[Paper Review] Asynchronous Subgradient-Push.

Mahmoud Assran, Michael Rabbat|arXiv (Cornell University)|Mar 23, 2018

Distributed Control Multi-Agent Systems31 references17 citations

TL;DR

This paper proposes Asynchronous Subgradient-Push, a distributed optimization algorithm for multi-agent networks where agents minimize a sum of local convex functions asynchronously. It achieves convergence to a neighborhood of the global minimum under asynchrony, with exact convergence when agents operate synchronously, outperforming synchronous first-order methods in speed, robustness, and scalability.

ABSTRACT

We consider a multi-agent framework for distributed optimization where each agent in the network has access to a local convex function and the collective goal is to achieve consensus on the parameters that minimize the sum of the agents' local functions. We propose an algorithm wherein each agent operates asynchronously and independently of the other agents in the network. When the local functions are strongly-convex with Lipschitz-continuous gradients, we show that a subsequence of the iterates at each agent converges to a neighbourhood of the global minimum, where the size of the neighbourhood depends on the degree of asynchrony in the multi-agent network. When the agents work at the same rate, convergence to the global minimizer is achieved. Numerical experiments demonstrate that Asynchronous Subgradient-Push can minimize the global objective faster than state-of-the-art synchronous first-order methods, is more robust to failing or stalling agents, and scales better with the network size.

Motivation & Objective

• To address the challenge of distributed optimization in multi-agent networks with unreliable or variable update timing.
• To design an algorithm that allows agents to operate independently and asynchronously while still converging to the global minimum.
• To analyze the impact of asynchrony on convergence accuracy and to quantify the neighborhood size around the global minimum.
• To improve upon existing synchronous first-order methods in terms of speed, fault tolerance, and scalability.

Proposed method

• Each agent maintains a local estimate of the optimal parameter and updates it using subgradients of its local convex function.
• Agents communicate their estimates over a directed network using a push-sum protocol to maintain balance in information exchange.
• Updates are performed asynchronously, without requiring coordination or synchronization across agents.
• The algorithm uses a diminishing step size rule to ensure convergence despite asynchrony.
• The network topology is modeled as a time-varying directed graph, allowing for dynamic and unreliable communication.
• Convergence is analyzed using Lyapunov functions and stochastic approximation techniques under weak assumptions on agent update timing.

Experimental results

Research questions

• RQ1 How does asynchrony affect the convergence behavior of distributed subgradient methods in multi-agent optimization?
• RQ2 Can an asynchronous algorithm achieve convergence to the global minimum, and if not, how close can it get?
• RQ3 What is the relationship between the degree of asynchrony and the size of the neighborhood around the global minimum?
• RQ4 How does the performance of the asynchronous algorithm compare to state-of-the-art synchronous first-order methods in practice?
• RQ5 To what extent does the algorithm remain robust to failing or stalling agents in large-scale networks?

Key findings

• A subsequence of the iterates at each agent converges to a neighborhood of the global minimum, with the neighborhood size increasing with the degree of asynchrony.
• When agents operate at the same rate (i.e., synchronously), the algorithm converges exactly to the global minimizer.
• Numerical experiments show that Asynchronous Subgradient-Push minimizes the global objective faster than state-of-the-art synchronous first-order methods.
• The algorithm is more robust to failing or stalling agents compared to synchronous alternatives.
• The method scales better with increasing network size due to its asynchronous and decentralized nature.
• The convergence neighborhood is quantitatively bounded based on the maximum delay in agent updates.