Dimension-Adaptive Batched Lipschitz Narrowing Without Knowing the Zooming Dimension

Abstract

The Appropriately Combined Edge-length (ACE) sequence in A-BLiN depends on the zooming dimension $d_z$. This note removes that dependence. The next edge length is selected from the number of cubes that survive the preceding elimination. The resulting Count-Adaptive BLiN algorithm does not use $d_z$ or the zooming constant $C_z$, yet it attains ${\tilde{\mathcal{O}}}_d(T^{(d_z+1)/(d_z+2)})$ regret with ${\mathcal{O}}_d(\log \log T)$ batches. Together with the adaptive-grid lower bound in Theorem 10 of the original paper, the optimal batch complexity remains ${\Theta }_d(\log \log T)$ when $d_z$ is unknown.