# LMD3: Language Model Data Density Dependence

arXiv: https://arxiv.org/abs/2405.06331 · HTML: https://mlsanity.com/paper/2405.06331/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 29 references that are in the map.
- First posted on arXiv 10 May 2024.

## Builds on (most cited first, 23 of 23)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Billion-scale similarity search with GPUs](https://mlsanity.com/paper/1702.08734/) (2017)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Quantifying Memorization Across Neural Language Models](https://mlsanity.com/paper/2202.07646/) (2022)
- [Aligning AI With Shared Human Values](https://mlsanity.com/paper/2008.02275/) (2020)
- [Large Language Models Struggle to Learn Long-Tail Knowledge](https://mlsanity.com/paper/2211.08411/) (2022)
- [Paraphrasing evades detectors of AI-generated text, but retrieval is an effective defense](https://mlsanity.com/paper/2303.13408/) (2023)
- [Improving Text Embeddings with Large Language Models](https://mlsanity.com/paper/2401.00368/) (2023)
- [TRAK: Attributing Model Behavior at Scale](https://mlsanity.com/paper/2303.14186/) (2023)
- [Rethinking Benchmark and Contamination for Language Models with Rephrased Samples](https://mlsanity.com/paper/2311.04850/) (2023)
- [Datamodels: Predicting Predictions from Training Data](https://mlsanity.com/paper/2202.00622/) (2022)
- [On the Reliability of Watermarks for Large Language Models](https://mlsanity.com/paper/2306.04634/) (2023)
- [Scaling Laws and Interpretability of Learning from Repeated Data](https://mlsanity.com/paper/2205.10487/) (2022)
- [D4: Improving LLM Pretraining via Document De-Duplication and Diversification](https://mlsanity.com/paper/2308.12284/) (2023)
- [Investigating Data Contamination for Pre-training Language Models](https://mlsanity.com/paper/2401.06059/) (2024)
- [Datasheet for the Pile](https://mlsanity.com/paper/2201.07311/) (2022)
- [Data Similarity is Not Enough to Explain Language Model Performance](https://mlsanity.com/paper/2311.09006/) (2023)
- [DEANN: Speeding up Kernel-Density Estimation using Approximate Nearest Neighbor Search](https://mlsanity.com/paper/2107.02736/) (2021)

## Built on by (most cited first, 7 of 7)

- [Generalization v.s. Memorization: Tracing Language Models' Capabilities Back to Pretraining Data](https://mlsanity.com/paper/2407.14985/) (2024)
- [Unleashing the Power of Data Tsunami: A Comprehensive Survey on Data Assessment and Selection for Instruction Tuning of Language Models](https://mlsanity.com/paper/2408.02085/) (2024)
- [How Much Can We Forget about Data Contamination?](https://mlsanity.com/paper/2410.03249/) (2024)
- [Toward Preference-aligned Large Language Models via Residual-based Model Steering](https://mlsanity.com/paper/2509.23982/) (2025)
- [Leveraging Open Knowledge for Advancing Task Expertise in Large Language Models](https://mlsanity.com/paper/2408.15915/) (2024)
- [Interrogating LLM design under a fair learning doctrine](https://mlsanity.com/paper/2502.16290/) (2025)
- [FictionalQA: A Dataset for Studying Memorization and Knowledge Acquisition](https://mlsanity.com/paper/2506.05639/) (2025)

## Abstract (arXiv)

We develop a methodology for analyzing language model task performance at the individual example level based on training data density estimation. Experiments with paraphrasing as a controlled intervention on finetuning data demonstrate that increasing the support in the training distribution for specific test queries results in a measurable increase in density, which is also a significant predictor of the performance increase caused by the intervention. Experiments with pretraining data demonstrate that we can explain a significant fraction of the variance in model perplexity via density measurements. We conclude that our framework can provide statistical evidence of the dependence of a target model's predictions on subsets of its training data, and can more generally be used to characterize the support (or lack thereof) in the training data for a given test task.