  One fundamental problem in decentralized multi-agent optimization is the
trade-off between gradient/sampling complexity and communication complexity. We
propose new algorithms whose gradient and sampling complexities are graph
topology invariant while their communication complexities remain optimal. For
convex smooth deterministic problems, we propose a primal dual sliding (PDS)
algorithm that computes an $\epsilon$-solution with
$O((\tilde{L}/\epsilon)^{1/2})$ gradient and
$O((\tilde{L}/\epsilon)^{1/2}+\|\mathcal{A}\|/\epsilon)$ communication
complexities, where $\tilde{L}$ is the smoothness parameter of the objective
and $\mathcal{A}$ is related to either the graph Laplacian or the transpose of
the oriented incidence matrix of the communication network. The results can be
improved to $O((\tilde{L}/\mu)^{1/2}\log(1/\epsilon))$ and
$O((\tilde{L}/\mu)^{1/2}\log(1/\epsilon) + \|\mathcal{A}\|/\epsilon^{1/2})$
respectively with $\mu$-strong convexity. We also propose a stochastic variant,
the primal dual sliding (SPDS) algorithm for problems with stochastic
gradients. The SPDS algorithm utilizes the mini-batch technique and enables the
agents to perform sampling and communication simultaneously. It computes a
stochastic $\epsilon$-solution with $O((\tilde{L}/\epsilon)^{1/2} +
(\sigma/\epsilon)^2)$ sampling complexity, which can be improved to
$O((\tilde{L}/\mu)^{1/2}\log(1/\epsilon) + \sigma^2/\epsilon)$ with strong
convexity. Here $\sigma^2$ is the variance. The communication complexities of
SPDS remain the same as that of the deterministic case. All the aforementioned
gradient and sampling complexities match the lower complexity bounds for
centralized convex smooth optimization and are independent of the network
structure. To the best of our knowledge, these gradient and sampling
complexities have not been obtained before for decentralized optimization over
a constraint feasible set.
