left(e_{\text{task}},e_{\text{
condition}},e_{\text{demo}}\right)$,
where $E$ is the prompt template, $e_{\text{task}}$ is the task element, $e_{\text{condition}}$ is the condition element, and $e_{\text{demo}}$ is the anchor data element.
The conceptual framework of this modeling is straightforward. $S_{\text{gen}}$ essentially constitutes a modification of the output generated by $M$ in response to the prompt $p$, where the prompt $p$ is defined by the downstream task $\mathcal{T}$ and the anchor data $S_{\text{anchor}}$.
The specifics of the generation process are thus governed by the prompt $p$ and $M$.

We enhance our understanding of synthetic data generation by reevaluating the distributional relationships among the anchor data $S_{\text{anchor}}$, the prompt $p$, and the synthetic data $S_{\text{gen}}$ produced.
We postulate that the anchor data $S_{\text{anchor}}$ is sampled from distribution $\mathcal{D}$ associated with the downstream task, and the generation process is influenced by both the prompt $p$ and the generative model $M$.
Consequently, $S_{\text{gen}}$ represents a modification of $M$’s output in response to the prompt $p$:
$S_{\text{gen}}=M(p)+\epsilon$,
where $\epsilon$ is a noise term for the measurement of revision, such as the data curation process.

The prompt $p$ is intricately linked to the downstream task $\mathcal{T}$ and the anchor data $S_{\text{anchor}}$. We postulate that $S_{\text{anchor}}$ forms the core of the prompt $p$, upon which a task-specific transformation function $\phi_{\mathcal{T}}$ is applied. Consequently, the prompt $p$ can be mathematically modeled as
$p=\phi_{\mathcal{T}}(S_{\text{anchor}})$,
where $\phi_{\mathcal{T}}$ is a reversible function that maps the anchor data to the prompt, consists of all the task-relevant transformation, like the template and other customized settings for more faithful and diverse generation.

Figure: Figure 2: The simulation of the distribution relationships with GMMs. “$\bullet$” represents the anchor data sampled from distributions colored blue, and “$\bullet$” represents the synthetic data sampled from distributions colored orange.
Refer to caption: x2.png

For simplicity, we note that $S_{\text{anchor}}\sim\mathcal{D}$, $p\sim\mathcal{D}_{p}$, $M(p)\sim\mathcal{D}_{M}(\cdot|p)$, and $S_{\text{gen}}\sim\mathcal{D}_{\text{gen}}$,
and comprehensive details about the relationships between the distributions are listed in Appendix C.
The overall synthetic data generation process in our modeling is depicted in Figure 1 (a).
This illustration enhances our understanding of the connection between the generation process and distributions.

The lower part of Figure 1 (a) details the specific stages of data generation. Initially, the anchor data $S_{\text{anchor}}$ undergoes a transformation via the function $\phi_{\mathcal{T}}$ to constitute the prompt $p$, which in turn is used by the LLM $M$ to generate the synthetic data $S_{\text{gen}}$, incorporating noise $\epsilon$.
The upper portion of Figure 1 (a) delineates the corresponding process of distribution shift.
$S_{\text{anchor}}$ is derived from distribution $\mathcal{D}$, and the prompt $p$ emerges from distribution $\mathcal{D}_{p}$ conditioned on $\phi_{\mathcal{T}}$.
The LLM $M$ produces the output $M(p)$ from the conditional distribution $\mathcal{D}_{M}(\cdot|p)$, and the final synthetic data $S_{\text{gen}}$ is sampled from $\mathcal{D}_{\text{gen}}$, representing a convolution of $\mathcal{D}_{M}$ and $\mathcal{D}_{\epsilon}$ also conditioned on $p$.

Given that $\mathcal{D}_{p}$ relates solely to $\phi_{\mathcal{T}}$ and $\mathcal{D}$ (or $S_{\text{anchor}}$), and $\mathcal{D}_{\text{gen}}$ related only to $M$ and $p$, the transition from $S_{\text{anchor}}$ to $p$ to $S_{\text{gen}}$, (i.e. $S_{\text{anchor}}\rightarrow p\rightarrow S_{\text{gen}}$), constitutes a Markov chain.
Figure 1 (b) provides a comprehensive view of the distributions and the nature of the distribution shift discussed.
Specifically, $\mathcal{D}$ is denoted as the orange circle, and $\mathcal{D}_{M}$ is denoted as the blue circle. After $M$ being prompted on $p$, the conditioned distribution $\mathcal{D}_{M}(\cdot|p)$ is denoted as all blue areas, and the final $\mathcal{D}_{\text{gen}}$ is represented as the deep blue area after the compression on $\epsilon$.
This illustration aids in understanding that the generation process essentially compresses the output distribution of $M$, $\mathcal{D}_{M}$, towards the post-training target distribution $\mathcal{D}$, based on the conditions imposed by the prompt $p$ and noise $\epsilon$.

To provide a clearer visualization, we simulate the distribution relationships using GMMs, the result is depicted in Figure 2.
The anchor data is represented by blue dots, and their distribution is illustrated by blue ellipses. In contrast, synthetic data is represented by orange.
The distributions of $S_{\text{gen}}$ are visualized as an effort to encompass the distributions of $S_{\text{anchor}}$.
However, since $S_{\text{gen}}$ is derived from the model $M$, which incorporates more complex distribution components, the distribution of $S_{\text{gen}}$ not only attempts to mirror $S_{\text{anchor}}$ but also extends beyond, covering broader areas.

### 3.3 Bridging the Generalization Capability

Subsection 3.2 offers an exhaustive examination of the synthetic data generation process, which is pivotal for elucidating the generalization error associated with the under-aligned LLM $\pi$ when applied to synthetic data $S_{\text{gen}}$.
This subsection endeavors to correlate the generalization error of $\pi$ on the synthetic data $S_{\text{gen}}$ with the synthetic data generation process as previously delineated.

Given the focus on the alignment task performance, and considering that $\pi$ is a pre-trained LLM, then is
subsequently trained on synthetic data sampled from $\mathcal{D}_{\text{gen}}$, the generalization error of post-trained LLM $\pi^{S_{\text{gen}}}$ is delineated as
$\operatorname{Err}(\pi^{S_{\text{gen}}})=\left|R_{\mathcal{D}}(\pi^{S_{\text{
gen}}})-\widehat{R}_{S_{\text{gen}}}(\pi^{S_{\text{gen}}})\right|$,
where $\mathcal{D}$ is the real distribution of the post-training task. $R_{\mathcal{D}}(\pi^{S_{\text{gen}}})=\mathbb{E}_{z\sim\mathcal{D}}\left[\ell(
\pi^{S_{\text{gen}}},z)\right]$ denotes the true error of $\pi^{S_{\text{gen}}}$ on the distribution $\mathcal{D}$, and $\widehat{R}_{S_{\text{gen}}}(\pi^{S_{\text{gen}}})=\frac{1}{n}\Sigma_{z\in S_{
\text{gen}}}\left[\ell(\pi^{S_{\text{gen}}},z)\right]$ denotes the empirical error of $\pi^{S_{\text{gen}}}$ on the synthetic data.
Similar like Zheng et al. (), and by the definition of the synthetic data generation process, we can simplify the above upper bound as the following lemma:

###### Lemma 3.1 .

Assume that $\pi$ is with a loss function $\ell$ bounded by $C$, given an i.i.d. synthetic dataset $S_{\text{gen}}$ generated as the above defined, then the following synthetic data training generalization error upper bound holds:

$$ $\operatorname{Err}(\pi^{S_{\text{gen}}})\leq C\underbrace{\left(D_{\text{TV}}( \mathcal{D},\mathcal{D}_{M})+D_{\text{TV}}(\mathcal{D}_{M},\mathcal{D}_{\text{ gen}})\right)}_{\text{Distributions' Divergence}}+\underbrace{\left|R_{ \mathcal{D}_{\text{gen}}}(\pi^{S_{\text{gen}}})-\widehat{R}_{S_{\text{gen}}}( \pi^{S_{\text{gen}}})\right|}_{\text{Generalization Error w.r.t. synthetic data}}.$ (1) $$

The proof is referred to the Appendix D.
The divergences can be defined as the task divergence ($D_{\text{TV}}(\mathcal{D},\mathcal{D}_{M})$) and the generation divergence ($D_{\text{TV}}(\mathcal{D}_{M},\mathcal{D}_{\text{gen}})$), which is denoted in Figure 1 (b). The task divergence is determined by the ability of the LLM $M$ and the relevance with the task $\mathcal{T}$.
The generation divergence is determined by the generation process including the prompt engineering and the data curation. In the training practice, the two divergences are controlled by either the strong ability of $M$ or the strict prompt engineering, this partially explains why synthetic data is effective.

###### Lemma 3.1 .

## 4 Main Result

In preceding sections, we established a comprehensive modeling for synthetic data generation and elucidated the connection between this process and the generalization error as delineated in Lemma 3.1. This section delves deeper into the implications of the synthetic data generation process on the generalization capabilities.

### 4.1 Information Gain & Reverse-Bottleneck

Figure: Figure 3: Illustration about the reverse bottleneck effect and comparison with classic ML process. Left: the similarity between the forward process of synthetic data generation and classic ML. Right: the difference between the information flow of the two process, where synthetic data generation gains information from $M$, constituting a reverse-bottleneck.
Refer to caption: x3.png

To enhance our understanding of the synthetic data generation process, we delineate a suite of concepts pertaining to the information-flow within this process.
Initially, we introduce the notion of synthetic factors, which represent the fundamental elements that influence the formation of $S_{\text{gen}}$.

###### Definition 4.1 .

(Synthetic factors.) Assume that the synthetic data $S_{\text{gen}}=M(p)+\epsilon$ is derived from two factors, i.e. $M(p)=h(e_{p})+g(e_{M})$. The $e_{p}$ represents the factor w.r.t. prompt $p$ and the $e_{M}$ represents the factor w.r.t. applied LLM $M$.

Following this framework, we proceed to introduce the concept of information gain within the context of the synthetic data generation process.

###### Definition 4.2 .

(Information gain.) The information gain in the synthetic data generation process is defined as:

$$ $\Delta I=H(M(p))-I\left(h(e_{p}),M(p)\right).$ (2) $$

The information gain, denoted as $\Delta I$, serves as a metric for assessing the enhancement of information in the synthetic data generation process. It quantifies the incremental information content from the prompt $p$ to the synthetic data $S_{\text{gen}}$, specifically, the information introduced by the LLM $M$.

In alignment with the classical information bottleneck theory, we also introduce the concept of a compression bottleneck, which is defined in the context of synthetic factors.

###### Definition 4.3 .

(Compression bottleneck.) We consider the compression bottleneck of the synthetic data towards the post-trained model parameter $W$ as:

$$ $B_{\text{syn}}=I(e_{M},W)+I(e_{p},W).$ (3) $$

Having delineated the concepts of information gain and compression bottleneck, we now advance our discussion to clarify the information flow within the synthetic data generation process, introducing the notion of a reverse-bottleneck effect.
This framework acknowledges that the distribution $\mathcal{D}_{p}$ is directly influenced by $\phi_{\mathcal{T}}$ and $\mathcal{D}_{\text{anchor}}$ (or $S_{\text{anchor}}$),
while $\mathcal{D}_{\text{gen}}$ pertains solely to $M$ and $p$.
Consequently, the sequence $S_{\text{anchor}}\rightarrow p\rightarrow e_{p}\rightarrow W$ constitutes a Markov chain.
Similarly, the process $M(p)\rightarrow e_{M}\rightarrow W$ also forms a Markov chain.

The former Markov chain, as depicted in the left part of Figure 3, parallels a classical machine learning (ML) process,
in which the input $X$ is transformed into a latent representation $Z$ via an encoder, and then $Z$ is further decoded into the output $Y$ through a decoder.
Similarly, in the synthetic data generation process, the input $S_{\text{anchor}}$ is converted to $p$ (which is often assumed as latent in practical applications) via $\phi_{\mathcal{T}}$, and subsequently $p$ is transformed into $S_{\text{gen}}$ by $M$.
However, the presence of the latter Markov chain introduces a crucial distinction between the two processes from an information flow perspective due to the prior knowledge embedded by $M$.
As illustrated in the right part of Figure 3,
unlike classic ML process, the synthetic data generation process leverages $M$ to facilitate information gains, thereby enriching the informational content of $S_{\text{gen}}$.

This perspective emphasizes the distinctive dynamics and augmented capabilities of the synthetic data generation process in terms of capturing and utilizing information.
Subsequently, we aim to analyze the relationship between the information gain and the generalization error of the model after training on the synthetic data.

###### Definition 4.1 .

###### Definition 4.2 .

###### Definition 4.3 .

### 4.2 Information-Flow Generalization Error Upper Bound

In this subsection, we endeavor to derive the upper bounds of the generalization error from an information-flow perspective, employing the concepts previously defined.
We initiate our analysis with a classical information upper bound applicable to deep neural networks, as elaborated in Lemma 4.4 (Zhang et al., ).

###### Lemma 4.4 .

For a deep neural network with $L$ hidden layers, input $S$, and parameters $W$. The loss function is $\sigma$-sub-Gaussian with respect to $(W,Z)$ given any $w$, if all $L$ hidden layers are contraction layers, the expected generalization error can be bounded as follows,

$$ $\mathbb{E}\left[R(W)-R_{S}(W)\right]\leq\exp\left(-\frac{L}{2}\log\frac{1}{ \eta}\right)\sqrt{\frac{2\sigma^{2}}{n}I(S,W)}.$ (4) $$

Lemma 4.4 establishes a connection between the expected generalization error and the mutual information between training data $S$ and learned model parameters $W$.
Despite network depth $L$ and instance volume $n$, the principal constraint is imposed by the mutual information term.

Accordingly, in scenarios where post-training is with synthetic data, the generalization error is inherently constrained by the mutual information between the synthetic data $S_{\text{gen}}$ and LLM parameters after training, denoted as $I(S_{\text{gen}},W)$.
Characterizing this term presents a significant challenge due to the difficulty in measuring mutual information accurately.
To address this, we introduce an analytical upper bound for $I(S_{\text{gen}},W)$ in Lemma 4.5 to facilitate a more comprehensive understanding of the dynamics influencing model performance in post-training.

###### Lemma 4.5 .

(Information-flow upper bound.) Given a synthetic dataset $S_{\text{gen}}$ defined above, and model parameters $W$ learned from $S_{\text{gen}}$, the mutual information term $I(S_{\text{gen}},W)$ can be bounded by the following inequality:

$$ $I(S_{\text{gen}},W)\leq-\Delta I+B_{\text{syn}}+H(e_{M})+\delta_{\epsilon,p},$ (5) $$

where $\delta_{\epsilon,p}$ indicates the efficiency during the data curation and model prompting process, which is detailed in the proof in Appendix E.
Together with Lemma 4.4, we can further derive an upper bound for a training procedure with relation to the synthetic data defined above in Lemma 4.6.

###### Lemma 4.6 .

(Generalization error upper bound w.r.t. synthetic data.) For a deep neural network $\pi$ with $L$ hidden layers, the parameters $W$ are optimized from synthetic data $S_{\text{gen}}$ described aboved. The loss function is $\sigma$-sub-Gaussian with respect to $(W,Z)$ given any $w$, if all $L$ hidden layers are contraction layers, the expected generalization error can be bounded as follows:

$$ $\mathbb{E}\left|R_{\mathcal{D}_{\text{gen}}}(\pi^{S_{\text{gen}}})-\widehat{R} _{S_{\text{gen}}}(\pi^{S_{\text{gen}}})\right|\leq\exp\left(-\frac{L}{2}\log \frac{1}{\eta}\right)\sqrt{\frac{2\sigma^{2}\left[-\Delta I+B_{\text{syn}}+H(e _{M})+\delta_{\epsilon,p}\right]}{n}}.$ (6) $$

Lemma 4.6 delineates a quantifiable upper bound for the expected generalization error in relation to synthetic data.
Beyond basic configuration parameters such as network depth $L$ and data size $n$,
this upper bound is determined by four key factors outlined in the corresponding remarks.

Remark 1. $\Delta I$ quantifies the information gain during the data generation process. This bound demonstrates that an increase in information extracted from the model $M$ enhances the quality of the generated data.

Remark 2. $B_{\text{syn}}$ denotes the compression bottleneck, which is defined as the mutual information between synthetic factors and the model parameters $W$. A more pronounced compression of this term leads to improved generalization performance.

Remark 3. $H(e_{M})$ represents the entropy associated with the synthetic factor relative to the model $M$. Intuitively, reducing this entropy by choosing a model $M$ more aligned with the specific tasks can substantially enhance downstream generalization.

Remark 4. $\delta_{\epsilon,p}$ concerns the efficiency during the data curation and model prompting process, highlighting the impact of noise and other data degradation factors on the overall data utility.

These factors collectively influence the generalization performance, indicating that a better generalization ability can be achieved by enhancing the information gain, reducing the compression bottleneck, minimizing the entropy, and balancing the efficiency.
Finally, by integrating the insights from Lemma 3.1, the overall upper bound of the expected generalization error in the LLM post-training with synthetic data can be derived as a comprehensive boundary in Theorem 4.7.

###### Theorem 4.7 .

(Synthetic data post-training upper bound.) For the same condition as lemma 4.6 and a synthetic data generation process described above, the generalization error of the model $\pi$ post-trained on the synthetic data can be bounded as:

$$ $\displaystyle\mathbb{E}(\operatorname{Err}(\pi^{S_{\text{gen}}}))$ $\displaystyle\leq C\underbrace{\left(D_{\text{TV}}(\mathcal{D},\mathcal{D}_{M} )+D_{\text{TV}}(\mathcal{D}_{M},\mathcal{D}_{\text{gen}})\right)}_{\text{ Distributions' Divergence}}$ (7) $\displaystyle+\underbrace{\exp\left(-\frac{L}{2}\log\frac{1}{\eta}\right)\sqrt {\frac{2\sigma^{2}\left[-\Delta I+B_{\text{syn}}+H(e_{M})+\delta_{\epsilon,p} \right]}{n}}}_{\text{Generalization Error w.r.t. synthetic data}}.$ $$

###### Lemma 4.4 .

###### Lemma 4.5 .

###### Lemma 4.6 .

###### Theorem 4.7 .

### 4.3 Generalization Gain with Synthetic Data

Theorem 4.7 establishes a general upper bound for the generalization error of LLMs post-trained with synthetic data. In this section, our objective is to analyze the generalization gains achieved by using synthetic data compared to scenarios devoid of synthetic data.

We commence our analysis with the anchor data $S_{\text{anchor}}$. Analogous to the definition of $\operatorname{Err}(\pi^{S_{\text{gen}}})$, the generalization error of an LLM that has been post-trained on $S_{\text{anchor}}$ is defined as $\operatorname{Err}(\pi^{S_{\text{anchor}}})=\left|R_{\mathcal{D}}(\pi^{S_{
\text{anchor}}})-\widehat{R}_{S_{\text{anchor}}}(\pi^{S_{\text{anchor}}})\right|$.
It is logically sound to assume that $S_{\text{anchor}}$ is sampled from the distribution $D$. Building upon Lemma 4.4 and assume that $S_{\text{anchor}}$ comprises $m$ instances, we can derive the subsequent result in Lemma 4.8.

###### Lemma 4.8 .

(Anchor data post-training upper bound.) For the same condition as lemma 4.6, the generalization error of the model $\pi$ post-trained on the anchor data can be bounded as:

$$ $\mathbb{E}(\operatorname{Err}(\pi^{S_{\text{anchor}}}))\leq\exp\left(-\frac{L} {2}\log\frac{1}{\eta^{{}^{\prime}}}\right)\sqrt{\frac{2\sigma^{2}}{m}I(S_{ \text{anchor}},W^{{}^{\prime}})}.$ (8) $$

Given that $m<<n$ typically applies in real-world scenarios, Lemma 4.8 often represents a less stringent upper bound compared to Lemma 4.4, this results in potentially poorer generalization when relying solely on $S_{\text{anchor}}$ rather than utilizing synthetic data.

But a pertinent question arises: do other aspects of synthetic data generation, beyond the influence of data size, also contribute to improvements in generalization performance?
Our focus is on examining how various elements within the synthetic data process impact generalization during post-training.
It is inappropriate, however, to directly compare other components across these two bounds due to variations in loss and training data specifics, which affect the parameters $\eta$ and $W$ differently,
where $\eta$ represents a measure of information compression and is challenging to quantify accurately (Zhang et al., ). Thus, our analysis primarily centers on the mutual information terms $I(S_{\text{anchor}},W^{{}^{\prime}})$ and $I(S_{\text{gen}},W)$.
To systematically evaluate the generalization capabilities conferred by synthetic data in relation to these mutual information metrics, we introduce a definition for generalization gain measurement as Definition 4.9.

###### Definition 4.9 .

(Generalization Gain via Mutual Information, GGMI.) GGMI is defined as the difference between the mutual information terms in the two generalization upper bounds:

$$ $\operatorname{GGMI}=I(S_{\text{anchor}},W^{{}^{\prime}})-I(S_{\text{gen}},W).$ (9) $$

A larger upper bound for the GGMI signifies greater potential generalization benefits when utilizing synthetic data. To elucidate the impact of synthetic data on model generalization, we isolate the influence of $W^{{}^{\prime}}$ and establish that the GGMI can be effectively bounded.
This relationship is formalized in Theorem 4.10, which demonstrates that GGMI is constrained by $\Delta I$—the measure of information gain—and various entropy components solely associated with $W$.

###### Theorem 4.10 .

(Upper bound of GGMI.) Given the synthetic data generation above, $W^{{}^{\prime}}$ is parameterized by training with $S_{\text{anchor}}$, and $W$ is parameterized by training with $S_{\text{gen}}$, the GGMI can be bounded as follows:

$$ $\operatorname{GGMI}\leq\Delta I-(\alpha+1)H(S_{\text{anchor}}|W)+2\Delta H+H(S _{\text{gen}}|W)+\epsilon_{W,p},$ (10) $$

where $\Delta H=H\left(S_{\text{anchor}}\right)-H\left(S_{\text{gen}}\right)$, $\epsilon_{W,p}=H(S_{\text{anchor}}|W)-H(S_{\text{anchor}}|M(p))$, it is assumed that $H(S_{\text{anchor}}|W^{{}^{\prime}})=\alpha H(S_{\text{anchor}}|W)$, $\alpha\geq 0$.

The proof is referred to Appendix F.
Consequently, we proceed to conduct a thorough analysis of each component specified in Theorem 4.10.

Remark 1. $\Delta I$ represents the information gain derived from the model $M$. An increase in this information gain typically leads to improved generalization capability for $\pi^{S_{\text{gen}}}$ compared to $\pi^{S_{\text{anchor}}}$, as the model leverages additional insights to enhance performance.

Remark 2. $H(S_{\text{anchor}}|W)$ indicates the conditional entropy between the anchor data $S_{\text{anchor}}$ and the model parameters $W$. For a larger upper bound of GGMI, it is encouraged to decrease this value by strengthen the relevance between model parameters $W$ and anchor data $S_{\text{anchor}}$.

Remark 3. $\Delta H$ denotes the entropy decrease when generating synthetic data $S_{\text{gen}}$ from anchor data $S_{\text{anchor}}$.
It implies that more uncertainty is eliminated during synthetic data generation leads to more generalization ability.

Remark 4. $H(S_{\text{gen}}|W)$ reflects the conditional entropy between the synthetic data $S_{\text{gen}}$ and the model parameters $W$. Weakening the relevance between these two entities is encouraged to ensure that the model learns the general pattern of synthetic data thus leading to better generalization.

Remark 5. $\epsilon_{W,p}$ denotes the effect of information compression by the training algorithm. A more pronounced compression effect typically results in a higher value, suggesting that efficient data representation contributes positively to model efficacy.

As emphasized in (Long et al., ), the generation of synthetic data typically focuses on two primary objectives: faithfulness and diversity. These objectives are associated with $H(S_{\text{anchor}}|W)$ and $\Delta I$, respectively.
Specifically, $H(S_{\text{anchor}}|W)$, which quantifies the relevance between $W$ and $S_{\text{anchor}}$, as presented in Theorem 4.10, encourages the model to align with the information derived from the original real data. This alignment is crucial for achieving the objective of faithfulness.
In addition, $\Delta I$ serves as a measurement of the additional information introduced by the generative model $M$.
Given that $M$ is typically pre-trained on a more extensive dataset, $\Delta I$ in Theorem 4.10 promotes the objective of diversity by facilitating greater information gain from $M$.

Figure: Figure 4: KL Gap with different components settings. By default, we set $K=J=L=2$, and vary each of them from $2$ to $15$ to observe the corresponding change of KL Gap. An increase of KL Gap is observed when $J$ increases, while a decrease is observed when $K$ and $L$ increase. The shading indicates the standard deviation of 100 rounds of random settings.
Refer to caption: x4.png

###### Lemma 4.8 .

###### Definition 4.9 .

###### Theorem 4.10 .

### 4.4 Verification with GMM Simulation

Building upon the simulation settings, we offer a straightforward validation of the theoretical results discussed above.
Specifically, we first fit a GMM $\pi$ comprising $K+J+L$ components to both $S_{\text{anchor}}$ and $S_{\text{gen}}$, yielding $\pi^{S_{\text{anchor}}}$ and $\pi^{S_{\text{gen}}}$ respectively.
We then introduce a metric termed KL Gap, defined as $D_{KL}(\pi^{S_{\text{anchor}}}||G)-D_{KL}(\pi^{S_{\text{gen}}}||G)$, which represents the difference of KL-divergence between the fitted GMMs ($\pi^{S_{\text{anchor}}}$ and $\pi^{S_{\text{gen}}}$) and the ground-truth GMM $G$.
A larger KL Gap corresponds to a greater GGMI, indicating enhanced generalization benefits from synthetic data.

To control the variables outlined in Theorem 4.10, we adjust the number of components in the GMM $M$ and the ground-truth GMM $G$. The result is illustrated in Figure 4.
Generally, increasing $J$ facilitates the scaling of $\Delta I$, resulting in a larger upper bound for GGMI.
In contrast, larger $K$ amplifies the influence of anchor data within the post-training target distribution, thereby increasing the $H(S_{\text{anchor}}|W)$ term and tightening the upper bound of GGMI.
Additionally, while an increase in $L$ enhances $H(S_{\text{gen}}|W)$ due to the introduction of greater diversity, it concurrently leads to a reduction in $\Delta H$.
As a result, we observe a trade-off manifested as a decrease in the KL Gap in our simulation outcomes.

## 5 Conclusion

In this paper, we have conducted a detailed analysis of synthetic data utilization in post-training large language models (LLMs). We present a comprehensive modeling of the current synthetic data generation process, focusing on its distributional aspects, which further connects the generalization capabilities of post-trained models.
We introduce a novel reverse-bottleneck framework, allowing us to derive a measurable upper bound on generalization errors. Our analysis reveals that the pivotal constraint on generalization ability is influenced by the information gain from the generative model $M$. Additionally, we present the Generalization Gain via Mutual Information (GGMI), showing that larger information gains enhance the generalization capability of post-trained models. We emphasize the importance of balancing faithfulness and diversity during post-training stages,
providing a theoretical foundation for existing methodologies.

Unfortunately, due to limitations in computational resources, we are unable to validate our findings within real-world LLM settings.
Looking ahead, future research should focus on developing adaptive models that respond to the evolving characteristics of synthetic data. This includes enhancing generative models and fine-tuning parameters for specific learning scenarios, as well as exploring various generative models to better replicate real-world data complexities while improving model performance.

## Appendix A Definition of Notations

We summarize the notations used in subsection 3.1 and provide their specific definitions.

First, we define the notations about the concept related to information entropy.

###### Definition A.1 .

(Entropy of a random variable.) The entropy of a random variable $X$ is defined as:

$$ $H(X)=-\sum_{x}p(x)\log p(x),$ $$

For continual random variable, the entropy is defined as:

$$ $H(X)=-\int p(x)\log p(x)dx.$ $$

The entropy is a measurement of the uncertainty of the random variable, and the larger the entropy, the more uncertain the random variable is.
It can also be considered as the average information content of the random variable.

###### Definition A.2 .

(Conditional entropy of a random variable.) The conditional entropy of a random variable $X$ given another random variable $Y$ is defined as:

$$ $H(X|Y)=-\sum_{x,y}p(x,y)\log p(x|y).$ $$

For continual random variable, the conditional entropy is defined as:

$$ $H(X|Y)=-\int p(x,y)\log p(x|y)dxdy.$ $$

The conditional entropy is a measurement of the uncertainty of the random variable $X$ given the information of the random variable $Y$.
It can also be considered as the average information content of the random variable $X$ with $Y$ given.

Building upon the definitions above, we can further define the concepts we used in the main text with relation to information theory,
including relative entropy, total variation distance, and mutual information.

###### Definition A.3 .

(Relative entropy or Kullback-Leibler divergence.) The relative entropy or Kullback-Leibler divergence between two probability distributions $p$ and $q$ is defined as:

$$ $D_{\text{KL}}(p\|q)=\sum_{x}p(x)\log\frac{p(x)}{q(x)}.$ $$

The relative entropy serves as a measurement of the difference between two probability distributions.

###### Definition A.4 .

(Total variation distance.) The total variation distance between two probability distributions $p$ and $q$ on a finite or countable set $E$ is defined as:

$$ $\displaystyle D_{\text{TV}}(p,q)$ $\displaystyle=\operatorname{sup}_{A\in E}\left|p(A)-q(A)\right|$ $\displaystyle=\frac{1}{2}\sum_{x\in E}\left|p(x)-q(x)\right|.$ $$

The total variation distance is also a measurement of the difference between two probability distributions.

###### Definition A.5 .

(Mutual information.) The mutual information between two random variables $X$ and $Y$ is defined as:

$$ $I(X,Y)=H(X)-H(X|Y).$ $$

The mutual information is a measurement of the amount of information that one random variable contains about another random variable.
The larger the mutual information, the more information the two random variables share.

###### Definition A.1 .

###### Definition A.2 .

###### Definition A.3 .

###### Definition A.4 .

###### Definition A.5 .

## Appendix B Details of Experimental Settings

Figure: Figure 5: Illustration of the setup of the GMMs for simulation.
Refer to caption: x5.png

We utilize Gaussian Mixture Models (GMMs) to simulate the data generation process, as illustrated in Figure 5.
Overall, we use a gt-GMM to simulate the ground-truth, or the post-training target distribution, and a GMM $M$ to simulate the generative model applied in the data generation process.

Their are three parts of components in the GMMs: the anchor sample part with $K$ components, the unsampled part with $J$ components, and task-irrelevant part with $L$ components.
It is assumed that the post-training target distribution is a combination of the anchor sample part and the unsampled part, thus the gt-GMM contains $K+J$ components from the anchor sample part and the unsampled part,
which is denoted as blue in Figure 5.
However, the anchor data is only sampled from the anchor sample part. This is a reasonable assumption for the real-world scenario, since the anchor data is sparse and hard to cover the whole post-training task distribution.

Additionally, the generative model $M$ is assumed to be a GMM with $K+J+L$ components.
Except for the post-training target distribution, $M$ also contains a task-irrelevant part, which is denoted as orange in Figure 5.
This is due to the fact that the generative model is always pre-trained on a larger scale of data,
and may not be perfectly aligned with the post-training target distribution, and may introduce task-irrelevant components in the synthetic data generation process.

Building upon the settings above, we sample from the anchor sample part components of the gt-GMM to generate the anchor data $S_{\text{anchor}}$,
and sample from the generative model $M$ to generate the synthetic data $S_{\text{gen}}$. In the experiment, we set the dimension of the data to be $d=2$,
and $K=J=L=2$ by default, to facilitate the visualization and analysis of the data generation process.

For simulation in the main text, we set the number of initial anhcor data $N=50$ for each anchor sample part component,
and resample the $1000$ data points for both GMM fitted on $S_{\text{anchor}}$ and $S_{\text{gen}}$.
For the simulation to evaluate the KL Gap, the results are averaged over the $100$ rounds, where for each round, we also resample the final data points for $100$ rounds.

## Appendix C Details of Synthetic Data Generation Modeling

In this section, we elaborate on the modeling aspects of synthetic data generation, particularly focusing on the distributions of the prompt $p$ and synthetic data $S_{\text{gen}}$,
which are central to the process of generating synthetic data for training large language models (LLMs).

Distribution of $p$:
The prompt $p$ is is derived from the transformation function $\phi_{\mathcal{T}}$, applied to the anchor data $S_{\text{anchor}}$.
This function is assumed to be reversible, allowing us to explore its properties in the context of data generation:

$$ $p=\phi_{\mathcal{T}}(S_{\text{anchor}}),$ $$

where $\phi_{\mathcal{T}}$ integrates various task-specific and conditional elements, defined as $e_{\text{task}}$ and $e_{\text{condition}}$. Assuming that $\phi_{\mathcal{T}}$ is reversible, we can derive the distribution of $p$ through the probability density function (PDF) of $\mathcal{D}_{\text{anchor}}$ (denoted as $f_{\mathcal{D}_{\text{anchor}}}$), the distribution of $p$ can be modeled as follows:

$$ $p\sim\mathcal{D}_{p}(\phi_{\mathcal{T}})=\mathcal{D}_{\phi_{\mathcal{T}}^{-1}},$ $$

where the PDF of $\mathcal{D}_{\phi_{\mathcal{T}}^{-1}}$ is expressed as:

$$ $f_{\phi_{\mathcal{T}}^{-1}}(x)=f_{\mathcal{D}_{\text{anchor}}}(\phi_{\mathcal{ T}}^{-1}(x))\left|\operatorname{det}\left(\frac{\partial\phi_{\mathcal{T}}^{-1 }}{\partial\mathbf{x}}\right)\right|,$ $$

which indicates how changes in $\mathcal{D}_{\text{anchor}}$ influence the distribution of $p$ through the transformation function, taking into account the Jacobian determinant of the inverse transformation.

Distribution of $S_{\text{gen}}$:
The synthetic data $S_{\text{gen}}$ is the output of the large language model $M$ when prompted with $p$, typically augmented with noise $\epsilon$ to introduce variability and improve robustness. Assuming that the output of $M$ follows a specific distribution $\mathcal{D}_{M}$, based on the conditioning on $p$, we represent the distribution of $M(p)$ as:

$$ $M(p)\sim\mathcal{D}_{M}(\cdot\mid p),$ $$

The distribution of $S_{\text{gen}}$ then combines the model’s output with noise, which is mathematically characterized by the convolution of $\mathcal{D}_{M}(\cdot|p)$ and $\mathcal{D}_{\epsilon}$:

$$ $S_{\text{gen}}\sim\mathcal{D}_{\text{gen}}(M,p)=\mathcal{D}_{M}(\cdot|p)* \mathcal{D}_{\epsilon},$ $$

where $*$ is the convolution operator, integrating the noise distribution $\mathcal{D}_{\epsilon}$ into the output distribution of the model. This convolution reflects how noise impacts the precision and variability of the generated synthetic data, thus affecting the overall utility and effectiveness of the synthetic data in model training.

Through these detailed formulations, we aim to provide a more granular understanding of how synthetic data is modeled and generated, facilitating better integration and utilization in LLM training processes. This deeper insight into the synthetic data generation mechanics enables more targeted and effective training strategies, optimizing the performance of large language models in diverse applications.

## Appendix D Proof of Lemma 3.1

###### Proof.

Similar like Zheng et al. (), we can further decompose the generalization error into the following three components:

$$ $\displaystyle\operatorname{Err}(\pi^{S_{\text{gen}}})$ $\displaystyle\leq\left|R_{\mathcal{D}}(\pi^{S_{\text{gen}}})-R_{\mathcal{D}_{M }}(\pi^{S_{\text{gen}}})\right|+\left|R_{\mathcal{D}_{M}}(\pi^{S_{\text{gen}}} )-R_{\mathcal{D}_{\text{gen}}}(\pi^{S_{\text{gen}}})\right|$ (11) $\displaystyle+\left|R_{\mathcal{D}_{\text{gen}}}(\pi^{S_{\text{gen}}})- \widehat{R}_{S_{\text{gen}}}(\pi^{S_{\text{gen}}})\right|.$ $$

For the first item in lemma, we have:

$$ $\displaystyle\left|R_{\mathcal{D}}(\pi^{S_{\text{gen}}})-R_{\mathcal{D}_{M}}( \pi^{S_{\text{gen}}})\right|$ $\displaystyle=\left|\int_{\mathbf{z}}\ell(\pi^{S_{\text{gen}}},\mathbf{z}) \left(\mathbb{P}_{\mathcal{D}}(\mathbf{z})-\mathbb{P}_{\mathcal{D}_{M}}( \mathbf{z})\right)d\mathbf{z}\right|$ (12) $\displaystyle\leq\int_{\mathbf{z}}\left|\ell(\pi^{S_{\text{gen}}},\mathbf{z}) \left(\mathbb{P}_{\mathcal{D}}(\mathbf{z})-\mathbb{P}_{\mathcal{D}_{M}}( \mathbf{z})\right)\right|d\mathbf{z}$ $\displaystyle\leq C\int_{\mathbf{z}}\left|\mathbb{P}_{\mathcal{D}}(\mathbf{z}) -\mathbb{P}_{\mathcal{D}_{M}}(\mathbf{z})\right|$ $\displaystyle\lesssim CD_{\text{TV}}(\mathcal{D},\mathcal{D}_{M}).$ $$

Similarly, for the second item in lemma, we have:

$$ $\displaystyle\left|R_{\mathcal{D}_{M}}(\pi^{S_{\text{gen}}})-R_{\mathcal{D}_{ \text{gen}}}(\pi^{S_{\text{gen}}})\right|$ $\displaystyle=\left|\int_{\mathbf{z}}\ell(\pi^{S_{\text{gen}}},\mathbf{z}) \left(\mathbb{P}_{\mathcal{D}_{M}}(\mathbf{z})-\mathbb{P}_{\mathcal{D}_{\text{ gen}}}(\mathbf{z})\right)d\mathbf{z}\right|$ (13) $\displaystyle\leq\int_{\mathbf{z}}\left|\ell(\pi^{S_{\text{gen}}},\mathbf{z}) \left(\mathbb{P}_{\mathcal{D}_{M}}(\mathbf{z})-\mathbb{P}_{\mathcal{D}_{\text{ gen}}}(\mathbf{z})\right)\right|d\mathbf{z}$ $\displaystyle\leq C\int_{\mathbf{z}}\left|\mathbb{P}_{\mathcal{D}_{M}}(\mathbf {z})-\mathbb{P}_{\mathcal{D}_{\text{gen}}}(\mathbf{z})\right|$ $\displaystyle\lesssim CD_{\text{TV}}(\mathcal{D}_{M},\mathcal{D}_{\text{gen}}).$ $$

Together with Eq. (11), Eq. (12), and Eq. (13), we have:

$$ $\displaystyle\operatorname{Err}(\pi^{S_{\text{gen}}})$ $\displaystyle\leq\left|R_{\mathcal{D}}(\pi^{S_{\text{gen}}})-R_{\mathcal{D}_{M }}(\pi^{S_{\text{gen}}})\right|+\left|R_{\mathcal{D}_{M}}(\pi^{S_{\text{gen}}} )-R_{\mathcal{D}_{\text{gen}}}(\pi^{S_{\text{gen}}})\right|$ (14) $\displaystyle+\left|R_{\mathcal{D}_{\text{gen}}}(\pi^{S_{\text{gen}}})- \widehat{R}_{S_{\text{gen}}}(\pi^{S_{\text{gen}}})\right|$ $\displaystyle\leq CD_{\text{TV}}(\mathcal{D},\mathcal{D}_{M})+CD_{\text{TV}}( \mathcal{D}_{M},\mathcal{D}_{\text{gen}})+\left|R_{\mathcal{D}_{\text{gen}}}( \pi^{S_{\text{gen}}})-\widehat{R}_{S_{\text{gen}}}(\pi^{S_{\text{gen}}})\right|$ $\displaystyle=C\left(D_{\text{TV}}(\mathcal{D},\mathcal{D}_{M})+D_{\text{TV}}( \mathcal{D}_{M},\mathcal{D}_{\text{gen}})\right)+\left|R_{\mathcal{D}_{\text{ gen}}}(\pi^{S_{\text{gen}}})-\widehat{R}_{S_{\text{gen}}}(\pi^{S_{\text{gen}}} )\right|.$ $$

This finishes the proof.

∎

###### Proof.

## Appendix E Proof of Lemma 4.5

###### Proof.

Considering the Markov chain $M(p)\rightarrow S_{\text{gen}}\rightarrow W$, according to the properties of mutual information, we have:

$$ $H(S_{\text{gen}})\leq H(M(p)).$ (15) $$

Furtherly, the following inequality can be derived:

$$ $I(S_{\text{gen}},W)\leq I(M(p),W).$ (16) $$

Building upon equation (16), we can derive the following equations:

$$ $\displaystyle I(S_{\text{gen}},W)$ $\displaystyle=I(M(p),W)-\delta_{\epsilon}$ (17) $\displaystyle\leq I(M(p),W),$ $$

where $\delta_{\epsilon}$ is the information loss due to the noise $\epsilon$ in the data curation process.

Since $h(\cdot)$ and $g(\cdot)$ are deterministic functions which decrease the entropy of random variables, we have:

$$ $H(h(e_{p}))\leq H(e_{p}),\quad H(g(e_{M}))\leq H(e_{M}).$ (18) $$

Accordingly, the following inequalities can be derived:

$$ $\displaystyle I(h(e_{p}),W)$ $\displaystyle=H(h(e_{p}))-H(h(e_{p})|W)$ (19) $\displaystyle\leq H(e_{p})-H(e_{p}|W)$ $\displaystyle=I(e_{p},W).$ $$

Similarly, we have:

$$ $\displaystyle I(g(e_{M}),W)$ $\displaystyle=H(g(e_{M}))-H(g(e_{M})|W)$ (20) $\displaystyle\leq H(e_{M})-H(e_{M}|W)$ $\displaystyle=I(e_{M},W).$ $$

This is because the deterministic functions $h(\cdot)$ and $g(\cdot)$ decrease the information content, and make the information a subset of the original random variables.

Then we consider the upper bound of $I(M(p),W)$ according to the result above:

$$ $\displaystyle I(M(p),W)$ $\displaystyle=I(h(e_{p})+g(e_{M}),W)$ (21) $\displaystyle\leq I(h(e_{p}),W)+I(g(e_{M}),W)$ $\displaystyle\leq I(e_{p},W)+I(e_{M},W)$ $$

For further analysis, we consider the following assumption related to the efficiency of the model utilizing the prompt:

###### Lemma E.1 .

(Efficiency of the model prompting.) For the model $M$ utilizing the prompt $p$, with $\lambda\geq 1$, we have:

$$ $H(e_{p})\leq\lambda I(e_{p},M(p)).$ (22) $$

Lemma E.1 indicates that the entropy of $e_{p}$ is upper bounded by the mutual information between the synthetic factor $e_{p}$ and the model output $M(p)$ by a factor of $\lambda$.
In other words, the efficiency of the model utilizing the prompt is reflected in the value of $\lambda$, which quantifies the extent to which the model can leverage the information contained in the prompt.
For example, a larger $\lambda$ indicates a smaller $I(e_{p},M(p))$, which implies that the $M(p)$ contains more information from the prompt $p$.

Building upon Lemma E.1, we can further derive the deduction following equation (21):

$$ $\displaystyle I(M(p),W)$ $\displaystyle\leq I(e_{p},W)+I(e_{M},W)$ (23) $\displaystyle=H(e_{p})-H(e_{p}|W)+I(e_{M},W)$ $\displaystyle=H(M(p))-H(M(p))+H(e_{p})-H(e_{p}|W)+I(e_{M},W)$ $\displaystyle\leq-H(M(p))+I(e_{p},M(p))-I(e_{p},M(p))$ $\displaystyle\quad+\lambda I(e_{p},M(p))+H(M(p))-H(e_{p}|W)+I(e_{M},W)$ $\displaystyle\leq-\Delta I+I(e_{M},W)+H(M(p))-H(e_{p}|W)+(\lambda-1)I(e_{p},M( p))$ $\displaystyle\leq-\Delta I+B_{\text{syn}}+H(e_{M}).$ $$

###### Lemma E.2 .

(Entropy gap upper bound) The difference between the entropy of $M(p)$ and $e_{p}$ can be upper bounded by the following inequality:

$$ $H(M(p))-H(e_{p})\leq H(e_{M}).$ (24) $$

The proof of Lemma E.2 is listed in equation (25):

$$ $\displaystyle H(M(p))-H(e_{p})$ $\displaystyle=H(h(e_{p})+g(e_{M}))-H(e_{p})$ (25) $\displaystyle\leq H(e(p))+H(g(e_{M}))-H(e_{p})$ $\displaystyle\leq H(e_{p})+H(e_{M})-H(e_{p})$ $\displaystyle=H(e_{M}).$ $$

Building upon Lemma E.2, we can further deduce the following inequality following equation (23):

$$ $\displaystyle I(M(p),W)$ $\displaystyle\leq-\Delta I+I(e_{M},W)+H(M(p))-H(e_{p}|W)+(\lambda-1)I(e_{p},M( p))$ (26) $\displaystyle\leq-\Delta I+B_{\text{syn}}-I(e_{p},W)+H(M(p))-H(e_{p}|W)+( \lambda-1)I(e_{p},M(p))$ $\displaystyle=-\Delta I+B_{\text{syn}}+H(M(p))-H(e_{p})+(\lambda-1)I(e_{p},M(p))$ $\displaystyle\leq-\Delta I+B_{\text{syn}}+H(e_{M})+(\lambda-1)I(e_{p},M(p)).$ $$

Together with equations (17) and (26), we have:

$$ $\displaystyle I(S_{\text{gen}},W)$ $\displaystyle=I(M(p),W)-\delta_{\epsilon}$ (27) $\displaystyle\leq-\Delta I+B_{\text{syn}}+H(e_{M})+(\lambda-1)I(e_{p},M(p))- \delta_{\epsilon}$ $\displaystyle\leq-\Delta I+B_{\text{syn}}+H(e_{M})+\delta_{\epsilon,p},$ $$

where $\delta_{\epsilon,p}=(\lambda-1)I(e_{p},M(p))-\delta_{\epsilon}$.

This finishes the proof.
∎

###### Proof.

###### Lemma E.1 .

###### Lemma E.2 .

## Appendix F Proof of Theorem 4.10

###### Proof.

Considering the Markov chain $h(e_{p})\rightarrow M(p)\rightarrow S_{\text{gen}}$, we have:

$$ $H(M(p))\geq H(S_{\text{gen}}).$ (28) $$

In addition, according to the properties of mutual information, we have:

$$ $I(S_{\text{anchor}},M(p))\geq I\left(h(e_{p}),M(p)\right).$ (29) $$

Building upon the inequalities above, we can derive the following equations:

$$ $\displaystyle\Delta I$ $\displaystyle=H(M(p))-I\left(h(e_{p}),M(p)\right)$ (30) $\displaystyle\geq H(S_{\text{gen}})-I(S_{\text{anchor}},M(p))$ $\displaystyle=I(S_{\text{gen}},W)+H(S_{\text{gen}}|W)-I(S_{\text{anchor}},M(p)).$ $$

Based on the assumptions mentioned above, we also have:

$$ $\displaystyle I(S_{\text{anchor}};W^{{}^{\prime}})$ $\displaystyle=H(S_{\text{anchor}})-H(S_{\text{anchor}}|W^{{}^{\prime}})$ (31) $\displaystyle=H(S_{\text{anchor}})-\alpha H(S_{\text{anchor}}|W)$ $\displaystyle=I(S_{\text{anchor}},W)+(1-\alpha)H(S_{\text{anchor}}|W).$ $$

Furthermore, based on the definitions, we have:

$$ $\displaystyle I(S_{\text{anchor}},M(p))$ $\displaystyle=H(S_{\text{anchor}})-H(S_{\text{anchor}}|M(p))$ (32) $\displaystyle=I(S_{\text{anchor}},W)+H(S_{\text{anchor}}|W)-H(S_{\text{anchor} }|M(p))$ $\displaystyle=I(S_{\text{anchor}},W)+\epsilon_{W,p}.$ $$

By the definition of GGMI, and with equation (31), the following result can be deduced:

$$ $\displaystyle\operatorname{GGMI}=$ $\displaystyle I(S_{\text{anchor}},W^{{}^{\prime}})-I(S_{\text{gen}},W)$ (33) $\displaystyle=$ $\displaystyle I(S_{\text{anchor}},W)+(1-\alpha)H(S_{\text{anchor}}|W)-I(S_{ \text{gen}},W)$ $\displaystyle=$ $\displaystyle I(S_{\text{gen}},W)+H(S_{\text{gen}}|W)-I(S_{\text{anchor}},M(p))$ $\displaystyle-I(S_{\text{gen}},W)-H(S_{\text{gen}}|W)+I(S_{\text{anchor}},M(p))$ $\displaystyle+I(S_{\text{anchor}},W)+(1-\alpha)H(S_{\text{anchor}}|W)-I(S_{ \text{gen}},W).$ $$

Subsequently, together with equations (30) and (32), we can further deduce that:

$$ $\displaystyle\operatorname{GGMI}\leq$ $\displaystyle\Delta I-2I(S_{\text{gen}},W)-H(S_{\text{gen}}|W)+I(S_{\text{ anchor}},W)$ (34) $\displaystyle+(1-\alpha)H(S_{\text{anchor}}|W)+I(S_{\text{anchor}},M(p))$ $\displaystyle=$ $\displaystyle\Delta I-2I(S_{\text{gen}},W)-H(S_{\text{gen}}|W)$ $\displaystyle+2I(S_{\text{anchor}},W)+(1-\alpha)H(S_{\text{anchor}}|W)+ \epsilon_{W,p}$ $\displaystyle=$ $\displaystyle\Delta I-2H(S_{\text{gen}})+H(S_{\text{gen}}|W)$ $\displaystyle+2H(S_{\text{anchor}})-(\alpha+1)H(S_{\text{anchor}}|W)+\epsilon_ {W,p}.$ $$

Finally, together with all the deduce and definition above, we have:

$$ $\operatorname{GGMI}\leq\Delta I-(\alpha+1)H(S_{\text{anchor}}|W)+2\Delta H+H(S _{\text{gen}}|W)+\epsilon_{W,p},$ (35) $$

This finishes the proof.
∎

###### Proof.

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 3
presentation: 2
questions: See above
soundness: 2
strengths: 1. The motivation is easy to follow and the figure is well represented.

2. The equation derived from previous literature, such as Lemma 3.1 and 4.4 is clear. 

3. The paper tries to inject the LLM synthetic generation perspective using previously derived framework, showing a potential direction to understanding the synthetic data.
summary: The paper introduces the framework of understanding the LLM synthetic data generation process, by incorporating the reverse-bottleneck perspective. Using a reverse-bottleneck perspective, the authors model the generation of synthetic data and propose an information-theoretic metric called Generalization Gain via Mutual Information (GGMI) to assess the generalization performance of post-trained models. It also reports the information gain and generalization bound from previous literature.
weaknesses: 1. The paper lacks the experimental support, which cannot convince the readers of their assumption and derivations.

2. The paper is not organized in a good shape for guiding the readers for understanding their contributions. Instead, it enumerates the equations of generalization errors and gains, but lacking sufficient experimental results to support this.

3. Some derivations may be challenging for readers unfamiliar with advanced information theory, which may requires providing backgrounds about this. Since the authors did not conduct real-world experiments, it is unsure whether this can generalize to real-world settings. The framework assumes ideal conditions for synthetic data and LLMs, which may not generalize well to complex real-world scenarios with noisy data or biases.

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: Here are suggested experiments that could help validate the paper's hypotheses and theoretical frameworks:
1. LLM-based Validation Experiments :
     - ⁠Compare different sizes of anchor data and their corresponding synthetic data generation
     - ⁠Measure the relationship between model size and synthetic data quality
     - ⁠Test various prompting strategies and their impact on synthetic data diversity
     - Track information gain across different LLM architectures 
2. Practical Application Questions:
    - How can practitioners use your theoretical bounds to improve their synthetic data generation process?
    - What specific guidance would you give for prompt engineering based on your theoretical findings?
3. Validation Questions:
    - How would your bounds change with different LLM architectures or sizes?
    -  ⁠What metrics would you recommend for measuring the quality of synthetic data in practice?
soundness: 3
strengths: 1. The theoretical modeling of synthetic data and how it interacts with the output distribution of the LLM is intuitive and complete. Great job there!
2. The math behind generalization error bounds is solid.
3. The connection to the classical information bottleneck theory is novel and very intuitive, clearly justifies why synthetic data helps better performance on downstream task generalization.
summary: The paper does a good job of establishing a theoretical rationale behind using synthetic data for LLM post-training processes like SFT. The modeling of synthetic data, focusing on distributional aspects is elaborate and properly justified as well. It establishes an intuitive information-theoretic based upper bound on the generalization error for an under-aligned LLM fine-tuned on synthetic data, and also delineates how this upper bound compares to the one when the LLM is tuned on just the real world samples (anchor data) - showcasing how the former proves to be a more stringent upper bound, leading to better generalization capabilities. Definitely helps bridge a gaping gap in LLM research!
weaknesses: 1. The paper presents good theoretical foundations of "why" it is important to use synthetic data in post-training alignment and "how" it aids better generalization capabilities of the model. But it fails to address some really important "hows" that it promises in the Introduction section :
   - How does this study help in developing tailored synthetic data generation - that more effectively address specific gaps in training data, thereby enhancing the overall performance and generalization capabilities of large language models
   - How does this translate to better post-training practices involving synthetic datasets, from a practical perspective.
2. The assumption that the transformation function φT is reversible (Section 3.2) is unclear for real LLM prompting scenarios, as task-to-prompt relationships are often many-to-many.A better delineation of this point would be helpful.
3. ⁠The relationship between "synthetic factors" and actual LLM generation processes is not well defined.
4. The connection between the theoretical "information gain" concept and practical improvements in LLM performance is not clearly established. 
5. I think its very crucial to address how the "diversity" and "quality" in synthetic data samples are a big part of why task generalization actually works when models are post trained on synthetic data. These should be adequately quantified and incorporated in the upper bound equations as well, for a more complete picture.

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confidence: 3
contribution: 4
initial_rating: 8
presentation: 4
questions: - How does task difficulty affect generalization capability and the derived upper bounds? Are there any factors in the documented theoretical expressions that model or are associated with task complexity?
- Lemma 4.8: η' and W' are not defined prior to their usage in the expression; for clarity, it would be useful to define them as other variables have been.
soundness: 3
strengths: - This work has a sound, well-researched theoretical basis
- The provided mathematical proofs are comprehensive and well-structured.
- The experimental setup is well-described and the theory is supported by some experimental results.
summary: This work provides a theoretical framework to analyse the nature and impact of synthetic data on the post-training process of LLMs. This work also proposes a "reverse-bottleneck" framework to visualize information flow in the synthetic generation process. On the basis of these theoretical principles, this work proceeds to mathematically derive useful upper bounds for model generalization when trained with synthetic data, in order to better quantify performance expectations.
weaknesses: - This work could benefit from additional explanation of how diversity of generated data influences generalization capabilities, as well as a clearer explanation of the intuition behind how ∆I (the information gain derived from the model M) is associated with diversity (lines 483-485) and not just more unique but similar data points, possibly with examples.
-  The dimension of data on which these experiments are performed is 2; however, real-world data is often more complex and higher-dimensional. It would be illustrative to conduct the experiments documented in this paper on data of higher dimensions to reinforce that the theory holds for datasets of varying dimensionality.
- Additionally, given that this analysis is aimed at studying the behaviour of LLMs, the theoretical proofs presented in this work would benefit from conducting the same experiments either with LLMs or other non-GMM models to corroborate the theoretical inferences.

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confidence: 3
contribution: 4
initial_rating: 6
presentation: 3
questions: 1. How would synthetic data that has not been processed by an LLM impact the proposed theory? For instance, data generated through specific rules, such as template-based concatenation or converting game-playing sequences into text.
2. Does this theoretical analysis still hold for non-text, multimodal data? Would the same principles apply, or are there limitations in extending the theory to such data types?
soundness: 3
strengths: 1. The paper provides a detailed theoretical analysis of the introduction of synthetic data, clearly explaining the information gain that synthetic data can bring.
2. The authors present thorough and well-supported mathematical proofs to substantiate their claims, offering a solid foundation for their arguments.
3. The reverse-bottleneck theory mathematically captures the essence of synthetic data's impact, offering valuable insights and guidance for the development of synthetic data methodologies.
summary: This paper explores the use of synthetic data in post-training tasks for large language models. It presents a theoretical framework to analyze the effects of synthetic data on model generalization, focusing on the reverse-bottleneck effect. The authors introduce key metrics like Generalization Gain via Mutual Information (GGMI) and propose a detailed modeling of the synthetic data generation process. They also provide upper bounds on generalization error based on information theory.
weaknesses: 1. The paper primarily focuses on theoretical analysis with minimal experimental validation. It is recommended to include empirical experiments to support the proposed theory.
2. The use of GMM models and non-real numerical data raises questions about whether the findings can effectively reflect real-world LLM outcomes.
3. The theory mainly emphasizes the performance benefits of introducing synthetic data, but lacks analysis in other areas. For example, it could explore potential drawbacks of over-reliance on synthetic data in LLMs, as highlighted in studies like https://www.nature.com/articles/s41586-024-07566-y.

Meta-review: The paper proposes a theoretical framework to analyze the role of synthetic data in the post-training of LLMs. 

 *strengths*
- Solid theoretical foundations and well-structured mathematical proofs to support the claims.
- Novel concepts like the reverse-bottleneck framework and GGMI that provide valuable insights into the role of synthetic data.
- Good job in establishing the theoretical rationale for using synthetic data in LLM post-training.

*weaknesses*
- The lack of extensive empirical validation on real-world LLM settings, which raises concerns about the applicability of the theoretical findings.
- The focus on the performance benefits of synthetic data, without a thorough analysis of potential downsides or limitations.
- The need for more concrete guidance on how the theoretical insights can be leveraged in practical LLM post-training workflows.

Overall,  it is marginally above the acceptance threshold: the reviewers acknowledged the solid theoretical contributions of the paper, but felt that more work is needed to bridge the gap between the theory and real-world LLM applications.

Decision: Accept (Poster)