  We consider the problem of decentralized consensus optimization, where the
sum of $n$ smooth and strongly convex functions are minimized over $n$
distributed agents that form a connected network. In particular, we consider
the case that the communicated local decision variables among nodes are
quantized in order to alleviate the communication bottleneck in distributed
optimization. We propose the Quantized Decentralized Gradient Descent (QDGD)
algorithm, in which nodes update their local decision variables by combining
the quantized information received from their neighbors with their local
information. We prove that under standard strong convexity and smoothness
assumptions for the objective function, QDGD achieves a vanishing mean solution
error under customary conditions for quantizers. To the best of our knowledge,
this is the first algorithm that achieves vanishing consensus error in the
presence of quantization noise. Moreover, we provide simulation results that
show tight agreement between our derived theoretical convergence rate and the
numerical results.
