  Communication efficiency is a major bottleneck in the applications of
distributed networks. To address the problem, the problem of quantized
distributed optimization has attracted a lot of attention. However, most of the
existing quantized distributed optimization algorithms can only converge
sublinearly. To achieve linear convergence, this paper proposes a novel
quantized distributed gradient tracking algorithm (Q-DGT) to minimize a finite
sum of local objective functions over directed networks. Moreover, we
explicitly derive the update rule for the number of quantization levels, and
prove that Q-DGT can converge linearly even when the exchanged variables are
respectively one bit. Numerical results also confirm the efficiency of the
proposed algorithm.
