  Decentralized strategies are of interest for learning from large-scale data
over networks. This paper studies learning over a network of geographically
distributed nodes/agents subject to quantization. Each node possesses a private
local cost function, collectively contributing to a global cost function, which
the proposed methodology aims to minimize. In contrast to many existing
literature, the information exchange among nodes is quantized. We adopt a
first-order computationally-efficient distributed optimization algorithm (with
no extra inner consensus loop) that leverages node-level gradient correction
based on local data and network-level gradient aggregation only over nearby
nodes. This method only requires balanced networks with no need for stochastic
weight design. It can handle log-scale quantized data exchange over possibly
time-varying and switching network setups. We analyze convergence over both
structured networks (for example, training over data-centers) and ad-hoc
multi-agent networks (for example, training over dynamic robotic networks).
Through analysis and experimental validation, we show that (i) structured
networks generally result in a smaller optimality gap, and (ii) logarithmic
quantization leads to smaller optimality gap compared to uniform quantization.
