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 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(ℓ3 log3 k)-approximation algorithm. On the other hand, we prove a lower bound of Ω(ℓ3 log3 k/ log2 (ℓ log k)). Previously, only an Ω(ℓ log k) lower bound was known [Bhattacharya, Eube, Röglin, Schmidt; ESA 2020] and there was no known upper bound.
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(ℓ3 log3 k)-approximation algorithm. On the other hand, we prove a lower bound of Ω(ℓ3 log3 k/ log2 (ℓ log k)). Previously, only an Ω(ℓ log k) lower bound was known [Bhattacharya, Eube, Röglin, Schmidt; ESA 2020] and there was no known upper bound.
Original language | English |
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Title of host publication | Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) |
Editors | Nikhil Bansal, Viswanath Nagarajan |
Publisher | Society for Industrial and Applied Mathematics |
Publication date | 2023 |
Pages | 1012-1070 |
ISBN (Electronic) | 978-1-61197-755-4 |
DOIs | |
Publication status | Published - 2023 |
Event | 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA23) - Florence, Italy Duration: 22 Jan 2023 → 25 Jan 2023 |
Conference
Conference | 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA23) |
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Land | Italy |
By | Florence |
Periode | 22/01/2023 → 25/01/2023 |
ID: 382690810