[Paper Review] Distributed Nonconvex Multiagent Optimization Over Time-Varying Networks

Ying Sun, Gesualdo Scutari+1 more|arXiv (Cornell University)|Jul 1, 2016

Distributed Control Multi-Agent Systems24 references17 citations

TL;DR

This paper proposes SONATA, a novel broadcast-based distributed algorithm for nonconvex constrained optimization over time-varying directed networks. It combines Successive Convex Approximation (SCA) with gradient tracking and a new in-network broadcast protocol, achieving convergence to stationary solutions without requiring double-stochastic consensus matrices or knowledge of the graph sequence—making it the first such framework for arbitrary time-varying digraphs.

ABSTRACT

We study nonconvex distributed optimization in multiagent networks where the communications between nodes is modeled as a time-varying sequence of arbitrary digraphs. We introduce a novel broadcast-based distributed algorithmic framework for the (constrained) minimization of the sum of a smooth (possibly nonconvex and nonseparable) function, i.e., the agents' sum-utility, plus a convex (possibly nonsmooth and nonseparable) regularizer. The latter is usually employed to enforce some structure in the solution, typically sparsity. The proposed method hinges on Successive Convex Approximation (SCA) techniques coupled with i) a tracking mechanism instrumental to locally estimate the gradients of agents' cost functions; and ii) a novel broadcast protocol to disseminate information and distribute the computation among the agents. Asymptotic convergence to stationary solutions is established. A key feature of the proposed algorithm is that it neither requires the double-stochasticity of the consensus matrices (but only column stochasticity) nor the knowledge of the graph sequence to implement. To the best of our knowledge, the proposed framework is the first broadcast-based distributed algorithm for convex and nonconvex constrained optimization over arbitrary, time-varying digraphs. Numerical results show that our algorithm outperforms current schemes on both convex and nonconvex problems.

Motivation & Objective

• To address the challenge of distributed nonconvex optimization in multiagent networks with time-varying, directed communication topologies.
• To develop a scalable, fully distributed algorithm that does not require double-stochastic consensus matrices or prior knowledge of the network sequence.
• To enable efficient computation and information sharing among agents using a novel broadcast protocol.
• To establish convergence guarantees for nonconvex problems under minimal assumptions on network structure.
• To outperform existing methods in both convex and nonconvex settings through empirical validation.

Proposed method

• The algorithm uses Successive Convex Approximation (SCA) to handle nonconvexity by iteratively solving convex subproblems.
• Each agent tracks the gradients of other agents' functions using a local gradient tracking mechanism.
• A new in-network broadcast protocol enables efficient dissemination of information and distribution of computation without requiring feedback from neighbors.
• The method relies on column-stochastic consensus matrices rather than double-stochastic ones, broadening applicability to arbitrary digraphs.
• The algorithm is designed to operate in a fully distributed manner, with agents only knowing their own local functions and no global network knowledge.
• The framework supports both smooth nonconvex and convex nonsmooth regularizers, enabling structured solutions like sparsity.

Experimental results

Research questions

• RQ1 Can a distributed algorithm achieve convergence to stationary solutions for nonconvex multiagent optimization over arbitrary time-varying digraphs without requiring double-stochasticity?
• RQ2 How can agents estimate the gradients of other agents' functions in a fully distributed and scalable manner?
• RQ3 What is the impact of using a broadcast-based protocol instead of traditional consensus schemes in directed and time-varying networks?
• RQ4 Can the proposed algorithm outperform existing schemes in both convex and nonconvex optimization problems under practical network constraints?
• RQ5 What are the convergence properties of the algorithm under minimal assumptions on network topology and agent knowledge?

Key findings

• SONATA achieves asymptotic convergence to stationary solutions for nonconvex problems over arbitrary time-varying digraphs without requiring double-stochastic consensus matrices.
• The algorithm outperforms subgradient-push and other state-of-the-art schemes in terms of convergence speed and consensus accuracy on both convex and nonconvex problems.
• Numerical results show that SONATA reaches consensus and convergence significantly faster than subgradient-push, especially in nonconvex settings.
• The SCA-based scheme with partial linearization and proximal terms improves convergence over simple linearization in the target localization problem.
• In the target localization problem with 30 sensors and 5 targets, SONATA achieved faster convergence than a benchmark gradient algorithm within 200 iterations.
• The algorithm is robust in noisy environments and maintains fast convergence even when distance measurements are corrupted by i.i.d. Gaussian noise.