ToEdit: How to Synthesize Text Data to Avoid Model Collapse?
Under review as a conference paper at ICLR 2025

TOEDIT: HOW TO SYNTHESIZE TEXT DATA TO AVOID
MODEL COLLAPSE?

Anonymous authors
Paper under double-blind review

ABSTRACT

We explore model collapse caused by synthetic data, where AI models trained on
such data experience a gradual decline in performance. Our initial analysis exam-
ines language model pretraining on mixed human and synthetic data, highlighting
performance degradation. Further statistical analysis reveals distributional shifts
and an over-concentration of n-gram features caused by synthetic data. Inspired
by these insights, we propose token-level editing on human data, to obtain semi-
synthetic data instead of fully using model outputs. As a proof of concept, we
theoretically demonstrate that token-level editing can prevent model collapse, as
the test error is constrained by a finite upper bound. We conducted extensive ex-
periments on pretraining, continual pretraining, and supervised fine-tuning of lan-
guage models. The results validate our theoretical proof that token-level editing
improves data quality and enhances model performance.

1

INTRODUCTION

As generative artificial intelligence (AI) such as ChatGPT (Achiam et al., 2023) and Stable Diffu-
sion (Rombach et al., 2021) are now widely used in our daily lives, training next-generation language
models within an ecosystem of synthetic and human data will be inevitable. How will synthetic data
influence AI training? Recent studies have given rise to two opposing viewpoints: some argue that
synthetic data is the future of AI training, while others claim it leads to model collapse. From a prac-
tical perspective, numerous synthetic datasets have been proved to boost the capabilities of language
models, like mathematics (Trinh et al., 2024; LI et al., 2024), biomedicine (Zhang et al., 2024),
alignment abilities (Ouyang et al., 2022; Cui et al., 2023) and so on. From a theoretical perspective,
training models iteratively on their own synthetic outputs results in the continuous accumulation of
errors, manifesting as a degenerative process for model learning (Shumailov et al., 2024), i.e., model
collapse. Furthermore, model collapse leads to a breakdown of scaling laws, ultimately rendering
the incremental computational effort ineffective (Dohmatob et al., 2024b).

There are two key questions that require further investigation: (1) Beyond the highly filtered syn-
thetic data in post-training, what is the impact of general synthetic data on language model training,
and how does it differ from human data? (2) How can we prevent model collapse when synthesizing
data, thereby producing higher-quality data?

In this paper, we answer the first question through data mixture pre-training with synthetic and
human data, which shows the non-iterative model collapse. Subsequent statistical analysis on dis-
tribution and features indicates coverage collapse and over-concentrates n-gram features of syn-
thetic data. Based on the above insights, we answer the second question by proposing a token-level
editing (ToEdit), which can avoid model collapse in theory and produce high-quality data across
experiments, including pre-training, continual pre-training, and supervised fine-tuning in practical.

Remarkable recent works provide a solid foundation for our work. Shumailov et al. (2024); Dohma-
tob et al. (2024a) identify the model collapse phenomenon and provide the first theoretical frame-
work based on linear regression. Gerstgrasser et al. (2024) demonstrated that if synthetic data is
accumulated while retaining the initial real data, the test error will be bounded, thus breaking model
collapse. Building on the above frameworks, we prove that our token-level editing can also avoid
model collapse. Additionally, Dohmatob et al. (2024b) indicated missing long tails of synthetic data
lead to scaling law cutoff, which motivated us to explore data mixture pretraining and statistical
analysis.

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Under review as a conference paper at ICLR 2025

Figure 1: Model Collapse of Synthetic Data. ① the model continuously trains on its previously
generated data, leading to a gradual decline in model performance, i.e., model collapse. ② We
use the trained model for token-level editing rather than purely synthesizing data. In this case, we
can preserve the distribution coverage, thereby avoiding model collapse and obtaining better data
compared to the initial data. Specifically, ① starting from real data (xo, yo), the test error Etest
increases as f0 is iteratively trained on synthetic data (y1, y2, . . . , yn). Our method, ② ToEdit,
utilizes f0 and an operation matrix mi to edit the data, achieving a fixed upper bound. Theoretical
details are provided in § 3

Contributions. We summarize the key contributions of this work as follows:

• We discover non-iterative model collapse through pre-training GPT-2 on a mixture of synthetic
and human data (§ 2.1). Specifically, we find that directly mixing general synthetic data, without
iterative training, leads to performance degradation.

• We conduct distributional statistical analysis to uncover that synthetic data cause distribution cov-
erage collapse and n-gram features over-concentrate. Further data selection struggled to correct
the distribution(§ 2.2)

• We propose token-level editing, which can be proved to avoid model collapse (§ 3) and pro-
duce high-quality data across scenarios of pre-training, continual pre-training and supervised fine-
tuning of language models (§ 4).

2 NON-ITERATIVE MODEL COLLAPSE

In this section, we investigate non-iterative synthetic data mixture training and explore the reasons
behind non-iterative model collapse. Non-iterative refers to training a model directly on data synthe-
sized by other models. Compared to previous iterative model collapse, non-iterative settings more
closely reflect real-world model training scenarios.

2.1 HUMAN AND SYNTHETIC DATA MIXTURE PRE-TRAINING

Setup We define the mixing ratio between human and synthetic data as α, where 0 ≤ α ≤ 1. The
total amount of training data Dtotal is expressed as a combination of human data Dhuman and synthetic
data Dsynthetic, represented by the formula:

Dtotal = αDhuman + (1 − α)Dsynthetic

(1)

We use Dolma (Soldaini et al., 2024) as
dia (Ben Allal et al., 2024) as

We use Cosmope-
the source synthetic data, which is distilled from

source human data.

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𝑚1𝑦1+1−𝑚1𝑦0𝑓0① Complete Synthetic Data ⇢𝑬𝒕𝒆𝒔𝒕=𝝈𝟐𝒅𝑻−𝒅−𝟏×𝒏⇢ Model Collapse 𝑥0𝑦0Training𝑓0𝑓1𝑦1𝑓𝑛𝑦2𝑓2……② Token-Level Editing ⇢𝑬𝒕𝒆𝒔𝒕≤𝝈𝟐𝒅𝑻−𝒅−𝟏×𝟐⇢Avoiding Model Collapse and Obtaining Better Data𝑥0𝑦0𝑦1𝑓𝑛……𝑓1TrainingTrainingTrainingTraining…………Source Real Data: 𝑥0,𝑦0𝑓0𝑦1Data Synthesizing𝑓0𝑦1Data EditingTest Error 𝐸𝑡𝑒𝑠𝑡Editing Operation Matrix miNumber of Iterations𝑖∈{1,…,𝑛}Synthetic Data: (𝑦1,𝑦2,…,𝑦𝑛)Data Size TInput Dimensions 𝑑Trained Model 𝑓𝑖Label Noise Scalar 𝜎Under review as a conference paper at ICLR 2025

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Figure 2: Non-Iterative Model Collapse. Training language models from scratch on AI-synthesized
data or a mixture of human and synthetic data leads to performance degradation. This degradation
is positively correlated with the proportion of synthetic data used in training. A. We pretrain GPT-
2 Small (124M) on human (Dolma (Soldaini et al., 2024)) and synthetic (Cosmopedia (Ben Allal
et al., 2024)) data. As the proportion of synthetic data increases, the model’s loss decreases. B. As
the proportion of synthetic data increases, the PPL also rises. This trend remains consistent across
different validation sets.

Mixtral-8x7B-Instruct-v0.1 (Jiang et al., 2024). Using the data mixture of 50B tokens,
we train two models from scratch, including GPT-2 (Radford et al., 2019) and OLMo (Groeneveld
et al., 2024).

Finding I: General synthetic data harm the language models pre-training. Previous massive
works have proved synthetic data can boost language models’ capability, including instruction fol-
lowing (Wang et al., 2022a), reasoning (Zhu et al., 2023; Trinh et al., 2024), alignment (Cui et al.,
2023), biomedicine (Zhang et al., 2024) and so on. However, as illustrated in Figure 2, the PPL of
real-world validation sets is inversely proportional to the proportion of synthetic data. Compared
with prior studies, we mix synthetic data in pre-training, not supervised fine-tuning and RLHF,
which are downstream tasks. Before a language model reaches a certain level of learning, that is,
when training from scratch, synthetic data is unlikely to help the model learn and may even hinder its
learning. When synthetic data incorporates some human data into training data, the model collapse
can be alleviated. Compared to previous works on iterative model collapse (Shumailov et al., 2024;
Dohmatob et al., 2024a;b), the non-iterative damage caused by synthetic data is more concerning
and relevant to the training of next-generation language models.

2.2 WHY DO SYNTHETIC DATA FAIL IN LANGUAGE MODEL PRE-TRAINING?

We conduct three statistical analyses: (1) sample-level distribution, (2) feature-based overlap, and
(3) distribution-reference data selection. From the following experiments, we can summarize that
compared with human data, synthetic data not only lacks long tails but also coverage collapse. It is
hard to use human data as a reference to filter synthetic because the features in synthetic data are
condensed heavily.

Setup We conducted statistical and feature-based analyses to explore why synthetic data fails in
pre-training. (1) We leverage a prior distribution P to estimate the human and synthetic data. We use
Llama-3-8B (AI@Meta, 2024) and StableLM-Zephyr-3B (Bellagente et al., 2024). Different priors
consistently yield the same results. (2) We analyze the n-gram features of human and synthetic data
from a feature-based perspective, such as n-gram response values. (3) Based on the distribution of
real data, we sample data from the synthetic dataset that closely matches the real data distribution in
an attempt to filter the synthetic data.

Finding II.i Synthetic data distribution not only misses long tails, but also causes coverage
collapse. Figure 3 and 9 illustrate that the PPL of synthetic data is confined to the lower 25% of the

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A. GPT-2 Pre-Training Loss B. GPT-2 PPL on Validation SetsUnder review as a conference paper at ICLR 2025

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Figure 3: PPL distribution of human and synthetic data estimated by Llama-3-8B. The synthetic
data lacks the long tail of the human data and is also concentrated within the first 25% of the human
data distribution. A. Distribution of human data is sharp with a long tail, spanning a wide range
from 0 to over 100. B. The values are concentrated within a much narrower range, mostly between
0 and 12. The experiment uses Dolma v6 and Cosmopedia as human and synthetic data, each with
sampled 6B tokens. More results in Figure 9.

human data, failing to capture the full range and complexity of real data distributions. Specifically, as
illustrated in Figure 3A, human data exhibit a wide distribution in the range [1, 100+], characterized
by a sharp peak and a pronounced long tail. In contrast, as shown in Figure 3B, the synthetic data
is confined to a narrower range of [0, 14], displaying a smoother distribution. Further results of
StabLM are shown in Figure 9. While the absolute PPL ranges estimated by different models may
vary, the relative shapes and proportional ranges of these two distributions remain consistent. This
phenomenon provides evidence that when scaling up to larger synthetic datasets, there is a notable
absence of the long tail. Furthermore, we also observe a more severe coverage collapse. This
limited coverage reduces the data’s ability to generalize well and may contribute to model collapse
in Figure 2.

Finding II.ii Synthetic data over-concentrates N-
gram features. Based on the above distribution
estimate, we further analyze why synthetic data fails
at the feature level. Figure 10 and 11 demonstrate
that synthetic data exhibits higher frequencies in cer-
tain bi-grams than human data. To further exam-
ine feature-level differences, we hash unigram and
bigram features into 10,000 hash buckets. As il-
lustrated in Figure 4, the response range of human
data is noticeably broader, while the features of syn-
thetic data are primarily concentrated in a few spe-
cific buckets. This indirectly supports our earlier
observation of feature over-concentration. We then
expanded the hash bucket range to 1,000 × 20,000
buckets and used a locality-sensitive hashing method
to differentiate the features more precisely. How-
ever, the results remained similar. As shown in Fig-
ure 12, the majority of the response values are close
to zero. The features of synthetic data are difficult to
distinguish.

Finding II.iii Distribution shifting cannot be mit-
igated through data selection.
Inspired by recent
data selection works (Xie et al., 2023; Albalak et al.,
2024), we try to leverage the human data features
as reference distribution to select synthetic samples.
We implement importance sampling in DSIR (Xie

4

Figure 4: Uni/Bi-gram feature distribution
across 10,000 hash buckets.

A. Human Data PPL Distribution Estimated by Llama-3-8BB. Synthetic Data PPL Distribution Estimated by Llama-3-8BUnder review as a conference paper at ICLR 2025

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Figure 5: A. Embedding visualization using t-SNE and sentence-transformers. B. pretraining results
for selected synthetic data and other data mixtures.

et al., 2023) to filter synthetic data. As shown in
Figure 5A, the sampled data still fails to align with real data in the embedding space, even at the
boundary regions of the synthetic data. As illustrated in Figure 5B, the training results of selected
synthetic samples still fluctuates around the original performance of the synthetic data, indicating
that even biased sampling cannot correct the distributional shift.

2.3 PROPOSED STRATEGY

Following these lessons so far, due to the properties of coverage collapse and feature overconcen-
tration of synthetic data, our best option is to use totally human data and avoid the inclusion of
synthetic data. However, we are still wondering how we can use synthetic data to improve human
data. We arrive at a general guideline for synthetic data: full synthetic data will result in model
collapse, so we need to keep the main human data distribution. In that case, we propose token-level
editing, which leverages a prior distribution to edit the data. Our method can maintain the source
distribution and improve the source data, called semi-synthetic data.

3 TOKEN-LEVEL DATA EDITING

In this section, we introduce token-level data editing to obtain semi-synthetic data. Furthermore, we
provide theoretical analysis and proof that our method’s test squared error has a finite upper bound,
independent of the number of iterations. In this case, our method not only avoids model collapse but
also obtains better performance.

3.1 METHOD

We formulate data synthesis as follows: assuming P is a prior distribution, given a sequence of
tokens x = (x1, . . . , xt), the full synthetic data is y = (y1, . . . , yn). The synthesis process is
derived as:

P (y1, . . . , yn | x1, . . . , xt) =

n
(cid:89)

i=1

P (yi | y1, . . . , yi−1, x1, . . . , xt).

(2)

This conditional probability formulation outlines the generation of synthetic data conditioned on the
given token sequence. Then the synthetic data is used to train models.

Inspired by previous studies of data selection (Mindermann et al., 2022; Ankner et al., 2024; Lin
In
et al., 2024), prior distribution can be a pointer to indicate the useless or learnable samples.

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A. EmbeddingVisualization between Human, Synthetic, and DSIR-Selected Data using t-SNEB. PPLResults for OLMo-237M Pretraining on Selected Synthetic Data and Data MixturesUnder review as a conference paper at ICLR 2025

this case, we use a pre-trained language model to infer the pretraining corpus. As illustrated in
Figure 6, even a model pre-trained on trillions of tokens can not fit the pretraining corpus perfectly.
Specifically, 75% is under 0.6 probability. The tokens with both high and low probabilities are the
most concentrated, suggesting the potential for data filtering. We leverage this U-shape distribution
as a pointer to resample tokens. Specifically, we use a language model as prior distribution to
compute each token’s conditional probability P (·|x) if the probability exceeds a certain threshold
P (·|x) ≥ p, it indicates that this token is easy to learn, and we proceed with resampling at that point.

Token-level Editing doesn’t generate the whole sequence but leverages conditional probability
P (xi | x1, . . . , xi−1) to revise the input sequence. In this way, we can avoid using purely synthetic
data while modifying the dataset to preserve human long-tail features, aiming to obtain higher-
quality semi-synthetic data. Token-level Editing can be formulated as follows:

xi′ =

(cid:26)xi,
˜xi,

if P (xi | x1, . . . , xi−1) < p
if P (xi | x1, . . . , xi − 1) ≥ p

(3)

Where x′
i is the final token in the edited sequence. ˜xi is a token resampled from a prior distribution.
We can control the size of p that balances between retaining the structure of human data and avoiding
overfitting to the synthetic data.

Compute conditional probability P (xi | x1, . . . , xi−1)
if P (xi | x1, . . . , xi−1) ≥ p then

Algorithm 1 Token-level Editing
1: Input: Sequence of tokens x = (x1, . . . , xt), prior distribution P , probability threshold p
1, . . . , x′
2: Output: Edited sequence x’ = (x′
t)
3: for each token xi in sequence x do
4:
5:
6:
7:
8:
9:
end if
10:
11: end for
12: Return: Edited sequence x’ = (x′

Resample token ˜xi from prior distribution
Set x′

i ← xi

i ← ˜xi

Set x′

else

1, . . . , x′
t)

3.2 THEORETICAL ANALYSIS

To investigate more mathematical insights, we
utilize an analytical framework of the lin-
ear model and adopt notions in prior
re-
search (Mobahi et al., 2020; Dohmatob et al.,
2024a; Gerstgrasser et al., 2024). This the-
oretical framework primarily considers a lin-
ear model that iteratively trains on its own
generated data, similar to pipelines like self-
play and self-distillation, but without complex
constraints.
It simply involves training con-
tinuously on the data generated by the previ-
ous generation of the model. Dohmatob et al.
(2024a) point out that with iterative training,
test errors accumulate progressively, eventually
leading to model collapse. Based on this theo-
retical framework, we incorporate our proposed
token-level editing into the framework and an-
alyze whether our method can prevent model
collapse.

Figure 6: U-shape token probability distribution
of Dolma-sampled V6 estimated by Qwen-0.5B-
Instruct (qwe, 2024).

Notation and Preliminaries For a given dis-
tribution PΣ,w,σ2 , the data (x, y) ∼ PΣ,w,σ2 on Rd ×R, where x is drawn from a multivariate normal

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Under review as a conference paper at ICLR 2025

distribution x ∼ N (0, Σ), ϵ is an independent noise term sampled from N (0, σ2), and the label y is
given by the linear model y = x · w∗ + ϵ.

Iterative Data Editing Process We utilize the model obtained from the previous round of training
to make a limited number of modifications. Specifically, we resample and replace data points with
relatively high confidence. The editing operations are defined by the matrices {M1, M2, . . . , Mn}.
The iterative data synthesis and model-fitting process can be formalized as follows:

PΣ,w∗,σ2 → PΣ, ˆw1,σ2 → . . . → PΣ, ˆwn,σ2,
where n is the number of iterations. The detailed process of data editing and iterations is described
as follows:
For n = 1, we begin by initializing the covariates/features as ˜X1 = X. The target values are defined
as ˜Y1 = ˆY1 = Xw∗ + E1, where E1 ∼ N (0, σ2IT ). The linear model is then fitted by solving for
ˆw1 = ˜X †
˜Y1. To proceed to the next iteration, we resample the data, obtaining ˆY2 = X ˆw1 + E2,
1
with E2 ∼ N (0, σ2IT ).
For n ≥ 2, the input covariates/features remain as ˜X ⊤
using the edited targets, following the equation ˜Y ⊤
model is then fitted by computing ˆwn = ˜X †
n
yielding ˆYn+1 = X ˆwn + En+1, where En+1 ∼ N (0, σ2IT ).
The matrix Mn is a diagonal matrix, where some elements on the diagonal are 1 and others are 0.
The multiplication by M can be interpreted as an operation that selectively modifies certain data
points (those corresponding to 1s) while retaining others (those corresponding to 0s). Then, the data
editing process can be formulated as follows:

n = X, while the target values are updated
n = Mn−1 ˆYn + (1 − Mn−1) ˜Yn−1. The linear
˜Yn. Finally, the data is resampled for the next iteration,

n = Mn−1 ˆYn + (1 − Mn−1) ˜Yn−1
˜Y ⊤
(4)
where ˜Yn−1 is the data after editing in the n−1 generation, and ˆYn is the synthetic data from the n-th
generation. This process can be described as: firstly, synthesizing labels for all inputs; secondly, the
M matrix determining which data is edited and which is retained. For a matrix A with full column
rank, its Moore-Penrose pseudo-inverse is A+ = (A⊤A)−1A⊤. The noise terms E1, E2, . . . , En
are independent of each other and the covariates/features. Since X has full column rank, ˜Xn retains
this property for all n ≥ 1.

Test Error Model collapse is ultimately reflected through test error, and here we follow previous
work (Gerstgrasser et al., 2024) to define the standard test error. For any linear estimator ˆw derived
from the training data, we evaluate the test error using the standard method as follows:

Etest(w) def= E (cid:2)(xT

testw − ytest)2(cid:3) − σ2 = E[∥w − w∗∥2
Σ]

(5)

where the expectation is computed with respect to the training data, while the test pair (xtest, ytest) is
sampled from PΣ,w∗,σ2 independently of the training set.

3.3 TEST ERROR UNDER DATA EDITING

Our goal is to derive an analytical expression for the test error of the n-th model in the data editing
setting. As indicated by the test error in Eq. 5, this requires two steps: (1) establishing the relation-
ship between the fitted linear parameters wn and the true parameters w∗, and (2) simplifying the test
error expression. We start by establishing the formulation between wn and w∗. Proofs are detailed
in App. B.

Theorem 1 In the data editing setting, ∀n ≥ 1, the fitted linear parameters ˆwn+1 can be derived
as:

(cid:32)

(cid:33)

n
(cid:88)

i=1

MiEi+1

ˆwn+1 = w∗ + (X ⊤X)−1X ⊤

E1 +

7

(6)

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Under review as a conference paper at ICLR 2025

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where, w∗ is the true parameter, X is the original design matrix, Ei is the extra noise added at the
i’th iteration, and Mi is an idempotent diagonal matrix, defining the editing operation.

Theorem 2 Consider an n + 1 fold data editing process with T ≥ d + 2 samples per iteration and
def
= Id), the test error for the ridgeless linear model ˆwn learned on the edited
isotropic features (Σ
data up to iteration n + 1, is bounded by:

Etest( ˆwn+1) ≤

2σ2d
T − d − 1

(7)

Furthermore, assuming the editing operation satisfies ||Mi|| = ||Mi−1||η with η ∈ (0, 1), the test
error can be further bounded by:

Etest( ˆwn+1) ≤

σ2d
T − d − 1

+ σ2(cid:113)

E [tr ((X ⊤X)−2)] ·

(cid:112)E [tr(M1)]
1 − η

(8)

Recalling that the cause of model collapse (Dohmatob et al., 2024a): training iteratively on synthetic
data leads to an accumulation of error over iterations, as shown in the following equation:

Ecollapse
test

( ˆwn) =

σ2d
T − d − 1

× n

(9)

Compared Eq. 7 with Eq. 9, the error in data editing is bounded by a fixed value, preventing contin-
uous error accumulation and thus avoiding model collapse. Combining the above theoretical deriva-
tions and statistical analysis of synthetic data (§ 2.1), the underlying reason is that our approach
retains the coverage of the initial distribution. We move away from pure data synthesis toward
token-level data editing, which allows us to obtain better data while avoiding model collapse. More-
over, remarkable previous studies (Dohmatob et al., 2024b; Gerstgrasser et al., 2024) pointed out
similar conclusions. They indicated mixing real data with synthetic data will break model collapse
and provide an upper bound under data accumulation. Different from their work, our data editing
aims to yield better data, enabling synthetic data to perform well both in theory and practice, not
only avoiding model collapse.

4 EXPERIMENTS

To validate our proposed method, we conduct experiments across three stages of language model
training including: pre-training, continual pre-training (CPT) and supervised fine-tuning (SFT).

4.1

IMPLEMENTATION

We use the Llama-3-8B (AI@Meta, 2024) as a prior distribution to estimate the token distribution
in each text sample. The modification probability is set to p = 0.99. This means that we resample
tokens in positions where the probability exceeds p, and the resampling is based on the conditional
probability given the preceding context. The entire process of our method requires only a single for-
ward pass, without auto-regressive generation. We integrate the fast inference engine vLLM (Kwon
et al., 2023), allowing the entire data editing process to be completed on a single 4090 GPU. After
completing the data editing, we compared the original data and the edited data on language model
training performance across pre-training, CPT, and SFT. Here, we used top-k as the sampling strat-
egy with k = 8. We also experimented with top-p and rejection sampling, which produced similar
results.

4.2 DATASETS AND MODELS

Here, we provide an overview of our experimental setup. More training details are presented in
Appendix D. As for pre-training, we pre-train the 1B OLMo model (Groeneveld et al., 2024)
from scratch, using Dolma-sampled V6 (6B tokens) as the pre-training corpus. Dolma (Soldaini

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Under review as a conference paper at ICLR 2025

et al., 2024) is the largest open-source pre-training corpus available. We use 8 general tasks in
lm-evaluation-harness (Gao et al., 2024) to evaluate for pre-training models. As for continual pre-
training, we follow Cheng et al. (2024b) to continual pre-train the OLMo-1B (Groeneveld et al.,
2024) and Llama-3-8B (AI@Meta, 2024) on Biomedicine, Finance and Math. Each domain corpus
contains 1B tokens. Correspondingly, we evaluate the continual pre-training models using 15 down-
stream tasks, with 5 tasks from each domain. As for supervised fine-tuning, we fine-tune Llama-3-
8B on instruction tuning tasks. We use natural-instructions (Wang et al., 2022b), as fine-tuning data,
which consists of over 1500 tasks. We evaluate the SFT models using 5 downstream tasks designed
to measure instruction-following capabilities. All Llama-3-8B experiments use LoRA (Hu et al.,
2021), while the OLMo-1B model is trained with full parameters.

Table 1: Domain specific tasks performance for continual pretraining models. CPT indicates contin-
ual pre-training. ∆ indicates training with our edited data. Our method shows consistent improve-
ments across three domains on OLMo-1B and Llama-3-8B.

Models

OLMo-1B
CPT
∆ ToEdit

LLama-3-8B
CPT
∆ ToEdit

Models

OLMo-1B
CPT
∆ ToEdit

LLama-3-8B
CPT
∆ ToEdit

MQP

52.59
52.29
54.59

66.80
72.29
76.39

HeadLine

69.00
70.31
71.77

81.28
85.68
83.83

Biomedicine

ChemProt

PubMedQA

17.2
21.00
22.40

28.59
29.4
30.2

FPB

47.03
49.78
51.39

63.58
54.22
61.61

51.40
58.50
65.00

60.8
69.1
65.3

RCT

32.70
34.90
34.50

73.85
72.65
73.30

Finance

FiQA SA

ConvFinQA

48.05
40.36
46.06

81.60
81.88
80.82

Math

4.83
18.72
18.85

52.88
67.78
67.31

USMLE Average

28.90
27.49
27.96

40.61
36.76
37.23

NER

62.19
60.44
62.97

72.53
67.43
67.62

36.63
38.83
40.89

54.13
56.04
56.48

Average

46.22
47.92
50.21

70.37
71.40
72.24

Models

Arc-Challenge

GPQA

GSM8K

MATH

MMLU

Average

OLMo-1B
CPT
∆ ToEdit

28.67
28.41
28.92

24.23
24.03
28.12

1.67
1.52
2.20

0.00
0.10
0.10

26.56
27.23
23.63

16.23
16.26
16.59

4.3 RESULTS

Table 1, 2, and 3 respectively demonstrate the effectiveness of our method in continual pre-training,
pre-training, and fine-tuning tasks. Across these three stages of language model training, our method
enhances the model’s performance on downstream tasks without increasing the data size. Our
method further taps into the potential of existing data, also demonstrating that semi-synthetic data is
a viable path to obtaining higher-quality data.

Specifically, as shown in Table 1, our method shows consistent improvements over the source data
across OLMo-1B and LLaMA-3-8B. For instance, in the Biomedicine domain, the average score
for OLMo-1B increased from 36.63 to 40.89 with ToEdit, while LLaMA-3-8B saw an increase
from 54.13 to 56.48. Table 2 further supports the effectiveness of our approach in pre-training. The
average performance of OLMo-1B increases from 32.75 to 33.11, reflecting improved generalization
capabilities. While the improvement is modest, the consistent trend across tasks like PIQA, BoolQ,
and ARC-c highlights the broader applicability of our method.

As for SFT results in Table 3, using both the original and edited data, the results indicate a small
but consistent improvement. Specifically, ToEdit improves orignal FLAN V2, with average per-
formance increasing from 70.18 to 70.65. As for Natural Instructions, the average performance of

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LLaMA-3-8B improves from 69.34 to 69.70, with gains in tasks like Winogrande and SIQA. These
improvements, demonstrate the adaptability of our method to instruction-tuning tasks. For code-
related tasks, the improvements are particularly evident in ARC-Challenge and GPQA, indicating
better reasoning and code comprehension.

In summary, experiments on pretraining, continual pretraining, and SFT validate the effectiveness
and versatility of our method. More ablation studies and discussions can be found Appendix F
and E.

Table 2: General performance of the pre-trained base models. PT indicates we pre-train OLMo-1B
from scratch. Experimental results demonstrate that our method can also enhance the effectiveness
of pre-training.

PIQA BoolQ OBQA ARC-c ARC-e HellaSwag

SIQA Winogrande Average

OLMo-1B (PT)
∆ ToEdit

53.97
54.13

38.26
38.65

12.20
12.80

17.23
18.43

28.36
27.48

26.02
25.94

34.80
34.95

51.14
52.49

32.75
33.11

Table 3: Performance of the SFT models. We fine-tune LLaMA-3-8B using instruction tuning and
code reasoning tasks, comparing performance with the edited version produced by our method. The
experimental results indicate that our approach can enhance the data for instruction-tuning and code
reasoning tasks.

Models

PIQA BoolQ HellaSwag

SIQA Winogrande Average

Instruction Tuning

Natural Instructions

CoT

FLAN V2

Open Assistant 1

Llama-3-8B 79.82
∆ ToEdit
80.58

Llama-3-8B 79.87
∆ ToEdit
80.25

Llama-3-8B 80.79
∆ ToEdit
80.69

Llama-3-8B 79.65
∆ ToEdit
79.98

87.06
87.80

81.28
81.16

84.04
85.20

83.18
83.91

58.32
58.27

59.72
59.74

59.98
59.99

60.51
60.34

46.83
46.93

49.69
50.56

51.43
52.00

48.52
48.31

74.66
74.90

74.51
74.59

74.66
75.37

74.11
74.66

69.34
69.70

69.01
69.26

70.18
70.65

69.19
69.44

Models

ARC-c GPQA GSM8K MMLU Average

Code Reasoning

OSS-Instruct-75K

Evol-Instruct-110K

Llama-3-8B
∆ ToEdit

Llama-3-8B
∆ ToEdit

51.28
51.79

52.90
52.22

27.46
28.79

27.90
29.69

49.58
49.36

50.87
50.87

62.14
62.04

62.40
62.60

45.76
46.13

46.62
46.92

5 CONCLUSION

With the growing prevalence of generative AI models like ChatGPT (Achiam et al., 2023) and Stable
Diffusion (Rombach et al., 2021), when training next-generation AI models, it will be inevitable to
use a mixture of synthetic data and human data. Therefore, we focus on two key questions: (1)
What is the impact of synthetic data on language model pre-training, and what are the underlying
causes? (2) How can we prevent model collapse and synthesize high-quality data? We found that
synthetic data can impair the effectiveness of pre-training when mixed with human data, leading to
non-iterative model collapse. Statistical analysis reveals that synthetic data suffers from significant
distribution gaps and overly concentrated n-gram features. Based on this, we propose token-level
editing instead of relying purely on synthetic data. Specifically, we perform token resampling guided
by a trained prior. Moreover, our method can theoretically prevent model collapse. Experimentally,
our approach shows improvements over the source data across pre-training, continual pre-training,
and supervised fine-tuning.

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Under review as a conference paper at ICLR 2025

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A RELATED WORK

Model collapse Shumailov et al. (2024); Dohmatob et al. (2024a;b) demonstrate AI models
trained recursively on data generated by earlier versions of themselves over time can result in per-
formance degradation, ultimately rendering the AI model completely useless. This process can be
formulated as follows:

Etest( ˆwn+1) =

σ2d
T − d − 1

× n

This indicates that the error will continuously increase with the number of iterations n. Dohmatob
et al. (2024b) further pointed out that synthetic data also contribute to a truncation of the scaling law.
This phenomenon stems from the sampling strategy (e.g., Top-p) used during the language model’s
generation process. Gerstgrasser et al. (2024) further adjusted the data iteration setting by replacing
data replacement with data accumulation during the iterative process. They demonstrated that data
accumulation can prevent model collapse.
Inspired by the above work, we believe that training
language models on synthetic datasets will be inevitable in the future. Therefore, it is crucial to
theoretically discuss how to prevent model collapse. Building on the above theoretical framework,
we proved that token-level editing establishes an upper bound during the iterative process, thereby
preventing the continuous accumulation of errors.

Synthetic Data Phi-1/2 (Gunasekar et al., 2023) demonstrated the synthetic data boost training ef-
ficiency and performance compared with raw data in language model pre-training. Liu et al. (2024)
highlighted that synthetic data will play a crucial role in the development of AI. For example, syn-
thetic data can be used to construct highly specialized datasets, enhancing the performance of down-
stream tasks. Trinh et al. (2024) utilized synthetic math data to train a 125M language model, which
successfully solved 25 out of 30 selected problems from the International Mathematical Olympiad
(IMO) problem set. Zhang et al. (2024) developed a biomedical instruction dataset that was used
to train specialized bio-models, enabling them to excel in answering questions related to medical
exams and clinical scenarios. Eldan & Li (2023) introduced a novel synthetic dataset and evalua-
tion paradigm that enables small language models to generate coherent, diverse, and grammatically
sound stories. As outlined above, in the post-training stages of LLMs, synthetic data enhances the
ability of downstream tasks and aligns foundation models with humans. And Maini et al. (2024)
proposed rephrasing the pre-training data into a Wikipedia or Q/A style to achieve better alignment
with downstream tasks. Synthetic data is a powerful tool for training. Our approach is also based
on synthetic data methods. Instead of sampling data solely based on this prior, we modify the data
using the prior as a guide.

B PROOF

B.1 PROOF OF THEOREM 1

For n = 1, we have:

ˆw1 = ˜X †
1

˜Y1 = (X ⊤X)−1X ⊤(Xw∗ + E1) = w∗ + (X ⊤X)−1X ⊤E1

For n ≥ 1, we have:

˜Yn+1
ˆwn+1 = ˜X †
n+1
= ( ˜X ⊤
˜Xn+1)−1 ˜X ⊤
= (X ⊤X)−1X ⊤ ˜Yn+1

n+1

n+1

˜Yn+1

Recalling that:

˜Yi =

(cid:26)Xw∗ + E1,

Mi−1(X ˆwi−1 + Ei) + (1 − Mi−1) ˜Yi−1,

if i = 1
if 2 ≤ i ≤ n + 1

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(10)

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Under review as a conference paper at ICLR 2025

Substituting this ˜Yi into the expression for ˆwn+1:
We begin the data editing data process:

˜Y2 = M1(X ˆw1 + E2) + (1 − M1) ˜Y1

˜Y3 = M2(X ˆw2 + E3) + (1 − M2) ˜Y2

Then:

We have:

˜Y3 = M2(X ˆw2 + E3) + (1 − M2)

(cid:16)

M1(X ˆw1 + E2) + (1 − M1) ˜Y1

(cid:17)

= M2(X ˆw2 + E3) + (1 − M2)M1(X ˆw1 + E2) + (1 − M2)(1 − M1) ˜Y1

We can expand ˜Yn+1 by recursively substituting the previous expressions:

˜Yn+1 = Mn(X ˆwn + En+1) + (1 − Mn) ˜Yn
(cid:104)

(cid:105)

= Mn(X ˆwn + En+1) + (1 − Mn)
= Mn(X ˆwn + En+1) + (1 − Mn)Mn−1(X ˆwn−1 + En) + (1 − Mn)(1 − Mn−1) ˜Yn−1

Mn−1(X ˆwn−1 + En) + (1 − Mn−1) ˜Yn−1

(13)

...

n
(cid:88)

=









n
(cid:89)





(1 − Mj)

 Mi(X ˆwi + Ei+1)

 +

i=1

j=i+1

Recalling properties of Mi:


 ˜Y1

(1 − Mj)





n
(cid:89)

j=1

Mi(1 − Mi) = 0 and
MiMj = 0 for

(1 − Mi)Mi = 0
i ̸= j
(1 − Mi)(1 − Mj) = 1 − Mi − Mj

i ̸= j

for

Then we have:

˜Yn+1 =

=

n
(cid:88)

i=1
n
(cid:88)

i=1

(cid:32)

Mi(X ˆwi + Ei+1) +

1 −

Mi(X ˆwi + Ei+1) +

1 −

(cid:32)

n
(cid:88)

i=1
n
(cid:88)

i=1

(cid:33)

Mi

˜Y1

(cid:33)

Mi

(Xw∗ + E1)

= Xw∗ + E1 +

n
(cid:88)

i=1

Mi (X( ˆwi − w∗) + (Ei+1 − E1))

(14)

(15)

(16)

(17)
(18)
(19)
(20)

(21)

(22)

(23)

Substituting this back into the expression for ˆwn+1:

ˆwn+1 = (X ⊤X)−1X ⊤

Xw∗ + E1 +

(cid:34)

n
(cid:88)

i=1

Mi (X( ˆwi − w∗) + (Ei+1 − E1))

(24)

(cid:35)

= w∗ + (X ⊤X)−1X ⊤

E1 +

(cid:34)

n
(cid:88)

i=1

MiX( ˆwi − w∗) +

(cid:35)

Mi(Ei+1 − E1)

(25)

n
(cid:88)

i=1

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Under review as a conference paper at ICLR 2025

We can observe:

ˆw1 = (X ⊤X)−1X ⊤(Xw∗ + E1) = w∗ + (X ⊤X)−1X ⊤E1
ˆw2 = w∗ + (X ⊤X)−1X ⊤ (cid:0)M1X(X ⊤X)−1X ⊤E1 + M1E2 + (1 − M1)E1

(cid:1)

= w∗ + (X ⊤X)−1X ⊤ (E1 + M1E2)

We prove this Theorem 1 by induction.

Inductive Step: Assume the formula holds for n, we have:

ˆwn+1 = w∗ + (X ⊤X)−1X ⊤ (E1 + M1E2 + M2E3 + · · · + MnEn+1)

= w∗ + (X ⊤X)−1X ⊤

E1 +

(cid:32)

(cid:33)

MiEi+1

n
(cid:88)

i=1

Substitute ˆwi into ˆwn+1:

Then we can get:

ˆwn+1 = w∗ + (X ⊤X)−1X ⊤

E1 +





= w∗ + (X ⊤X)−1X ⊤

E1 +

= w∗ + (X ⊤X)−1X ⊤

E1 +

(cid:32)

where P = X(X ⊤X)−1X ⊤,



MjEj+1

 +

n
(cid:88)

i=1





MjEj+1





i−1
(cid:88)

j=1

i−1
(cid:88)

j=1



MiP

E1 +



Mi

Ei+1 +

(cid:33)

MiEi+1

n
(cid:88)

i=1

n
(cid:88)

i=1

n
(cid:88)

i=1

The above derivation aligns with Theorem 1, and the proof is complete.

B.2 PROOF OF THEOREM 2

Mi(Ei+1 − E1)



We substitute the Eq. 30 into Test Error Eq. 5:


Etest( ˆwn+1) = E



(cid:32)

(cid:13)
(cid:13)
(X ⊤X)−1X ⊤
(cid:13)
(cid:13)
(cid:13)

n
(cid:88)

i=1

MiEi+1





(cid:33)(cid:13)
2
(cid:13)
(cid:13)
(cid:13)
(cid:13)

Σ

(cid:32)

E1 +

(cid:33)⊤

MiEi+1

X(X ⊤X)−2X ⊤

E1 +



(cid:32)

= E



E1 +

n
(cid:88)

i=1

(cid:33)


MiEi+1

n
(cid:88)

i=1

= σ2E (cid:2)tr (cid:0)(X ⊤X)−1(cid:1)(cid:3) + σ2

= σ2E (cid:2)tr (cid:0)(X ⊤X)−1(cid:1)(cid:3) + σ2

n
(cid:88)

i=1
n
(cid:88)

i=1

E (cid:2)tr (cid:0)Mi(X ⊤X)−1Mi

(cid:1)(cid:3)

E (cid:2)tr (cid:0)(X ⊤X)−1Mi

(cid:1)(cid:3)

Further, by applying the Cauchy-Schwarz inequality (Rudin, 1976), we obtain:

Etest( ˆwn+1) ≤ σ2E (cid:2)tr (cid:0)(X ⊤X)−1(cid:1)(cid:3) + σ2(cid:113)

E [tr ((X ⊤X)−2)] ·

n
(cid:88)

i=1

(cid:112)E [tr(Mi)]

(39)

We refer to the following lemma (Dohmatob et al., 2024a), which is essential for proving Theorem
2:

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(30)



(31)

(32)

(33)

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(38)

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Under review as a conference paper at ICLR 2025

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Lemma 3 Let T and d be positive integers with T ≥ d + 2, and let X ∈ RT ×d be a random matrix
with i.i.d. rows from N (0, Σ) with Σ positive definite. Then, X has full rank a.s. Moreover, it holds
that:

EX

(cid:2)(X ⊤X)−1(cid:3) =

1
T − d − 1

Σ−1.

Etest

(cid:2)tr (cid:0)(X ⊤X)−1(cid:1))(cid:3) =

d
T − d − 1

Using Lemma 3, we have:

Then, we have:

Etest( ˆwn+1) = σ2E (cid:2)tr (cid:0)(X ⊤X)−1(cid:1)(cid:3) + σ2

n
(cid:88)

i=1

E (cid:2)tr (cid:0)(X ⊤X)−1Mi

(cid:1)(cid:3)

≤

σ2d
T − d − 1

+ σ2(cid:113)

E [tr ((X ⊤X)−2)] ·

n
(cid:88)

i=1

(cid:112)E [tr(Mi)]

(40)

(41)

(42)

(43)

In our setting, the data is incrementally modified over iterations and modifications decreases pro-
gressively. This behavior can be modeled by the sum of a geometric series, where the amount of
modified data decreases by a fixed ratio η with each iteration. Then, we assume the editing operation
as ||Mi|| = ||Mi−1||η, for i = 1, 2, . . . , n. Therefore, the test error for data editing can be bounded:

Etest( ˆwn+1) ≤

σ2d
T − d − 1

+ σ2(cid:113)

E [tr ((X ⊤X)−2)] ·

(cid:112)E [tr(M1)]
1 − η

(44)

Additionally, since Mi is not full-rank, as seen from Eq. 38, we can apply a more relaxed and
simplified bound, as follows:

Etest( ˆwn+1) ≤

2σ2d
T − d − 1

(45)

Thus, the above derivation satisfies the Theorem 2.

C MORE RESULTS OF HUMAN AND SYNTHETIC DATA MIXTURE TRAINING

We provide more training results for the human and synthetic data mixture. The main results and
analysis can be found in Sec 2.1. Except for GPT-2 pretraining, we also use the OLMo mod-
els (Groeneveld et al., 2024) for further experiments.

As shown in Figure 8, the training loss continues to decrease as the amount of synthetic data in-
creases, which is consistent with GPT-2 pretriaing in Figure 2. More synthetic data can lead to
better fitting. However, a lower loss does not necessarily mean a better model. As illustrated in
Figure 2B and 7, models that fits better perform worse in real world tasks.

Furthermore we follow Maini et al. (2024) to conduct more experiments including PPL results on
22 validation sets of Pile (Gao et al., 2020) and general understanding tasks. The additional results
in Table 5 and 6 are consistent with our findings. Specifically, the PPL increases as the proportion of
purely synthetic data grows, while the performance on downstream tasks similarly exhibits a gradual
decline with the increase in synthetic data.

D DETAILED EXPERIMENT SETTINGS

In this section, we describe our experiments settings detailed.

18

Under review as a conference paper at ICLR 2025

D.1 TRAINING

Pre-training We utilized both GPT-2 and OLMo models. The pre-training datasets included
Dolma, representing real data, and Cosmopedia, representing synthetic data. For GPT-2, we em-
ployed the official FSDP (Fully Sharded Data Parallel) framework provided by Torch for training.
For OLMo1, we used the official open-source computational code, which also incorporates the FSDP
framework alongside Flash Attention for acceleration.

Continual Pre-training We follow Cheng et al.
(2024b) to conduct continual pre-training on Bio, Fi-
nance, and Math domains. Specifically, PubMed Ab-
stracts from the Pile are utilized as the pre-training cor-
pora for the biomedicine domain. For the finance domain,
financial news data covering over 7,000 stocks from May
2022 to May 2023 is collected using the FinGPT frame-
work. We continue pre-training OLMo-1B and LLaMA-
3-8B on each domain. For implementation, we utilized
the official training framework for OLMo-1B, leveraging
Fully Sharded Data Parallel (FSDP) for continual pre-
training. For LLaMA, we adopted the LLaMA-Factory
framework to carry out the continual pretraining process.
Our experiments was primarily conducted on OLMo-1B and Llama-3-8B models, with Llama-3-8B
utilizing LoRA (Low-Rank Adaptation) for parameter-efficient fine-tuning. The data and evaluation
are given in this repo2. We conducted the continual pretraining on a total of 1B tokens.

Figure 7: GPT-2 perplexity (PPL) on
validation sets, trained from scratch.

used

Supervised Fine-tuning We
the Llama-
Factory (Zheng et al., 2024) framework to fine-tune
Llama-3-8B. As for general
instruction tuning tasks,
we adopt instruction tuning datasets from (Xia et al.,
2024) 3,
including CoT (Wei et al., 2022) , FLAN
V2 (Longpre et al., 2023), and Open Assistant 1 (Kopf
et al., 2023). As for code-related reasoning tasks, we
utilize OSS-Instruct-75K 4 and Evol-Instruct-110K 5.
These datasets provide sufficient diversity for verification
on fine-tuning.

D.2 EVALUATION

Pre-training We use PPL and downstream tasks to con-
duct analysis and performance test. As for PPL, it stands
for perplexity, a commonly used metric in NLP to evalu-
ate the quality of language models. It measures how well
a probabilistic model predicts a given dataset, with lower
values indicating better performance. Formally, the per-
plexity of a language model is calculated as:

Figure 8: OLMo-237M pretraining with
mixed human and synthetic data pro-
portions. We pretrain the OLMo-237M
model using a mixture of human data
(Dolma (Soldaini et al., 2024)) and
synthetic data (Cosmopedia (Ben Allal
et al., 2024)).

Alternatively, it can also be expressed as:

PPL = 2− 1

N

(cid:80)N

i=1 log2 P (xi)

(cid:32)

PPL = exp

−

(cid:33)

log P (xi)

1
N

N
(cid:88)

i=1

1https://github.com/allenai/OLMo
2https://github.com/microsoft/LMOps/tree/main/adaptllm
3https://huggingface.co/datasets/princeton-nlp/less_data
4https://huggingface.co/datasets/ise-uiuc/Magicoder-OSS-Instruct-75K
5https://huggingface.co/datasets/ise-uiuc/Magicoder-Evol-Instruct-110K

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Where N is the number of tokens in the dataset, and P (xi) is the predicted probability of the i-th
token. Perplexity essentially represents the exponential of the average negative log-likelihood of the
predicted tokens, indicating how “perplexed” the model is when making predictions.

As for downstream tasks, we use general understanding tasks in (Maini et al., 2024) to analyze model
collapse in Table 5 and general test tasks in (Cheng et al., 2024a) to test our methods in Table 2. All
downstream tasks we used can be found in (Gao et al., 2024)6.

Continual Pre-training We use the domain specific task in (Cheng et al., 2024b) to test domain
CPT performance. The test data and code can be found in here7.

Supervised Fine-tuning We utilize the general downstream tasks from (Cheng et al., 2024a) to
evaluate instruction-tuning performance and reasoning tasks to assess reasoning capabilities. All
downstream tasks we used can be found in (Gao et al., 2024)8.

Table 4: Performance impact of different p in BioMed.

Criteria

PubMedqa MQP

RCT USMLE ChemProt

Avg

Resampled Tokens p ≥ 0.99
Resampled Tokens p ≥ 0.999
Resampled Tokens p ≤ 0.1
Resampled Tokens p ≤ 0.01
Resampled Tokens p ≤ 0.001

64.5
63.6
62.4
65.4
64.2

55.73
55.4
51.47
54.91
56.39

30.95
29.09
25.6
28.19
35.0

27.65
28.12
29.14
27.80
27.80

14.6
16.2
10.0
11.0
12.4

38.686
38.482
35.722
37.46
39.158

E ABLATION STUDIES ON THE HYPER-PARAMETER p

We supplement 4 experiments on hyper-parameter p, including: (1) ablation studies of values , (2)
token percentage statistics, (3) comparisons of sampling strategies, and (4) an ablation study on
sampling size.

As Table 4 shows different p influences on BioMed, different values lead to fluctuations in data
performance. The Table 9 presents the distribution percentages across different probability value
ranges. As mentioned above, we need to refine the data while preserving mainly source distribution.
As shown in Figure 6, a larger p indicates fewer tokens will be resampled, while a smaller p results in
more tokens being resampled. Balancing performance and the preservation of data distribution, we
set p = 0.99 as threshold for our experiments. The Table 8 shows the results of different sampling
strategies. Specifically, to control variables, we set k = 8 for top-k sampling and p = 0.99 for
top-p sampling. We use reject sampling implementation in Liu et al. (2023). The results of reject
sampling, top-p, and top-k are comparable. However, top-p involves a dynamic sampling range, and
reject sampling requires multiple rounds of computation, leading to increased overhead. Considering
computational efficiency, we chose top-k for sampling. This aligns with our original objective of
maintaining minimal computational overhead. This aligns with our initial objective of minimizing
computational overhead as much as possible. The Table 7 shows the ablation study on sampling
size of top-k. The improvement achieved with larger values is relatively small. Therefore, we chose
k = 8 in our experiments.

F DISCUSSION

F.1 WHAT IS THE DIFFERENCE BETWEEN NON-ITERATIVE AND ITERATIVE MODEL

COLLAPSE?

We define ’non-iterative model collapse’ as the performance degradation caused by directly mixing
general synthetic data with real data, without iterative training. Theoretically, without additional reg-
ularization constraints to guide data generation, the variance of the model-generated data gradually

6https://github.com/EleutherAI/lm-evaluation-harness
7https://github.com/microsoft/LMOps/tree/main/adaptllm
8https://github.com/EleutherAI/lm-evaluation-harness

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Under review as a conference paper at ICLR 2025

Table 5: Comparison of human and synthetic data performance across downstream tasks in (Maini
et al., 2024).

TruthfulQA LogiQA Wino.

PIQA ARC-E BoolQ OBQA

Avg

Human Data
25% Synthetic Data
50% Synthetic Data
75% Synthetic Data
Synthetic Data

32.68
27.91
30.84
29.5
28.89

23.03
21.37
22.58
22.65
22.58

51.3
50.12
52.41
49.8
49.72

64.42
63.93
63.33
63.44
63

44.4
43.94
44.02
44.53
46.3

60.98
62.29
62.14
61.56
54.53

15
15.4
16
17.2
16.8

41.69
40.71
41.62
41.24
40.26

Table 6: PPL evaluation results on 22 vaildation using the testing framework in (Maini et al., 2024).
The PPL increases as the proportion of purely synthetic data grows.

ArXiv

BookCorpus2

Books3

DM Mathematics

Enron Emails

EuroParl

FreeLaw

GitHub

Gutenberg (PG-19)

HackerNews

NIH ExPorter

Human Data
25% Synthetic Data
50% Synthetic Data
75% Synthetic Data
Synthetic Data

22.26
21.86
22.50
24.35
35.60

25.39
26.32
28.01
31.19
43.72

22.87
23.87
25.75
28.98
47.72

10.84
11.05
10.84
11.81
17.25

23.50
24.85
26.56
30.30
66.97

30.73
35.02
41.99
56.32
129.75

12.04
12.84
14.02
16.03
29.62

4.15
4.35
4.67
5.30
12.00

16.88
17.99
19.70
22.75
50.14

32.54
33.80
36.12
40.44
87.95

23.53
23.76
24.61
26.19
39.48

OpenSubtitles OpenWebText2 PhilPapers

Pile-CC

PubMed Abstracts PubMed Central

StackExchange Ubuntu IRC USPTO Backgrounds Wikipedia (en) YoutubeSubtitles

Avg

Human Data
25% Synthetic Data
50% Synthetic Data
75% Synthetic Data
Synthetic Data

28.08
29.25
31.00
34.18
57.83

25.77
26.94
28.76
32.04
53.94

33.56
34.63
37.48
42.39
78.18

26.78
27.83
29.36
32.17
54.69

18.97
19.55
20.51
22.33
34.82

15.49
15.38
15.89
16.92
23.87

10.81
11.03
11.54
12.55
20.47

20.86
22.32
23.53
26.54
51.78

19.32
19.58
20.51
22.21
37.24

24.31
25.88
27.57
30.68
46.12

21.54
22.63
24.91
28.98
65.49

21.37
22.31
23.90
27.03
49.30

decreases during this process. The diversity of the generated data diminishes over time, ultimately
leading to the collapse of the model itself.

From a setting perspective: The difference
between the two lies in their scope. Non-
iterative model collapse is not confined to train-
ing on self-generated data, which allows it to
uncover broader properties of synthetic data.
For instance, in our experiments, we train GPT-2 on the Cosmopedia dataset in a single generation,
which was generated by Mixtral-8x7B-Instruct-v0.1. In contrast, iterative model collapse focuses
on training the model over multiple generations using self-generated data.

Table 7: Ablation study on sampling size k for
top-k.

PubMedQA MedMCQA MedQA (4 options)

Sampling Size (k)

k = 8
k = 64

26.13
28.14

24.82
27.34

64.5
63.8

Table 8: Results of different sampling strategies.

From a property perspective: The non-
iterative model collapse emphasizes the gap be-
tween human data and general purely synthetic
data, particularly regarding distributional prop-
erties and n-gram features. In contrast, the iter-
ative model collapse illustrates the iterative evolution of the model, resembling a self-play process.
This process illustrates the gradual evolution of self-generated data. It does not involve an analysis
of the differences in nature between self-generated and real data.

Top-k
Top-p
Reject Sampling

PubMedQA MedMCQA MedQA (4 options)

Sampling Strategy

24.82
25.61
28.20

26.13
27.11
28.90

64.5
63.8
64.5

They both ultimately lead to model collapse, driven by the same underlying cause—synthetic data,
though they investigate different aspects of synthetic data.

The most common setting is training a model on a mixture of human and synthetic data, where the
synthetic data is not generated by the model itself, and its exact origin may be unknown. Moreover,
there are already numerous popular datasets, such as UltraChat and OpenOrca, that combine syn-
thetic and real data to improve training diversity and robustness. Therefore, studying synthetic data
in the context of non-iterative model collapse is more realistic.

F.2 WHAT IS COVERAGE COLLAPSE?

‘Coverage collapse’ refers to a phenomenon in which the distribution of synthetic data covers a sig-
nificantly narrower range of values compared to human data, even when the data sizes are identical.
For instance, as shown in Figure 3, the PPL range of synthetic data is limited to [0, 14], whereas
the PPL range of human data extends from [0, 100]. Despite this disparity, the overall coverage,
represented by the area under the distribution curves, remains the same. This significant distribution
gap is what we define as ‘coverage collapse.’

F.3 HOW DOES THE DSIR WORK?

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Under review as a conference paper at ICLR 2025

Probability Range

Table 9: Token distribution across different prob-
ability ranges in BioMed dataset.

DSIR (Xie et al., 2023) works by estimating
importance weights for each data sample to
measure its relevance to the target distribution.
This involves three main steps: first, we lever-
age n-gram models to estimate two distribu-
tions of human and synthetic data, qf eat and
pf eat, which represent the target and raw distri-
butions, respectively. We use them to compute
the likelihood ratio for each sample. Next, we
calculate the importance weight for each sam-
ple zi as wi = ˆpfeat(zi)
ˆqfeat(zi) . The weight wi quanti-
fies how well the sample aligns with the target
distribution. Finally, we perform importance-
weighted sampling without replacement to se-
lect examples, ensuring that the selected data is more representative of the target distribution.

388,626,330
90,716,809
60,477,872
49,278,266
42,558,503
40,318,546
41,438,924
44,798,424
58,238,944
303,543,988

0.0-0.1
0.1-0.2
0.2-0.3
0.3-0.4
0.4-0.5
0.5-0.6
0.6-0.7
0.7-0.8
0.8-0.9
0.9-1.0

34.7%
8.1%
5.4%
4.4%
3.8%
3.6%
3.7%
4.0%
5.2%
27.1%

Percentage Token Count

We use DSIR in our data analysis as it allows for principled and computationally efficient selection
of synthetic data points that align with the target distribution. Moreover, the importance weight
also reflects the alignment between the n-gram features of synthetic data and human data. Using
DSIR, we can analyze the differences between synthetic and human data across n-gram feature
distributions and data matching. As shown in Figure 4, it is challenging to select synthetic data that
matches human data characteristics under the significant distribution difference. To obtain high-
quality synthetic data, it is essential to focus on improving the data synthesis methods.

Method

Table 10: Comparison of different synthetic data methods.
Result

Data Type

Approach

Cosmopedia (Ben Allal et al., 2024)
Rephrasing the Web (Maini et al., 2024)

Pure synthetic
Semi-synthetic Using a prompt and source content to guide LLMs

Using a prompt to induce data from LLMs.

Reveal non-iterative model collapse.
Improve training performance.

ToEdit (Ours)

Semi-synthetic Using the distribution of source content estimated
by LLMs (single forward pass) to replace tokens.

Improve training performance.

to reformat source content.

F.4 WHY DOES THE OBSERVED PROBABILITY DISTRIBUTION EXHIBIT FILTERING

POTENTIAL?

From the perspective of information theory, we can analyze the filtering potential of the U-shape
distribution as follows: We utilize the U-shape distribution in Figure 6 to re-sample tokens in the
high-probability region, aiming to adjust the U-shaped distribution toward a uniform distribution. By
doing so, we can maximize the information entropy. According to information theory, maximizing
information entropy is achieved when the distribution is uniform.

Lemma 1: Let X be a discrete random variable with n possible outcomes. If the probability of each
outcome is uniform, i.e., P (xi) = 1
n for all i ∈ {1, 2, . . . , n}, the Shannon entropy is maximized,
given by:

H(X) = −

n
(cid:88)

i=1

1
n

log

1
n

= log n.

This represents the maximum uncertainty achievable, implying that the dataset carries the maximum
possible information content. Thus, the uniform distribution, which assigns equal probability to
all outcomes, possesses the maximum information entropy. To leverage this property, we utilize
the U-shape distribution to re-sample tokens in the high-probability region, adjusting the U-shaped
distribution toward a uniform distribution. By doing so, we can maximize the information entropy.

From the perspective of language model learning, our method emphasizes the importance of
poorly learned data. Specifically, we resample easy tokens and encourage the model to focus on
learning more challenging ones. Our method can enhance the learning of underrepresented data by
resampling high-probability tokens.

F.5 NON-AUTOREGRESSIVE TOKEN REPLACEMENT MAY COMPROMISE TEXT COHERENCE.

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Under review as a conference paper at ICLR 2025

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When designing data synthesis algorithms, we
must balance synthesis efficiency and effec-
tiveness, considering both autoregressive and
non-autoregressive approaches. Autoregres-
sive methods leverage the inherent capabilities
of language models to generate coherent text
sequentially.
In contrast, non-autoregressive
methods resample individual tokens based on
their probability distributions. Since data synthesis is a prerequisite for model training, we aim to
ensure that the cost of data synthesis does not exceed the cost of training itself.

Table 11: Percentage of tokens requiring edits in
the Natural-Instructions dataset. The total number
of tokens is 4,671,834.

Generation 1 (source) Generation 2 Generation 3

Tokens (p > 0.99)
Percentage

549,519
11.76%

584,103
12.5%

517,433
11.08%

Specifically, our ToEdit modifies data using the probability distribution in a single forward pass.
For instance, if the generated sequence length is 1024, the computational cost of autoregressive
methods would be 1024 times higher than ours. This efficiency advantage is why our method can
run effectively on GPUs like the 3090 or 4090 series.

However, this efficiency may come at the cost of coherence, as resampled tokens may not fit seam-
lessly into a given sentence. To address this issue, we introduce a hyperparameter, resampling prob-
ability p, to control the resampling threshold. We perform sampling in high-probability regions,
focusing on tokens that are relatively easier to predict. We manually verify and tune on a small
validation set before applying it across all experiments. In our experiments, we set p = 0.99.

Additionally, we supplement more experiments and discussion about hyper-parameter p. As Table 4
shows, different values of p influence BioMed performance, leading to fluctuations in data quality.
Table 9 presents the distribution percentages of the token probabilities across different value ranges.
We need to refine the data while primarily preserving the source distribution. A larger p indicates
fewer tokens will be resampled, while a smaller p results in more tokens being resampled. Balancing
performance and the preservation of data distribution, we set p = 0.99 as the threshold for our
experiments.

F.6 GRADUAL DECLINE IN EDITING

We present the percentage statistics of edited tokens in Table 11, demonstrating that the edited
tokens indeed exhibit a progressive decrease. Specifically, We observe that the percentage of edited
tokens (above the threshold p > 0.99) decreases as the generation number increases. Theoretically,
this is a process of distribution shifting. When tokens (p > 0.99) are resampled, randomness is
introduced. The sampling process can select tokens with lower probabilities. Then, tokens (p >
0.99) is replaced, leading to a reduction of edited tokens in subsequent generations. The Table 11
provides real-world evidences for this pattern of decay.

F.7 COMPARISON WITH PURE SYNTHETIC DATA AND REFORMAT METHODS

Specifically, both Rephrasing the Web (Maini et al., 2024) and our token-level editing aim to refine
data while preserving the original distribution, producing semi-synthetic data. In contrast, purely
synthetic data in Cosmopedia lacks the long-tail distribution and overly concentrates on n-gram
features. Ultimately, semi-synthetic data enhances training performance, whereas purely synthetic
data results in model collapse. Moreover, replacing a whole real sample with synthetic data can
damage the performance.

The primary distinction between Cosmopedia, Rephrasing the Web (Maini et al., 2024), and our
approach lies in how much of the original human data distribution is preserved. We provide a
detailed comparison of these synthetic methods in Table 10.

F.8 MUST WE ASSUME THE DATA IS 100% HUMAN-AUTHORED?

We do not need to assume that the data is 100% human authored; In experimental settings, some
datasets used in our experiments include partially synthetic data:

• Datasets used in continual pretraining (e.g., Biomed, Finance) include partially synthetic
data, which has been reformatted into a reading comprehension structure (Cheng et al.,
2024b).

23

Under review as a conference paper at ICLR 2025

• OSS-Instruct-75K and Evol-Instruct-110K also contain samples synthesized by ChatGPT.

In the theoretical framework, synthetic data is generated iteratively through an n-generation process.
(1) If the starting point is a real distribution, our method preserves most of the initial distribution
to generate higher-quality data. (2) If the starting point is a mixture of synthetic and real data, the
modifications are minimal, ensuring the original distribution remains largely unaffected. Therefore,
applying our method in any generation i, we can further avoid issues, such as reduced variance and
diminished diversity, which are key factors contributing to model collapse.

In other words, whether the current data is fully real or a mix of real and synthetic, using it as
anchor data to synthesize data, our method builds upon the current data distribution to achieve
improvements, rather than causing model collapse.

In summary, we aim to improve the data synthesis method, specifically focusing on how to obtain
higher-quality data from the existing datasets. We do not need to assume that the data at hand is
100% human-generated. Our algorithm is designed to minimize excessive distribution truncation of
the original data.

G POTENTIAL APPLICATIONS AND FUTURE WORK

Based on the above discussion, our approach can be applied to optimize the current data, even if it
is a mixture of real and synthetic data. From the findings and proposed method in our paper, we can
influence future research in the following aspects:

Potential applications of our work:
(1) Data optimizations. We can quickly modify and optimize
the current data, using a trained language model with a single forward pass. (2) Regularization in
the data synthesizing process. When synthetic data becomes excessive, we can introduce real data
as an anchor to balance the issues of excessive homogeneity and tail distribution cut-off in synthetic
data, thereby preventing mode collapse.

Lessons from our work: The key to improving the quality of synthetic data lies in balancing long-
tail distribution preservation and optimizing synthetic data approaches. In other words, we should
focus on two questions: how to generate more informative synthetic data and how to integrate
it with real data effectively. Building on this foundation, future improvements can focus on two
aspects: first, obtaining more information gain by designing more efficient generation mechanisms
to inject valuable information into the synthetic data; and second, optimizing methods to reduce
noise during the synthesis process. This approach ensures that synthetic data retains its authenticity
while enhancing its utility in practical tasks.

Figure 9: PPL distribution of human and synthetic data estimated by StabLM-Zephyr-3B. This
indicates that different prior distributions yielded the same result, which is consistent with Figure 3.
The synthetic data lacks a long tail and is concentrated within a narrow portion of the distribution.

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A. Human Data PPL Distribution Estimated by StableLM-3BB. Synthetic Data PPL Distribution Estimated by StableLM-3BUnder review as a conference paper at ICLR 2025

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Figure 10: The top 40 bi-grams from separately sampled 1M subsets of Dolma, Cosmopedia, and
DSIR-selected datasets.

Figure 11: The top 64 bi-grams from separately sampled 1M subsets of Dolma, Cosmopedia, and
DSIR-selected datasets.

Table 12: PPL results of GPT-2 124M pretraining on pure Human or Synthetic data.

Data Type

Tokens Size

Epochs

Wikitext-103
RedPajama
Falcon-RefinedWeb
c4-en
mc4-en

Human Data (Dolma)

Synthetic Data (Cosmopedia)

8.4B

16.8B 25.2B 33.6B 42B

8.4B

16.8B

25.2B

33.6B

42B

1

2

43.62
40.18
54.85
45.87
61.00

38.57
35.84
49.10
41.00
54.44

3

36.11
33.97
46.93
39.10
52.11

4

34.89
32.74
45.43
37.95
50.38

5

1

2

3

4

5

34.55
32.34
44.90
37.56
49.74

169.38
116.37
146.97
128.25
171.44

147.73
103.25
132.60
114.41
153.70

135.23
99.27
127.68
109.73
150.28

131.78
96.81
124.32
107.53
145.44

128.05
96.03
122.69
106.55
144.99

25

Under review as a conference paper at ICLR 2025

Figure 12: Density sampling response values. This result further confirms the issue of feature
collapse in synthetic data.

Figure 13: PPL results for OLMo-237M pretraining on selected synthetic data and data mixtures.(bar
plot version for Figure 5B)

Table 13: PPL results of GPT-2 124M pretraining on mixture of human and synthetic data.

Synthetic Data Ratio

25%

50%

75%

Tokens Size

8.4B

16.8B 25.2B 33.6B 42B

8.4B

16.8B 25.2B 33.6B 42B

8.4B

16.8B 25.2B 33.6B 42B

Epochs

Wikitext-103
RedPajama
Falcon-RefinedWeb
c4-en
mc4-en

1

2

45.97
42.28
56.40
48.15
62.46

39.87
37.62
50.62
43.14
56.80

3

37.65
35.72
48.26
40.98
54.35

4

36.91
34.66
47.13
39.91
53.06

5

1

2

36.32
34.24
46.66
39.41
52.71

50.29
46.89
61.06
51.79
70.43

43.15
41.42
54.34
46.06
62.48

3

40.46
39.37
51.72
43.90
59.61

4

39.43
38.21
50.39
42.73
57.66

5

1

2

38.65
37.72
49.87
42.23
57.07

58.66
55.72
69.32
58.60
80.37

48.75
49.26
61.50
52.22
71.77

3

45.20
46.27
58.28
49.26
67.90

4

43.42
44.81
56.77
47.87
65.31

5

42.95
44.30
56.19
47.27
64.82

Table 14: PPL results of OLMo-237M pretraining on mixture of human and synthetic data.

Synthetic Data Ratio

0%

Unique Tokens
Training Tokens
Epochs

Wikitext-103
RedPajama
Falcon-RefinedWeb
c4-en
mc4-en
M2D2-Wiki
M2D2-S2ORC

8.4B
8.4B
1

187.36
175.38
165.17
123.88
208.91
88.24
86.15

25%

8.4B
8.4B
1

185.5
183.93
166.69
127.68
208.94
87.34
81.53

50%

8.4B
8.4B
1

260.08
236.33
199.68
147.69
263.35
107.77
97.61

75%

8.4B
8.4B
1

367.46
301.09
245.15
174.48
324.91
114.19
100.64

100%

DSIR (1M) DSIR (10M) Edu Classifier (1M) Edu Classifier (10M)

PPL Filter (1M)

PPL Filter (10M) Density Sampling (1M) Density Sampling (10M)

8.4B
8.4B
1

1605.73
907.91
523.93
410.19
800.40
189.06
204.22

0.6B
8.4B
14

1309.53
649.36
573.61
457.96
861.01
234.45
170.78

8.4B
8.4B
1

1757.03
916.51
510.96
404.63
823.12
183.17
496.40

0.75B
10.5B
14

1111.29
811.14
522.97
415.88
769.86
161.58
145.27

7.4B
7.4B
1

1612.95
1104.75
612.72
487.97
955.70
206.45
201.52

0.97B
13.68B
14

738.36
376.36
344.82
286.95
476.81
130.43
117.44

9B
9B
1

1193.25
645.82
449.86
367.44
662.00
162.08
163.38

0.6B
8.9B
14

1188.40
789.67
501.99
414.55
740.75
167.20
131.22

7.1B
7.1B
1

1753.89
896.18
560.92
457.71
844.53
205.50
192.97

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1364
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1366
1367
1368

1369
1370
1371
1372
1373
1374

1375
1376
1377
1378
1379
1380

1381
1382
1383
1384
1385
1386

1387
1388
1389
1390
1391

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1396
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1404
1405

1406
1407
1408
1409
1410
1411

1412
1413
1414
1415
1416
1417

1418
1419
1420
1421
1422

1423
1424
1425
1426
1427
1428

1429
1430
1431
1432
1433
1434

1435
1436
1437
1438
1439
1440

1441
1442
1443
1444
1445

1446
1447
1448
1449
1450
1451

1452
1453
1454
1455
1456
1457

Table 15: Dolma dataset statistics (v1.6), quoted from source (Soldaini et al., 2024).

Source

Doc Type

UTF-8 bytes (GB) Documents (millions) Unicode words (billions) Llama tokens (billions)

Common Crawl
The Stack
C4
Reddit
PeS2o
Project Gutenberg
Wikipedia, Wikibooks

web pages
code
web pages
social media
STEM papers
books
encyclopedic

Total

9,022
1,043
790
339
268
20.4
16.2

11,519

3,370
210
364
377
38.8
0.056
6.2

4,367

1,775
260
153
72
50
4.0
3.7

2,318

2,281
411
198
89
70
6.0
4.3

3,059

27

