The proliferation of Large Language Models (LLMs) necessitates efficient mechanisms to distinguish machine-generated content from human text. While statistical watermarking has emerged as a promising solution, existing methods suffer from two critical limitations: the lack of a principled approach for selecting sampling distributions and the reliance on fixed-horizon hypothesis testing, which precludes valid early stopping. In this paper, we bridge this gap by developing the first e-value-based watermarking framework, Anchored E-Watermarking, that unifies optimal sampling with anytime-valid inference. Unlike traditional approaches where optional stopping invalidates Type-I error guarantees, our framework enables valid, anytime-inference by constructing a test supermartingale for the detection process. By leveraging an anchor distribution to approximate the target model, we characterize the optimal e-value with respect to the worst-case log-growth rate and derive the optimal expected stopping time. Our theoretical claims are substantiated by simulations and evaluations on established benchmarks, showing that our framework can significantly enhance sample efficiency, reducing the average token budget required for detection by 13-15% relative to state-of-the-art baselines.

Ninedifferentdebiasing algorithms (and a baseline) have been evaluated with this dataset using the popular ResNet-18 network[36]. CelebA contains faces of celebrities with several binary task labelsandtwoprotected labels(genderandyouth). Table 3showsthe prediction results from a biased binary classifier and its bias values using the seven metrics. Without losing generality, we consider "Sport" the positive class in the binary classifier. Following the DP formula in Appendix A.2, for the "Sport" class, thePPRfemale is 45.0% (90 /200), andPPRmale is65.0%

Participants are not allowed to modify the training procedure.