Mini-Batch Kernel $k$-means

Abstract

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 $\tilde{O}(k{b}^2)$ 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 ${\hat{I}}^3$ bounds the norm of points in feature space and $\hat{I}\mu$ 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 ${\hat{I}}^3=1$. Taking $\hat{I}\mu =O(1)$ and $b=\hat{I}˜(\log n)$, the algorithm terminates in $O(1)$ iterations, with each iteration running in $\tilde{O}(k)$ time.

arXiv admin note: text overlap with arXiv:2304.00419