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arXiv · 2312.07930

Towards Optimal Statistical Watermarking

Baihe Huang·Hanlin Zhu·Banghua Zhu·Kannan Ramchandran·Michael I. Jordan·Jason D. Lee·Jiantao Jiao
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Abstract

We study statistical watermarking by formulating it as a hypothesis testing problem, a general framework which subsumes all previous statistical watermarking methods. Key to our formulation is a coupling of the output tokens and the rejection region, realized by pseudo-random generators in practice, that allows non-trivial trade-offs between the Type I error and Type II error. We characterize the Uniformly Most Powerful (UMP) watermark in the general hypothesis testing setting and the minimax Type II error in the model-agnostic setting. In the common scenario where the output is a sequence of $n$ tokens, we establish nearly matching upper and lower bounds on the number of i.i.d. tokens required to guarantee small Type I and Type II errors. Our rate of $\Theta(h^{-1} \log (1/h))$ with respect to the average entropy per token $h$ highlights potentials for improvement from the rate of $h^{-2}$ in the previous works. Moreover, we formulate the robust watermarking problem where the user is allowed to perform a class of perturbations on the generated texts, and characterize the optimal Type II error of robust UMP tests via a linear programming problem. To the best of our knowledge, this is the first systematic statistical treatment on the watermarking problem with near-optimal rates in the i.i.d. setting, which might be of interest for future works.

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BibTeXRIS

Baihe Huang, Hanlin Zhu, Banghua Zhu, Kannan Ramchandran, Michael I. Jordan, Jason D. Lee, Jiantao Jiao. 2023-12-13. Towards Optimal Statistical Watermarking. https://arxiv.org/abs/2312.07930

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