  In this paper, we consider a distributed constrained optimization problem
with delayed subgradient information over the time-varying communication
network, where each agent can only communicate with its neighbors and the
communication channel has a limited data rate. We propose an adaptive
quantization method to address this problem. A mirror descent algorithm with
delayed subgradient information is established based on the theory of Bregman
divergence. With non-Euclidean Bregman projection-based scheme, the proposed
method essentially generalizes many previous classical Euclidean
projection-based distributed algorithms. Through the proposed adaptive
quantization method, the optimal value without any quantization error can be
obtained. Furthermore, comprehensive analysis on convergence of the algorithm
is carried out and our results show that the optimal convergence rate
$O(1/\sqrt{T})$ can be obtained under appropriate conditions. Finally,
numerical examples are presented to demonstrate the effectiveness of our
algorithm and theoretical results.
