Distributed stochastic compositional optimization problems over directed networks

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 Ok-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}\left( k^{-1/2}\right) $$\end{document} and sample complexity O1ϵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}\left( \frac{1}{\epsilon ^2}\right) $$\end{document} for finding the (ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document})-stationary point. When the objective function is strongly convex, the convergence rate is improved to Ok-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}\left( k^{-1}\right) $$\end{document}. 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.