Towards Optimal Statistical Watermarking

• URL: http://arxiv.org/abs/2312.07930v3
• Date: Tue, 6 Feb 2024 21:01:28 GMT
• Title: Towards Optimal Statistical Watermarking
• Authors: Baihe Huang and Hanlin Zhu and Banghua Zhu and Kannan Ramchandran and Michael I. Jordan and Jason D. Lee and Jiantao Jiao
• Abstract summary: We study statistical watermarking by formulating it as a hypothesis testing problem. Key to our formulation is a coupling of the output tokens and the rejection region. We characterize the Uniformly Most Powerful (UMP) watermark in the general hypothesis testing setting.
• Score: 95.46650092476372
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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