未知缩放维数下的维度自适应批处理Lipschitz缩小算法
Dimension-Adaptive Batched Lipschitz Narrowing Without Knowing the Zooming Dimension
浏览论文内容
中文总结 AI 辅助
本研究提出Count-Adaptive BLiN算法，消除了A-BLiN算法对缩放维数$d_z$的依赖，在$d_z$未知时仍能达到最优批处理复杂度$\bigTheta_d(\text{log log } T)$，并保证了悔值性能。
中文摘要 AI 辅助
A-BLiN中适当组合的边长（ACE）序列依赖于缩放维数$d_z$，本研究消除了这种依赖关系。下一个边长由前一次消除后幸存的立方体数量选择，所得的Count-Adaptive BLiN算法不使用$d_z$或缩放常数$C_z$，却能达到$\tilde{\bigO}_d(T^{(d_z+1)/(d_z+2)})$的悔值，且批处理复杂度为$\bigO_d(\text{log log } T)$。结合原论文定理10的自适应网格下界，当$d_z$未知时，最优批处理复杂度仍为$\bigTheta_d(\text{log log } T)$。
英文摘要
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 $\widetilde{\mathcal O}_d(T^{(d_z+1)/(d_z+2)})$ 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 $Θ_d(\log\log T)$ when $d_z$ is unknown.
