  In this paper, we consider decentralized optimization problems where agents
have individual cost functions to minimize subject to subspace constraints that
require the minimizers across the network to lie in low-dimensional subspaces.
This constrained formulation includes consensus or single-task optimization as
special cases, and allows for more general task relatedness models such as
multitask smoothness and coupled optimization. In order to cope with
communication constraints, we propose and study an adaptive decentralized
strategy where the agents employ differential randomized quantizers to compress
their estimates before communicating with their neighbors. The analysis shows
that, under some general conditions on the quantization noise, and for
sufficiently small step-sizes $\mu$, the strategy is stable both in terms of
mean-square error and average bit rate: by reducing $\mu$, it is possible to
keep the estimation errors small (on the order of $\mu$) without increasing
indefinitely the bit rate as $\mu\rightarrow 0$. Simulations illustrate the
theoretical findings and the effectiveness of the proposed approach, revealing
that decentralized learning is achievable at the expense of only a few bits.
