[Paper Review] Distributed Optimization Over Dependent Random Networks

Adel Aghajan, Behrouz Touri|arXiv (Cornell University)|Oct 5, 2020

Stochastic Gradient Optimization Techniques4 citations

TL;DR

This paper establishes almost-sure convergence of averaging-based distributed optimization over dependent random networks, where weight matrices are row-stochastic almost surely and column-stochastic in expectation. The key contribution is a general convergence result that extends prior work by allowing dependent network realizations and robustly handling link failures, while providing new tools for analyzing convergence rates in random networks.

ABSTRACT

We study the averaging-based distributed optimization solvers over random networks. We show a general result on the convergence of such schemes using weight-matrices that are row-stochastic almost surely and column-stochastic in expectation for a broad class of dependent weight-matrix sequences. In addition to implying many of the previously known results on this domain, our work shows the robustness of distributed optimization results to link-failure. Also, it provides a new tool for synthesizing distributed optimization algorithms. {To prove our main theorem, we establish new results on the rate of convergence analysis of averaging dynamics over (dependent) random networks. These secondary results, along with the required martingale-type results to establish them, might be of interest to broader research endeavors in distributed computation over random networks.

Motivation & Objective

• To establish convergence of distributed optimization algorithms over time-varying, dependent random networks where network links may fail or change over time.
• To generalize existing results by relaxing the i.i.d. assumption on weight matrices to allow for temporal dependence in network topology.
• To provide a robust framework for distributed optimization that remains valid under link failures and non-i.i.d. network dynamics.
• To develop new analytical tools for convergence rate analysis of averaging dynamics in dependent random networks.
• To unify and extend prior results on distributed averaging and optimization under weaker stochastic assumptions.

Proposed method

• The authors analyze distributed averaging dynamics over random networks using a novel martingale-type approach to handle dependent weight sequences.
• They introduce a new measure of network diameter, diam(A), defined as the maximum over all pairs of nodes of the sum of half the absolute differences of row entries in the weight matrix A.
• The convergence analysis relies on bounding the disagreement metric d(x), defined as the maximum norm difference between any two agents' states, using the network's diameter and row-stochasticity.
• A key technical tool is a recursive expectation inequality involving products of random variables, proven via induction and conditional expectation bounds.
• The method extends to systems with arbitrary control inputs by decomposing the state evolution into averaging and gradient descent components.
• The proof leverages the fact that column-stochasticity in expectation ensures that the expected state converges to the average, even when individual matrices are not doubly stochastic.

Experimental results

Research questions

• RQ1 Can averaging-based distributed optimization converge almost surely when the network topology is random and temporally dependent?
• RQ2 Does the convergence result hold when weight matrices are only row-stochastic almost surely and column-stochastic in expectation, rather than doubly stochastic?
• RQ3 How does the rate of convergence depend on the network's structure and the dependence structure of the random weight matrices?
• RQ4 Can the framework be extended to handle arbitrary control inputs, such as gradient descent steps, in a distributed optimization setting?
• RQ5 What new analytical tools are required to study convergence in dependent random networks, and how do they differ from those used in i.i.d. settings?

Key findings

• The averaging-based distributed optimization solver converges almost surely to a global optimizer when weight matrices are row-stochastic almost surely and column-stochastic in expectation.
• The convergence result is robust to network link failures, as it does not require the weight matrices to be doubly stochastic or i.i.d.
• The paper establishes a new bound on the disagreement metric d(x) in terms of the network diameter diam(A), showing that d(Ax) ≤ diam(A) d(x).
• A new convergence rate analysis is developed for averaging dynamics over dependent random networks, which may be useful for broader applications in distributed computation.
• The authors prove a key inequality involving the product of random variables under conditional expectation, which enables the analysis of non-i.i.d. weight sequences.
• The framework generalizes and unifies prior results on distributed optimization and averaging, including those for gossip and push-sum algorithms, under a common theoretical umbrella.