Towards Tight Communication Lower Bounds for Distributed Optimisation

TL;DR

This paper establishes fundamental lower bounds on the communication required for distributed optimization, showing that a certain amount of bits must be exchanged to achieve a specified accuracy, and introduces a matching algorithm for quadratic objectives.

Contribution

It provides the first unconditional communication lower bounds for distributed optimization, applicable to both deterministic and randomized algorithms without structural assumptions.

Findings

Total communication lower bound of  Nd g d / N bits for psilon-approximate solutions.

The bounds are tight for quadratic objectives, with a new quantized gradient descent algorithm matching the lower bounds within constant factors.

Abstract

We consider a standard distributed optimisation setting where $N$ machines, each holding a $d$-dimensional function $f_i$, aim to jointly minimise the sum of the functions $\sum_{i = 1}^N f_i (x)$. This problem arises naturally in large-scale distributed optimisation, where a standard solution is to apply variants of (stochastic) gradient descent. We focus on the communication complexity of this problem: our main result provides the first fully unconditional bounds on total number of bits which need to be sent and received by the $N$ machines to solve this problem under point-to-point communication, within a given error-tolerance. Specifically, we show that $\Omega( Nd \log d / N\varepsilon)$ total bits need to be communicated between the machines to find an additive $\epsilon$-approximation to the minimum of $\sum_{i = 1}^N f_i (x)$. The result holds for both deterministic and randomised algorithms, and, importantly, requires no assumptions on the algorithm structure. The lower bound is tight under certain restrictions on parameter values, and is matched within constant factors for quadratic objectives by a new variant of quantised gradient descent, which we describe and analyse. Our results bring over tools from communication complexity to distributed optimisation, which has potential for further applications.