  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\|$ where
$x^*=\arg\min_{x\in \mathbb{R}^d}\|Ax-b\|_2$. One popular approach for solving
such $\ell_2$ 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 $\arg\min_{x\in
\mathbb{R}^d} \|SAx-Sb\|_2$. 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 a
super-linear in $d$ rows, $m=\Omega(\epsilon^{-2}d^{1+\gamma})$ for
$\gamma=\Theta(\sqrt{\frac{\log\log n}{\log d}})$ is required. We also 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 arguably much simpler and more general
than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for
tensor product of vectors.

Data Structures and Algorithms;Machine Learning;Machine Learning