Online $k$-means Clustering
Résumé
We study the problem of online clustering where a clustering algorithm has to assign a new point that arrives to one of $k$ clusters. The specific formulation we use is the $k$-means objective: At each time step the algorithm has to maintain a set of k candidate centers and the loss incurred is the squared distance between the new point and the closest center. The goal is to minimize regret with respect to the best solution to the $k$-means objective ($\mathcal{C}$) in hindsight. We show that provided the data lies in a bounded region, an implementation of the Multiplicative Weights Update Algorithm (MWUA) using a discretized grid achieves a regret bound of $\tilde{O}(\sqrt{T})$ in expectation. We also present an online-to-offline reduction that shows that an efficient no-regret online algorithm (despite being allowed to choose a different set of candidate centres at each round) implies an offline efficient algorithm for the $k$-means problem. In light of this hardness, we consider the slightly weaker requirement of comparing regret with respect to $(1 + \epsilon) \mathcal{C}$ and present a no-regret algorithm with runtime $O\left(T(\mathrm{poly}(log(T),k,d,1/\epsilon)^{k(d+O(1))}\right)$. Our algorithm is based on maintaining an incremental coreset and an adaptive variant of the MWUA. We show that na\"{i}ve online algorithms, such as \emph{Follow The Leader}, fail to produce sublinear regret in the worst case. We also report preliminary experiments with synthetic and real-world data.
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