Spatio-Temporal Communication Compression for Distributed Prime-Dual Optimization

In this paper, for the problem of distributed computing, we propose a general spatio-temporal compressor and discuss its compression methods. This compressor comprehensively considers both temporal and spatial information, encompassing many existing specific compressors. We use the average consensus algorithm as a starting point and further studies distributed optimization algorithms, the Prime-Dual algorithm as an example, in both continuous and discrete time forms. We find that under stronger additional assumptions, the spatio-temporal compressor can be directly applied to distributed computing algorithms, while its default form can also be successfully applied through observer-based differential compression methods, ensuring the linear convergence of the algorithm when the objective function is strongly convex. On this basis, we also discuss the acceleration of the algorithm, filter-based compression methods in the literature, and the addition of randomness to the spatio-temporal compressor. Finally, numerical simulations illustrate the generality of the spatio-temporal compressor, compare different compression methods, and verify the algorithm's performance in the convex objective function scenario.