Exact minimax regret rate and removal of logarithmic factors

Determine the exact minimax regret rate for online maximum-cardinality matching in growing trees with an unknown constant parameter in the uniform–preferential attachment family, and establish whether the logarithmic factors in the current $O(\sqrt n\,\log^2 n)$ upper bound can be removed.

Background

For uniform–preferential attachment with an unknown fixed parameter, the geometric-update policy achieves a uniform expected regret upper bound of $O(\sqrt n\,\log^2 n)$. The paper defines the minimax oracle regret over policies that know the horizon but not the attachment parameter and shows that this quantity is at most $O(\sqrt n\,\log^2 n)$.

No matching lower bound of order $\sqrt n$ is proved. Consequently, the exact minimax rate is unresolved, and it is also unknown whether the logarithmic factors arise from the analysis or are intrinsic to the learning problem. The paper notes that sharper control of near-zero decision margins might improve the existing calibration and learning guarantees.