Fixed-query logarithmic-power gap

Determine whether a fixed-query method can achieve a stochastic term with logarithmic power in the range $1\le p<3$, thereby improving on the $O(\sigma^2\log^3 N/N)$ stochastic terms of RAIN and RRSEG while respecting the lower-bound exclusion of every power $p<1$.

Background

The lower-bound corollary proves that no fixed-query method can uniformly guarantee an expected squared residual of order $O(L^2D^2/N^2+\sigma^2\log^pN/N)$ for any $p<1$. RRSEG attains the optimal deterministic $O(1/N^2)$ term but has stochastic logarithmic power $p=3$, while RAIN has the same logarithmic power and uses an increasing number of oracle queries per iteration. Consequently, the interval between the lower-bound threshold and the currently achieved upper bound remains unresolved.