A Nearly-Optimal Bound for Fast Regression with $\ell_∞$ Guarantee
Authors: Zhao Song, Mingquan Ye, Junze Yin, Lichen Zhang

Given a matrix $A\in \mathbb{R}^{n\times d}$ and a vector $b\in \mathbb{R}^n$, we consider the regression problem with $\ell_\infty$ guarantees: finding a vector $x’\in \mathbb{R}^d$ such that $||x’-x^* ||_\infty \leq \frac{\epsilon}{\sqrt{d}}\cdot ||Ax^*-b||_2\cdot ||A^\dagger||$ with $x^*$ being the optimal solution to the regression $||Ax-b||_2$. One popular approach for solving $\ell_2$ regression problem is via sketching: picking a structured random matrix $S\in \mathbb{R}^{m\times n}$ with $m\ll n$ and $SA$ can be quickly computed, solve the “sketched” regression problem $x’=\mathrm{argmin} ||SAx-Sb||_2$. In this paper, we show that in order to obtain such $\ell_\infty$ guarantee for $\ell_2$ regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that, there exists a distribution of dense sketching matrices with $m=\epsilon^{-2}d\log^3(n/\delta)$ such that solving the sketched regression problem gives the $\ell_\infty$ guarantee, with probability at least $1-\delta$. Moreover, the matrix $SA$ can be computed in time $O(nd\log n)$. Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP’17], in which $m=\Omega(\epsilon^{-2}d^{1+\gamma})$ for $\gamma\in (0, 1)$ is required. Moreover, we develop a novel analytical framework for $\ell_\infty$ guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML’21]. Our analysis is much simpler and more general than that of [Price, Song and Woodruff, ICALP’17]. Leveraging this framework, we extend the $\ell_\infty$ guarantee regression result to dense sketching matrices for computing fast tensor product of vectors.