# The Communication Complexity of Optimization

## TL;DR

This paper investigates the communication complexity of distributed optimization problems, providing tight bounds and demonstrating the limitations of sampling and sketching techniques, thus motivating the development of new optimization methods.

## Contribution

It offers the first tight bounds for communication complexity in distributed linear systems and optimization, highlighting the limitations of existing techniques and proposing new bounds for various problem settings.

## Key findings

- Communication complexity for linear systems is $ ilde{ heta}(d^2L + sd)$ (deterministic) and $ ilde{ heta}(sd^2L)$ (randomized).
- Sampling and sketching are suboptimal for distributed optimization in dependence on $d$ and $ ext{epsilon}$.
- New bounds for linear programming communication complexity, especially when coefficients are randomly perturbed.

## Abstract

We consider the communication complexity of a number of distributed optimization problems. We start with the problem of solving a linear system. Suppose there is a coordinator together with $s$ servers $P_1, \ldots, P_s$, the $i$-th of which holds a subset $A^{(i)} x = b^{(i)}$ of $n_i$ constraints of a linear system in $d$ variables, and the coordinator would like to output $x \in \mathbb{R}^d$ for which $A^{(i)} x = b^{(i)}$ for $i = 1, \ldots, s$. We assume each coefficient of each constraint is specified using $L$ bits. We first resolve the randomized and deterministic communication complexity in the point-to-point model of communication, showing it is $\tilde{\Theta}(d^2L + sd)$ and $\tilde{\Theta}(sd^2L)$, respectively. We obtain similar results for the blackboard model. When there is no solution to the linear system, a natural alternative is to find the solution minimizing the $\ell_p$ loss. While this problem has been studied, we give improved upper or lower bounds for every value of $p \ge 1$. One takeaway message is that sampling and sketching techniques, which are commonly used in earlier work on distributed optimization, are neither optimal in the dependence on $d$ nor on the dependence on the approximation $\epsilon$, thus motivating new techniques from optimization to solve these problems. Towards this end, we consider the communication complexity of optimization tasks which generalize linear systems. For linear programming, we first resolve the communication complexity when $d$ is constant, showing it is $\tilde{\Theta}(sL)$ in the point-to-point model. For general $d$ and in the point-to-point model, we show an $\tilde{O}(sd^3 L)$ upper bound and an $\tilde{\Omega}(d^2 L + sd)$ lower bound. We also show if one perturbs the coefficients randomly by numbers as small as $2^{-\Theta(L)}$, then the upper bound is $\tilde{O}(sd^2 L) + \textrm{poly}(dL)$.