Mini-Batch Kernel $k$-means
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912064278102016 |
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| author | Jourdan, Ben Schwartzman, Gregory |
| author_facet | Jourdan, Ben Schwartzman, Gregory |
| contents | We present the first mini-batch kernel $k$-means algorithm, offering an order of magnitude improvement in running time compared to the full batch algorithm. A single iteration of our algorithm takes $\widetilde{O}(kb^2)$ time, significantly faster than the $O(n^2)$ time required by the full batch kernel $k$-means, where $n$ is the dataset size and $b$ is the batch size. Extensive experiments demonstrate that our algorithm consistently achieves a 10-100x speedup with minimal loss in quality, addressing the slow runtime that has limited kernel $k$-means adoption in practice. We further complement these results with a theoretical analysis under an early stopping condition, proving that with a batch size of $\widetildeΩ(\max \{γ^{4}, γ^{2}\} \cdot ε^{-2})$, the algorithm terminates in $O(γ^2/ε)$ iterations with high probability, where $γ$ bounds the norm of points in feature space and $ε$ is a termination threshold. Our analysis holds for any reasonable center initialization, and when using $k$-means++ initialization, the algorithm achieves an approximation ratio of $O(\log k)$ in expectation. For normalized kernels, such as Gaussian or Laplacian it holds that $γ=1$. Taking $ε= O(1)$ and $b=Θ(\log n)$, the algorithm terminates in $O(1)$ iterations, with each iteration running in $\widetilde{O}(k)$ time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mini-Batch Kernel $k$-means Jourdan, Ben Schwartzman, Gregory Machine Learning Artificial Intelligence Data Structures and Algorithms We present the first mini-batch kernel $k$-means algorithm, offering an order of magnitude improvement in running time compared to the full batch algorithm. A single iteration of our algorithm takes $\widetilde{O}(kb^2)$ time, significantly faster than the $O(n^2)$ time required by the full batch kernel $k$-means, where $n$ is the dataset size and $b$ is the batch size. Extensive experiments demonstrate that our algorithm consistently achieves a 10-100x speedup with minimal loss in quality, addressing the slow runtime that has limited kernel $k$-means adoption in practice. We further complement these results with a theoretical analysis under an early stopping condition, proving that with a batch size of $\widetildeΩ(\max \{γ^{4}, γ^{2}\} \cdot ε^{-2})$, the algorithm terminates in $O(γ^2/ε)$ iterations with high probability, where $γ$ bounds the norm of points in feature space and $ε$ is a termination threshold. Our analysis holds for any reasonable center initialization, and when using $k$-means++ initialization, the algorithm achieves an approximation ratio of $O(\log k)$ in expectation. For normalized kernels, such as Gaussian or Laplacian it holds that $γ=1$. Taking $ε= O(1)$ and $b=Θ(\log n)$, the algorithm terminates in $O(1)$ iterations, with each iteration running in $\widetilde{O}(k)$ time. |
| title | Mini-Batch Kernel $k$-means |
| topic | Machine Learning Artificial Intelligence Data Structures and Algorithms |
| url | https://arxiv.org/abs/2410.05902 |