We analyze the convergence of decentralized consensus algorithm with delayed gradient information across the network. The nodes in the network privately hold parts of the objective function and collaboratively solve for the consensus optimal solution of the total objective while they can only communicate with their immediate neighbors. In real-world networks, it is often difficult and sometimes impossible to synchronize the nodes, and therefore they have to use stale gradient information during computations. We show that, as long as the random delays are bounded in expectation and a proper diminishing step size policy is employed, the iterates generated by decentralized gradient descent method converge to a consensual optimal solution. Convergence rates of both objective and consensus are derived. Numerical results on a number of synthetic problems and real-world seismic tomography datasets in decentralized sensor networks are presented to show the performance of the method.
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