Distributed Stochastic Compositional Optimization Problems over Directed Networks

TL;DR

This paper introduces a distributed stochastic compositional gradient descent method tailored for directed networks, achieving optimal convergence rates and demonstrating effectiveness on meta-learning and logistic regression tasks.

Contribution

It develops a novel distributed stochastic compositional optimization algorithm that handles directed networks and improves convergence rates with theoretical guarantees.

Findings

Achieves $ ext{O}(k^{-1/2})$ convergence rate for smooth objectives.

Improves to $ ext{O}(k^{-1})$ for strongly convex objectives.

Demonstrates empirical success on meta-learning and logistic regression.

Abstract

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.