Compressed Distributed Zeroth-Order Gradient Tracking for Nonconvex Optimization

This paper investigates a distributed nonconvex optimization problem, which aims to minimize the average of the local nonconvex cost functions through the information exchange. Note that the communication channel often has limited bandwidth and gradient information is typically unavailable. To overcome these limitations, we propose a distributed zeroth- order gradient tracking algorithm based on general compressed communication with bounded relative compression and the random gradient estimator. To the best of our knowledge, this is among the first compressed gradient tracking algorithm without the gradient information for distributed nonconvex optimization. Furthermore, we design an appropriate Lyapunov function to analyze its convergence property. When local cost functions are smooth, we show that the proposed algorithm can find a stationary point with a sublinear convergence rate of
$\mathcal{O}(1/\sqrt{T})$
, where
$\mathcal{T}$
represents the total number of iterations. Moreover, if the global cost function satisfies the condition, we show that the proposed algorithm achieves sublinear convergence to the global optimum with a rate of
$\mathcal{O}(1)$
. Finally, numerical simulations are conducted to verify the theoretical results.