  We study distributed optimization problems over a network when the
communication between the nodes is constrained, and so information that is
exchanged between the nodes must be quantized. Recent advances using the
distributed gradient algorithm with a quantization scheme at a fixed resolution
have established convergence, but at rates significantly slower than when the
communications are unquantized.
  In this paper, we introduce a novel quantization method, which we refer to as
adaptive quantization, that allows us to match the convergence rates under
perfect communications. Our approach adjusts the quantization scheme used by
each node as the algorithm progresses: as we approach the solution, we become
more certain about where the state variables are localized, and adapt the
quantizer codebook accordingly.
  We bound the convergence rates of the proposed method as a function of the
communication bandwidth, the underlying network topology, and structural
properties of the constituent objective functions. In particular, we show that
if the objective functions are convex or strongly convex, then using adaptive
quantization does not affect the rate of convergence of the distributed
subgradient methods when the communications are quantized, except for a
constant that depends on the resolution of the quantizer. To the best of our
knowledge, the rates achieved in this paper are better than any existing work
in the literature for distributed gradient methods under finite communication
bandwidths. We also provide numerical simulations that compare convergence
properties of the distributed gradient methods with and without quantization
for solving distributed regression problems for both quadratic and absolute
loss functions.
