  In this paper, we study distributed prime-dual flows for multi-agent
optimization with spatio-temporal compressions. The central aim of multi-agent
optimization is for a network of agents to collaboratively solve a system-level
optimization problem with local objective functions and node-to-node
communication by distributed algorithms. The scalability of such algorithms
crucially depends on the complexity of the communication messages, and a number
of communication compressors for distributed optimization have recently been
proposed in the literature. First of all, we introduce a general
spatio-temporal compressor characterized by the stability of the resulting
dynamical system along the vector field of the compressor. We show that several
important distributed optimization compressors such as the greedy sparsifier,
the uniform quantizer, and the scalarizer all fall into the category of this
spatio-temporal compressor. Next, we propose two distributed prime-dual flows
with the spatio-temporal compressors being applied to local node states and
local error states, respectively, and prove (exponential) convergence of the
node trajectories to the global optimizer for (strongly) convex cost functions.
Finally, a few numerical examples are present to illustrate our theoretical
results.
