  We present a new class of decentralized first-order methods for nonsmooth and
stochastic optimization problems defined over multiagent networks. Considering
that communication is a major bottleneck in decentralized optimization, our
main goal in this paper is to develop algorithmic frameworks which can
significantly reduce the number of inter-node communications. We first propose
a decentralized primal-dual method which can find an $\epsilon$-solution both
in terms of functional optimality gap and feasibility residual in
$O(1/\epsilon)$ inter-node communication rounds when the objective functions
are convex and the local primal subproblems are solved exactly. Our major
contribution is to present a new class of decentralized primal-dual type
algorithms, namely the decentralized communication sliding (DCS) methods, which
can skip the inter-node communications while agents solve the primal
subproblems iteratively through linearizations of their local objective
functions. By employing DCS, agents can still find an $\epsilon$-solution in
$O(1/\epsilon)$ (resp., $O(1/\sqrt{\epsilon})$) communication rounds for
general convex functions (resp., strongly convex functions), while maintaining
the $O(1/\epsilon^2)$ (resp., $O(1/\epsilon)$) bound on the total number of
intra-node subgradient evaluations. We also present a stochastic counterpart
for these algorithms, denoted by SDCS, for solving stochastic optimization
problems whose objective function cannot be evaluated exactly. In comparison
with existing results for decentralized nonsmooth and stochastic optimization,
we can reduce the total number of inter-node communication rounds by orders of
magnitude while still maintaining the optimal complexity bounds on intra-node
stochastic subgradient evaluations. The bounds on the subgradient evaluations
are actually comparable to those required for centralized nonsmooth and
stochastic optimization.
