Sharper Bounds for $\ell_p$ Sensitivity Sampling

In large scale machine learning, random sampling is a popular way to approximate datasets by a small representative subset of examples. In particular, sensitivity sampling is an intensely studied technique which provides provable guarantees on the quality of approximation, while reducing the number of examples to the product of the VC dimension $d$ and the total sensitivity $\mathfrak{S}$ in remarkably general settings. However, guarantees going beyond this general bound of $\mathfrak{S} d$ are known in perhaps only one setting, for $\ell_2$ subspace embeddings, despite intense study of sensitivity sampling in prior work. In this work, we show the first bounds for sensitivity sampling for $\ell_p$ subspace embeddings for $p\neq 2$ that improve over the general $\mathfrak{S} d$ bound, achieving a bound of roughly $\mathfrak{S}^{2/p}$ for $1\leq p<2$ and $\mathfrak{S}^{2-2/p}$ for $2root leverage score sampling algorithm achieves a bound of roughly $d$ for $1\leq p<2$, and that a combination of leverage score and sensitivity sampling achieves an improved bound of roughly $d^{2/p}\mathfrak{S}^{2-4/p}$ for $2 Cite this Paper BibTeX @InProceedings{pmlr-v202-woodruff23a, title = {Sharper Bounds for $\ell_p$ Sensitivity Sampling}, author = {Woodruff, David and Yasuda, Taisuke}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {37238--37272}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/woodruff23a/woodruff23a.pdf}, url = {https://proceedings.mlr.press/v202/woodruff23a.html}, abstract = {In large scale machine learning, random sampling is a popular way to approximate datasets by a small representative subset of examples. In particular, sensitivity sampling is an intensely studied t