Distributed Normal Map-based Stochastic Proximal Gradient Methods over Networks

TL;DR

This paper develops distributed stochastic proximal gradient algorithms for networked agents to optimize composite functions, achieving near-centralized convergence rates with state-of-the-art transient times.

Contribution

It introduces two novel algorithms, norM-DSGT and norM-ED, with proven convergence rates matching centralized methods under general variance conditions.

Findings

Both algorithms asymptotically match centralized stochastic proximal gradient rates.

Transient time for norM-ED is (n^3/(1-)^2), matching non-proximal ED.

Transient time for norM-DSGT is (max(n^3/(1-)^2, n/(1-)^3)), matching non-proximal DSGT.

Abstract

Consider $n$ agents connected over a network collaborating to minimize the average of their local cost functions combined with a common nonsmooth function. This paper introduces a unified algorithmic framework for solving such a problem through distributed stochastic proximal gradient methods, leveraging the normal map update scheme. Within this framework, we propose two new algorithms, termed Normal Map-based Distributed Stochastic Gradient Tracking (norM-DSGT) and Normal Map-based Exact Diffusion (norM-ED). We demonstrate that both methods can asymptotically achieve comparable convergence rates to the centralized stochastic proximal gradient descent method under a general variance condition on the stochastic gradients. Additionally, the number of iterations required for norM-ED to achieve such a rate (i.e., the transient time) behaves as $\mathcal{O}(n^{3}/(1-\lambda)^2)$ for minimizing composite objective functions, matching the performance of the non-proximal ED algorithm. Here $1-\lambda$ denotes the spectral gap of the mixing matrix related to the underlying network topology. To our knowledge, such a convergence result is state-of-the-art for the considered composite problem. Under the same condition, norM-DSGT enjoys a transient time of $\mathcal{O}(\max\{n^3/(1-\lambda)^2, n/(1-\lambda)^3\})$, which matches that of the non-proximal DSGT algorithm and norM-ED under the condition $(1-\lambda)^{-1}=\mathcal{O}(n^{2})$.