ℓ p -Regression in the Arbitrary Partition Model of Communication.

We consider the randomized communication complexity of the distributed ${\ell }_p$ -regression problem in the coordinator model, for $p\in (0,2]$ . In this problem, there is a coordinator and $s$ servers. The $i$ -th server receives $A^i\in {\{-M,-M+1,\dots ,M\}}^{n\times d}$ and $b^i\in {\{-M,-M+1,\dots ,M\}}^n$ and the coordinator would like to find a $(1+\epsilon )$ -approximate solution to ${{\text{min}}_{x\in {\text{R}}^n}\Vert \left({\sum }_i{A}^i\right)x-\left(\sum _i{b}^i\right)\Vert }_p$ . Here $M\le$ poly(nd) for convenience. This model, where the data is additively shared across servers, is commonly referred to as the arbitrary partition model. We obtain significantly improved bounds for this problem. For $p=2$ , i.e., least squares regression, we give the first optimal bound of $\widetilde{\text{Θ}}\left(s{d}^2+sd/ϵ\right)$ ) bits. For $p\in (1,2)$ , we obtain an $\tilde{O}\left(s{d}^2/\epsilon +sd/\text{poly}\left(\epsilon \right)\right)$ upper bound. Notably, for $d$ sufficiently large, our leading order term only depends linearly on $1/ϵ$ rather than quadratically. We also show communication lower bounds of $\text{Ω}\left(s{d}^2+sd/{\epsilon }^2\right)$ for $p\in (0,1]$ and $\text{Ω}\left(s{d}^2+sd/\epsilon \right)$ for $p\in (1,2]$ . Our bounds considerably improve previous bounds due to (Woodruff et al. COLT, 2013) and (Vempala et al., SODA, 2020).