# Towards Better Statistical Understanding of Watermarking LLMs

arXiv: https://arxiv.org/abs/2403.13027 · HTML: https://mlsanity.com/paper/2403.13027/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Placed in Watermarking generated content and models, one of 12 directions in LLM safety and security.
- Builds on 22 of its 49 references that are in the map.
- First posted on arXiv 19 Mar 2024.

## Placed in

- LLM safety and security / [Watermarking generated content and models](https://mlsanity.com/safety/watermarking-generated-content-and-models/)

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Robust Speech Recognition via Large-Scale Weak Supervision](https://mlsanity.com/paper/2212.04356/) (2022)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [HuggingFace's Transformers: State-of-the-art Natural Language Processing](https://mlsanity.com/paper/1910.03771/) (2019)
- [DetectGPT: Zero-Shot Machine-Generated Text Detection using Probability Curvature](https://mlsanity.com/paper/2301.11305/) (2023)
- [ELI5: Long Form Question Answering](https://mlsanity.com/paper/1907.09190/) (2019)
- [Defending Against Neural Fake News](https://mlsanity.com/paper/1905.12616/) (2019)
- [Release Strategies and the Social Impacts of Language Models](https://mlsanity.com/paper/1908.09203/) (2019)
- [GLTR: Statistical Detection and Visualization of Generated Text](https://mlsanity.com/paper/1906.04043/) (2019)
- [Can AI-Generated Text be Reliably Detected?](https://mlsanity.com/paper/2303.11156/) (2023)
- [Robust Distortion-free Watermarks for Language Models](https://mlsanity.com/paper/2307.15593/) (2023)
- [Provable Robust Watermarking for AI-Generated Text](https://mlsanity.com/paper/2306.17439/) (2023)
- [Automatic Detection of Generated Text is Easiest when Humans are Fooled](https://mlsanity.com/paper/1911.00650/) (2019)
- [GPT detectors are biased against non-native English writers](https://mlsanity.com/paper/2304.02819/) (2023)
- [Online Convex Optimization with Time-Varying Constraints](https://mlsanity.com/paper/1702.04783/) (2017)
- [Fast Algorithms for Online Stochastic Convex Programming](https://mlsanity.com/paper/1410.7596/) (2014)
- [Simple and Fast Algorithm for Binary Integer and Online Linear Programming](https://mlsanity.com/paper/2003.02513/) (2020)

## Built on by (most cited first, 16 of 16)

- [Watermarking Techniques for Large Language Models: A Survey](https://mlsanity.com/paper/2409.00089/) (2024)
- [Theoretically Grounded Framework for LLM Watermarking: A Distribution-Adaptive Approach](https://mlsanity.com/paper/2410.02890/) (2024)
- [Debiasing Watermarks for Large Language Models via Maximal Coupling](https://mlsanity.com/paper/2411.11203/) (2024)
- [Watermarking Language Models for Many Adaptive Users](https://mlsanity.com/paper/2405.11109/) (2024)
- [WAPITI: A Watermark for Finetuned Open-Source LLMs](https://mlsanity.com/paper/2410.06467/) (2024)
- [Recursive Learning Without Collapse: A Weighting-Based Stabilization Framework](https://mlsanity.com/paper/2502.18049/) (2025)
- [SAEMark: Steering Personalized Multilingual LLM Watermarks with Sparse Autoencoders](https://mlsanity.com/paper/2508.08211/) (2025)
- [PASA: A Principled Embedding-Space Watermarking Approach for LLM-Generated Text under Semantic-Invariant Attacks](https://mlsanity.com/paper/2605.10977/) (2026)
- [Segmenting Watermarked Texts From Language Models](https://mlsanity.com/paper/2410.20670/) (2024)
- [A note on conditional PAC-efficient reasoning in large language model routing](https://mlsanity.com/paper/2512.03057/) (2025)
- [A Statistical Hypothesis Testing Framework for Data Misappropriation Detection in Large Language Models](https://mlsanity.com/paper/2501.02441/) (2025)
- [Beyond Heuristic Tuning: Power-Calibrated LLM Watermarking](https://mlsanity.com/paper/2607.05694/) (2026)
- [Watermark Forensics for Generative Models: An Information-Theoretic Perspective](https://mlsanity.com/paper/2607.13003/) (2026)
- [Adaptive Testing for Segmenting Watermarked Texts From Language Models](https://mlsanity.com/paper/2511.06645/) (2025)
- [Flip, Don't Shuffle: Watermarking LLMs at the Speed of Inference](https://mlsanity.com/paper/2609.03844/) (2026)
- [textGrain watermarking is harder to detect: can sum-tests still do it optimally?](https://mlsanity.com/paper/2610.17655/) (2026)

## Abstract (arXiv)

In this paper, we study the problem of watermarking large language models (LLMs). We consider the trade-off between model distortion and detection ability and formulate it as a constrained optimization problem based on the red-green list watermarking algorithm. We show that the optimal solution to the optimization problem enjoys a nice analytical property which provides a better understanding and inspires the algorithm design for the watermarking process. We develop an online dual gradient ascent watermarking algorithm in light of this optimization formulation and prove its asymptotic Pareto optimality between model distortion and detection ability. Such a result guarantees an averaged increased green list probability and henceforth detection ability explicitly (in contrast to previous results). Moreover, we provide a systematic discussion on the choice of the model distortion metrics for the watermarking problem. We justify our choice of KL divergence and present issues with the existing criteria of ``distortion-free'' and perplexity. Finally, we empirically evaluate our algorithms on extensive datasets against benchmark algorithms.