Finite-Bit Quantization For Distributed Algorithms With Linear Convergence

This paper studies distributed algorithms for (strongly convex) composite optimization problems over mesh networks, subject to quantized communications. Instead of focusing on a specific algorithmic design, a black-box model is proposed, casting linearly-convergent distributed algorithms in the form of fixed-point iterates. While most existing quantization rules, such as the popular compression rule, rely on some form of communication of scalar signals (in practice quantized at the machine precision), this paper considers regimes operating under limited communication budgets, where communication at machine precision is not viable. To address this challenge, the algorithmic model is coupled with a novel random or deterministic Biased Compression (BC-)rule on the quantizer design as well as with a new Adaptive range Non-uniform Quantizer (ANQ) and communication-efficient encoding scheme, which implement the BC-rule using a finite number of bits (below machine precision). A unified communication complexity analysis is developed for the black-box model, determining the average number of bits required to reach a solution of the optimization problem within a target accuracy. In particular, it is shown that the proposed BC-rule preserves linear convergence of the unquantized algorithms, and a trade-off between convergence rate and communication cost under quantization is characterized. Numerical results validate our theoretical findings and show that distributed algorithms equipped with the proposed ANQ have more favorable communication complexity than algorithms using state-of-the-art quantization rules.