A Nearly Tight Analysis of Greedy k-means++
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The famous k-means++ algorithm of Arthur and Vassilvitskii [SODA 2007] is the most popular way of solving the k-means problem in practice. The algorithm is very simple: it samples the first center uniformly at random and each of the following k-1 centers is then always sampled proportional to its squared distance to the closest center so far. Afterward, Lloyd's iterative algorithm is run. The k-means++ algorithm is known to return a Θ(log k) approximate solution in expectation. In their seminal work, Arthur and Vassilvitskii [SODA 2007] asked about the guarantees for its following greedy variant: in every step, we sample ℓ candidate centers instead of one and then pick the one that minimizes the new cost. This is also how k-means++ is implemented in e.g. the popular Scikit-learn library [Pedregosa et al.; JMLR 2011]. We present nearly matching lower and upper bounds for the greedy k-means++: We prove that it is an O(ℓ^3 log^3 k)-approximation algorithm. On the other hand, we prove a lower bound of Ω(ℓ^3 log^3 k / log^2(ℓlog k)). Previously, only an Ω(ℓlog k) lower bound was known [Bhattacharya, Eube, Röglin, Schmidt; ESA 2020] and there was no known upper bound.
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