Sample Complexities of Estimating Gumbel--Max Watermark Proportions with and without Reduction to Pivotal Statistics
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Mathematics > Statistics Theory
Title:Sample Complexities of Estimating Gumbel--Max Watermark Proportions with and without Reduction to Pivotal Statistics
Abstract:Watermarking promises a statistical trace of large language model (LLM) use, but real documents, after editing or paraphrasing, rarely arrive as purely human-written or purely machine-generated. This motivates a quantitative question beyond detection: what proportion of a document is generated from a pre-specified watermarked LLM? We study this watermark proportion estimation problem under the Gumbel--max watermarking mechanism, treating the next-token prediction (NTP) distributions as unknown and arbitrary nuisance parameters subject to a non-degeneracy condition. We compare two observation regimes: in the full observation regime, the estimator observes the pseudorandom vector and the selected token at each position; under the more popular setting of pivotal reduction, it observes only a scalar pivot, which follows a one-dimensional Uniform--Beta mixture distribution. Under pivotal reduction, we develop a Laguerre-polynomial estimator and establish a matching information-theoretic lower bound for the sample complexity. For full observation, we introduce an event-counting estimator and show a matching lower bound, yielding a substantially smaller sample complexity. As our results imply, although reducing to pivotal statistics is an elegant and widely used procedure, it is not always sample-efficient for estimating the proportion of watermarks.
| Subjects: | Statistics Theory (math.ST); Information Theory (cs.IT); Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2607.00224 [math.ST] |
| (or arXiv:2607.00224v1 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2607.00224
arXiv-issued DOI via DataCite (pending registration)
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