Sharper Bounds for ℓp 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 S in remarkably general settings. However, guarantees going beyond this general bound of Sd are known in perhaps only one setting, for ℓ2 subspace embeddings, despite intense study of sensitivity sampling in prior work. In this work, we show the first bounds for sensitivity sampling for ℓp subspace embeddings for p > 2 that improve over the general Sd bound, achieving a bound of roughly S 2-2/p for 2 < p < ∞. Furthermore, our techniques yield further new results in the study of sampling algorithms, showing that the root leverage score sampling algorithm achieves a bound of roughly d for 1 ≤ p < 2, and that a combination of leverage score and sensitivity sampling achieves an improved bound of roughly d 2/p S 2-4/p for 2 < p < ∞. Our sensitivity sampling results yield the best known sample complexity for a wide class of structured matrices that have small ℓp sensitivity. * Our sensitivity sampling results for p ∈ [1, 2) are subsumed by an earlier sharper result of Chen-Derezinski [CD21], which shows that the sensitivity scores upper bound Lewis weights up to a factor of d 1-p/2 , which implies a sampling complexity bound of Õ(ε -2 d 1-p/2 S). We refer to Section 1.1 as well as the recent work of [MO23] for more details. Question 1.4. How many samples are necessary for sensitivity sampling to output an ℓ p subspace embedding (2)? 3 We write Ax for the ith entry of the vector Ax ∈ R n . 4 We write Õ(f ) to denote f poly log f .