  In the modern paradigm of multi-agent networks, communication has become one
of the main bottlenecks for decentralized optimization, where a large number of
agents are involved in minimizing the average of the local cost functions. In
this paper, we propose a robust compressed push-pull algorithm (RCPP) that
combines gradient tracking with communication compression. In particular, RCPP
is compatible with a much more general class of compression operators that
allow both relative and absolute compression errors. We show that RCPP achieves
linear convergence rate for smooth objective functions satisfying the
Polyak-{\L}ojasiewicz condition over general directed networks. Numerical
examples verify the theoretical findings and demonstrate the efficiency,
flexibility, and robustness of the proposed algorithm.
