  We study the distributed stochastic compositional optimization problems over
directed communication networks in which agents privately own a stochastic
compositional objective function and collaborate to minimize the sum of all
objective functions. We propose a distributed stochastic compositional gradient
descent method, where the gradient tracking and the stochastic correction
techniques are employed to adapt to the networks' directed structure and
increase the accuracy of inner function estimation. When the objective function
is smooth, the proposed method achieves the convergence rate
$\mathcal{O}\left(k^{-1/2}\right)$ and sample complexity
$\mathcal{O}\left(\frac{1}{\epsilon^2}\right)$ for finding the
($\epsilon$)-stationary point. When the objective function is strongly convex,
the convergence rate is improved to $\mathcal{O}\left(k^{-1}\right)$. Moreover,
the asymptotic normality of Polyak-Ruppert averaged iterates of the proposed
method is also presented. We demonstrate the empirical performance of the
proposed method on model-agnostic meta-learning problem and logistic regression
problem.
