  Actor-critic (AC) algorithms have been widely adopted in decentralized
multi-agent systems to learn the optimal joint control policy. However,
existing decentralized AC algorithms either do not preserve the privacy of
agents or are not sample and communication-efficient. In this work, we develop
two decentralized AC and natural AC (NAC) algorithms that are private, and
sample and communication-efficient. In both algorithms, agents share noisy
information to preserve privacy and adopt mini-batch updates to improve sample
and communication efficiency. Particularly for decentralized NAC, we develop a
decentralized Markovian SGD algorithm with an adaptive mini-batch size to
efficiently compute the natural policy gradient. Under Markovian sampling and
linear function approximation, we prove the proposed decentralized AC and NAC
algorithms achieve the state-of-the-art sample complexities
$\mathcal{O}\big(\epsilon^{-2}\ln(\epsilon^{-1})\big)$ and
$\mathcal{O}\big(\epsilon^{-3}\ln(\epsilon^{-1})\big)$, respectively, and the
same small communication complexity
$\mathcal{O}\big(\epsilon^{-1}\ln(\epsilon^{-1})\big)$. Numerical experiments
demonstrate that the proposed algorithms achieve lower sample and communication
complexities than the existing decentralized AC algorithm.
