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 $A^i\in\{-M, -M+1, \ldots, M\}^{n\times d}$ and $b^i\in\{-M, -M+1, \ldots, M\}^n$ and the coordinator would like to find a $(1+\epsilon)$-approximate solution to $\min_{x\in\mathbb{R}^n} \|(\sum_i A^i)x - (\sum_i b^i)\|_p$. Here $M \leq \mathrm{poly}(nd)$ 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 $\tilde{\Theta}(sd^2 + sd/\epsilon)$ bits. For $p \in (1,2)$,we obtain an $\tilde{O}(sd^2/\epsilon + sd/\mathrm{poly}(\epsilon))$ upper bound. Notably, for $d$ sufficiently large, our leading order term only depends linearly on $1/\epsilon$ rather than quadratically. We also show communication lower bounds of $\Omega(sd^2 + sd/\epsilon^2)$ for $p\in (0,1]$ and $\Omega(sd^2 + sd/\epsilon)$ for $p\in (1,2]$. Our bounds considerably improve previous bounds due to (Woodruff et al. COLT, 2013) and (Vempala et al., SODA, 2020).

翻译：我们研究了在协调者模型中分布式$\ell_p$-回归问题的随机通信复杂度，其中$p\in (0,2]$。该问题涉及一个协调者和$s$个服务器：第$i$个服务器接收$A^i\in\{-M, -M+1, \ldots, M\}^{n\times d}$和$b^i\in\{-M, -M+1, \ldots, M\}^n$，协调者需要找到$\min_{x\in\mathbb{R}^n} \|(\sum_i A^i)x - (\sum_i b^i)\|_p$的一个$(1+\epsilon)$近似解。为方便起见，这里$M \leq \mathrm{poly}(nd)$。该模型下数据以加性方式分布在各服务器上，通常称为任意划分模型。针对该问题，我们获得了显著改进的界。对于$p=2$（即最小二乘回归），我们首次给出了最优界$\tilde{\Theta}(sd^2 + sd/\epsilon)$比特。当$p \in (1,2)$时，我们得到上界$\tilde{O}(sd^2/\epsilon + sd/\mathrm{poly}(\epsilon))$。值得注意的是，对于足够大的$d$，我们的主导项仅与$1/\epsilon$呈线性依赖而非二次关系。此外，我们证明了通信下界：当$p\in (0,1]$时为$\Omega(sd^2 + sd/\epsilon^2)$，当$p\in (1,2]$时为$\Omega(sd^2 + sd/\epsilon)$。我们的结果显著改进了此前(Woodruff等, COLT 2013)和(Vempala等, SODA 2020)的界。