Coresets for Multiple $\ell_p$ Regression
This work addresses a bottleneck in scalable data analysis for multiple regression tasks, providing nearly optimal coreset constructions that are independent of the number of responses.
The paper tackles the problem of constructing coresets for multiple ℓ_p regression with m responses, achieving dimension-free sizes of Õ(ε^{-2}d) for p<2 and Õ(ε^{-p}d^{p/2}) for p>2, which approximate the objective up to (1±ε) relative error. It also applies these results to settle tight sample bounds for approximating ℓ_p Euclidean power means and to show that matrices have subsets of Õ(ε^{-1}k) rows for optimal ℓ_p subspace approximation.
A coreset of a dataset with $n$ examples and $d$ features is a weighted subset of examples that is sufficient for solving downstream data analytic tasks. Nearly optimal constructions of coresets for least squares and $\ell_p$ linear regression with a single response are known in prior work. However, for multiple $\ell_p$ regression where there can be $m$ responses, there are no known constructions with size sublinear in $m$. In this work, we construct coresets of size $\tilde O(\varepsilon^{-2}d)$ for $p<2$ and $\tilde O(\varepsilon^{-p}d^{p/2})$ for $p>2$ independently of $m$ (i.e., dimension-free) that approximate the multiple $\ell_p$ regression objective at every point in the domain up to $(1\pm\varepsilon)$ relative error. If we only need to preserve the minimizer subject to a subspace constraint, we improve these bounds by an $\varepsilon$ factor for all $p>1$. All of our bounds are nearly tight. We give two application of our results. First, we settle the number of uniform samples needed to approximate $\ell_p$ Euclidean power means up to a $(1+\varepsilon)$ factor, showing that $\tildeΘ(\varepsilon^{-2})$ samples for $p = 1$, $\tildeΘ(\varepsilon^{-1})$ samples for $1 < p < 2$, and $\tildeΘ(\varepsilon^{1-p})$ samples for $p>2$ is tight, answering a question of Cohen-Addad, Saulpic, and Schwiegelshohn. Second, we show that for $1<p<2$, every matrix has a subset of $\tilde O(\varepsilon^{-1}k)$ rows which spans a $(1+\varepsilon)$-approximately optimal $k$-dimensional subspace for $\ell_p$ subspace approximation, which is also nearly optimal.