Determine removable dependencies when model parameters are partially unknown

Determine which dependencies on the model parameters can be removed from non-adaptive 1-bit mean-estimation protocols when some parameters, including the moment and accuracy parameters, are only partially known to the learner.

Background

The paper allows the learner to choose the query rule using the known parameters k, lambda, sigma, the target accuracy, the confidence level, and the prescribed sample budget. It notes that related work discusses settings in which some of these parameters are only partially known, but does not characterize which parameter dependencies can be eliminated entirely. This leaves unresolved the design and optimality of 1-bit mean-estimation procedures with reduced prior parameter knowledge.

References

(Some discussion on partially unknown parameters is given in \citep{lau2026order}, but it remains open which dependencies can be removed entirely.)

— Non-Adaptive 1-Bit Mean Estimation: Minimax Rates and the Sample-Interval Tradeoff  (2609.08564 - Lau et al., 8 Sep 2026) in Section 1, Problem Setup, paragraph “Learner's goal”

As noted in~\citet{lau2026order}, several open problems still remain including settings where $(\sigma,)$ is unknown to the learner and multivariate settings.

— Non-Adaptive 1-Bit Mean Estimation: Minimax Rates and the Sample-Interval Tradeoff  (2609.08564 - Lau et al., 8 Sep 2026) in Section 6, Conclusion, final paragraph