Fundamental Limits of Multi-User Distributed Computing of Linearly Separable Functions

TL;DR

This paper investigates the fundamental limits of multi-user distributed computing for linearly separable functions, proposing optimal schemes that balance communication and computation in both real and finite fields.

Contribution

It introduces a novel distributed computing scheme with optimal performance guarantees and provides new converses establishing fundamental limits in this setting.

Findings

Achieves optimal tradeoff between communication and computation.

Provides a scheme that is optimal in the real field.

Establishes bounds in the finite field setting.

Abstract

This work establishes the fundamental limits of the classical problem of multi-user distributed computing of linearly separable functions. In particular, we consider a distributed computing setting involving $L$ users, each requesting a linearly separable function over $K$ basis subfunctions from a master node, who is assisted by $N$ distributed servers. At the core of this problem lies a fundamental tradeoff between communication and computation: each server can compute up to $M$ subfunctions, and each server can communicate linear combinations of their locally computed subfunctions outputs to at most $\Delta$ users. The objective is to design a distributed computing scheme that reduces the communication cost (total amount of data from servers to users), and towards this, for any given $K$, $L$, $M$, and $\Delta$, we propose a distributed computing scheme that jointly designs the task assignment and transmissions, and shows that the scheme achieves optimal performance in the real field under various conditions using a novel converse. We also characterize the performance of the scheme in the finite field using another converse based on counting arguments.