Towards Tight Communication Lower Bounds for Distributed Optimisation
This work addresses communication bottlenecks in large-scale distributed optimization, providing foundational lower bounds that are incremental in applying communication complexity tools to this domain.
The paper tackles the problem of communication complexity in distributed optimization, establishing the first unconditional lower bounds on the total bits needed for N machines to find an ε-approximation, showing a requirement of Ω(Nd log d / Nε) bits, which is tight under certain conditions and matched by a new quantized gradient descent variant for quadratic objectives.
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 $Ω( Nd \log d / N\varepsilon)$ total bits need to be communicated between the machines to find an additive $ε$-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.