  The paper studies the distributed stochastic compositional optimization
problems over networks, where all the agents' inner-level function is the sum
of each agent's private expectation function. Focusing on the aggregative
structure of the inner-level function, we employ the hybrid variance reduction
method to obtain the information on each agent's private expectation function,
and apply the dynamic consensus mechanism to track the information on each
agent's inner-level function. Then by combining with the standard distributed
stochastic gradient descent method, we propose a distributed aggregative
stochastic compositional gradient descent method. When the objective function
is smooth, the proposed method achieves the optimal convergence rate
$\mathcal{O}\left(K^{-1/2}\right)$. We further combine the proposed method with
the communication compression and propose the communication compressed variant
distributed aggregative stochastic compositional gradient descent method. The
compressed variant of the proposed method maintains the optimal convergence
rate $\mathcal{O}\left(K^{-1/2}\right)$. Simulated experiments on decentralized
reinforcement learning verify the effectiveness of the proposed methods.
