We study dynamic clustering problems from the perspective of online learning. We consider an online learning problem, called \textit{Dynamic $k$-Clustering}, in which $k$ centers are maintained in a metric space over time (centers may change positions) such as a dynamically changing set of $r$ clients is served in the best possible way. The connection cost at round $t$ is given by the \textit{$p$-norm} of the vector consisting of the distance of each client to its closest center at round $t$, for some $p\geq 1$ or $p = \infty$. We present a \textit{$\Theta\left( \min(k,r) \right)$-regret} polynomial-time online learning algorithm and show that, under some well-established computational complexity conjectures, \textit{constant-regret} cannot be achieved in polynomial-time. In addition to the efficient solution of Dynamic $k$-Clustering, our work contributes to the long line of research on combinatorial online learning.
翻译:我们从在线学习的角度研究动态集群问题。 我们考虑一个在线学习问题, 叫做\ textit{ Dynic $k$- clustering}, 即 $k$ 中心在一定空间里长期维持( 中心可能会改变位置), 例如动态变化的一套美元客户以最佳方式服务。 圆美元连接成本由矢量的 extit{ $p$- norm} 给出, 由每个客户到其最接近中心的距离构成, 以美元为圆数, 以美元计, 以1美元 或 $p=\ infty$ 。 我们提出\ textit{ $\ theta\ left (\ min( k,r)\ right) $- regret} 混合时间在线学习算法, 并表明, 在一些成熟的计算复杂度参数下, \ textititit{ conta- regretret} 无法在多元时间实现 。 除了动态 $k$- cloinstrualstal restial restialting restial restial resting rodustration.