  In this paper, we propose and analyze SPARQ-SGD, which is an event-triggered
and compressed algorithm for decentralized training of large-scale machine
learning models. Each node can locally compute a condition (event) which
triggers a communication where quantized and sparsified local model parameters
are sent. In SPARQ-SGD each node takes at least a fixed number ($H$) of local
gradient steps and then checks if the model parameters have significantly
changed compared to its last update; it communicates further compressed model
parameters only when there is a significant change, as specified by a (design)
criterion. We prove that the SPARQ-SGD converges as $O(\frac{1}{nT})$ and
$O(\frac{1}{\sqrt{nT}})$ in the strongly-convex and non-convex settings,
respectively, demonstrating that such aggressive compression, including
event-triggered communication, model sparsification and quantization does not
affect the overall convergence rate as compared to uncompressed decentralized
training; thereby theoretically yielding communication efficiency for "free".
We evaluate SPARQ-SGD over real datasets to demonstrate significant amount of
savings in communication over the state-of-the-art.
