[Paper Review] NEXT: In-Network Nonconvex Optimization

Paolo Di Lorenzo, Gesualdo Scutari|arXiv (Cornell University)|Feb 1, 2016

Distributed Control Multi-Agent Systems49 references4 citations

TL;DR

This paper proposes NEXT, a novel in-network algorithm for distributed nonconvex optimization over time-varying directed networks. It uses successive convex approximation with dynamic consensus to enable agents to collaboratively minimize a sum of smooth nonconvex functions and a convex regularizer, achieving convergence to stationary points under mild assumptions.

ABSTRACT

We study nonconvex distributed optimization in multi-agent networks with time-varying (nonsymmetric) connectivity. We introduce the first algorithmic framework for the distributed minimization of the sum of a smooth (possibly nonconvex and nonseparable) function - 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 techniques while leveraging dynamic consensus as a mechanism to distribute the computation among the agents: each agent first solves (possibly inexactly) a local convex approximation of the nonconvex original problem, and then performs local averaging operations. Asymptotic convergence to (stationary) solutions of the nonconvex problem is established. Our algorithmic framework is then customized to a variety of convex and nonconvex problems in several fields, including signal processing, communications, networking, and machine learning. Numerical results show that the new method compares favorably to existing distributed algorithms on both convex and nonconvex problems.

Motivation & Objective

• Address the lack of distributed algorithms for nonconvex optimization in multi-agent networks with time-varying, possibly nonsymmetric, connectivity.
• Enable decentralized computation where agents only know their local functions, a global regularizer, and operate under intermittent neighbor communication.
• Develop a provably convergent method for minimizing the sum of smooth (possibly nonconvex) functions plus a convex regularizer in a distributed setting.
• Overcome limitations of prior methods that require centralized coordination, full knowledge of global functions, or specific network topologies.
• Establish convergence to stationary solutions for general nonconvex problems in distributed networks with dynamic communication graphs.

Proposed method

• Formulate the distributed nonconvex optimization problem as minimizing the sum of local smooth nonconvex functions plus a global convex regularizer over a closed convex set.
• Apply successive convex approximation (SCA) to locally approximate the nonconvex objective with convex subproblems at each agent.
• Use dynamic consensus to distribute computation: agents perform local SCA steps and then exchange and average estimates with neighbors.
• Introduce a diminishing step size rule to ensure convergence while allowing inexact solutions to subproblems.
• Leverage the convexity of the regularizer and Lipschitz continuity of gradients to bound suboptimality and ensure convergence.
• Combine local descent steps with averaging operations to maintain network-wide consistency and drive iterates toward a stationary point.

Experimental results

Research questions

• RQ1 Can a distributed algorithm be designed for nonconvex optimization in time-varying directed networks without requiring a centralized fusion center?
• RQ2 Does the proposed method converge to a stationary solution of the nonconvex problem under general assumptions on the network and objective functions?
• RQ3 How does the algorithm perform in comparison to existing distributed convex and nonconvex methods in terms of convergence and scalability?
• RQ4 Can the algorithm handle nonseparable and nonsmooth regularizers while maintaining convergence guarantees?
• RQ5 What are the theoretical conditions under which the algorithm converges to a stationary point in a fully decentralized setting?

Key findings

• The algorithm achieves asymptotic convergence to a stationary solution of the nonconvex problem under standard assumptions, including time-varying directed graphs and local agent knowledge.
• Convergence is established even when subproblems are solved inexactly, provided the error is bounded and diminishes over time.
• The method ensures that the average of agents' estimates converges to a point where the local approximations are consistent with the global iterate.
• The sequence of objective values converges to a finite limit, and the distance between agents' local estimates and the average iterate tends to zero in the limit.
• The algorithm is applicable to a wide range of problems in signal processing, communications, networking, and machine learning, including LASSO, logistic regression, and matrix completion.
• Numerical results demonstrate that NEXT outperforms existing distributed algorithms on both convex and nonconvex problems in terms of convergence speed and solution quality.