Precision dependence in dimension-independent population-only statistical-query bounds

Determine whether the residual precision gap between the population-only lower bound of order \(\Omega(\sqrt{M}/[\varepsilon\log(1/\varepsilon)])\) and the \(\Omega(\sqrt{M}/\varepsilon^2)\) behavior available under stronger accuracy requirements can be removed for unrestricted statistical queries in the dimension-independent population-only setting.

Background

The paper distinguishes population accuracy from sample accuracy and post hoc generalization. It notes that the classical lower bound of Nissim et al. assumes both population accuracy and post hoc generalization, whereas the cited population-only result has weaker precision dependence, leaving a residual gap in the dependence on $\varepsilon$.

The unresolved issue concerns whether the population-only comparison can achieve the stronger inverse-quadratic precision dependence without imposing the additional post hoc generalization condition. This is a question about the cited classical statistical-query setting rather than the Pauli-prediction theorem proved in the paper.