  In this paper, we study the distributed nonconvex optimization problem, which
aims to minimize the average value of the local nonconvex cost functions using
local information exchange. To reduce the communication overhead, we introduce
three general classes of compressors, i.e., compressors with bounded relative
compression error, compressors with globally bounded absolute compression
error, and compressors with locally bounded absolute compression error. By
integrating them with distributed gradient tracking algorithm, we then propose
three compressed distributed nonconvex optimization algorithms. For each
algorithm, we design a novel Lyapunov function to demonstrate its sublinear
convergence to a stationary point if the local cost functions are smooth.
Furthermore, when the global cost function satisfies the Polyak--{\L}ojasiewicz
(P--{\L}) condition, we show that our proposed algorithms linearly converge to
a global optimal point. It is worth noting that, for compressors with bounded
relative compression error and globally bounded absolute compression error, our
proposed algorithms' parameters do not require prior knowledge of the P--{\L}
constant. The theoretical results are illustrated by numerical examples, which
demonstrate the effectiveness of the proposed algorithms in significantly
reducing the communication burden while maintaining the convergence
performance. Moreover, simulation results show that the proposed algorithms
outperform state-of-the-art compressed distributed nonconvex optimization
algorithms.
