NEXT: In-Network Nonconvex Optimization (1602.00591v1)

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.

• The paper introduces NEXT, a distributed algorithm that combines successive convex approximation with dynamic consensus to address nonconvex challenges.
• The algorithm employs local convex surrogate optimization and coordination among agents, ensuring convergence to stationary solutions.
• The paper demonstrates enhanced convergence speed and scalability in practical applications such as signal processing and machine learning.

An Overview of "NEXT: In-Network Nonconvex Optimization"

The paper, "NEXT: In-Network Nonconvex Optimization" by Paolo Di Lorenzo and Gesualdo Scutari, presents a novel framework for addressing nonconvex optimization problems within distributed multi-agent networks with time-varying connectivity. This work introduces an algorithm, named NEXT, which efficiently handles the minimization of a collective sum-utility function and a convex regularizer across agents in a distributed manner.

Core Contributions

1. Framework Development: The authors propose an algorithmic framework targeting nonconvex distributed optimization. By leveraging successive convex approximation (SCA) techniques and dynamic consensus, NEXT offers a method by which each agent solves a local problem based on a convex surrogate of its nonconvex utility. This yields an efficient way to approximate the original challenge in a distributed setting.
2. Algorithmic Details: The algorithm iteratively updates each agent's local estimate through two main steps: a local optimization step using the SCA followed by a consensus step that aligns estimates across neighboring agents. Convergence to stationary solutions is guaranteed through well-founded mathematical principles.
3. Handling Nonconvexity: Importantly, NEXT accommodates the nonconvex nature of the problem by not requiring global knowledge of the objective function or global connectivity of the network, thus broadening its applicability.
4. Practical Implementations: The paper tailors the NEXT framework to various applications, showcasing its versatility. These applications span signal processing, machine learning, and networking, illustrating the practical impact and effectiveness of the method.
5. Performance and Comparisons: Numerical results demonstrate that NEXT competes favorably with existing distributed algorithms, especially in terms of convergence speed and solution accuracy. It handles both convex and nonconvex problems seamlessly, showing better practical convergence.

Implications and Future Directions

Theoretical Implications:

The proposed approach advances the current understanding of distributed optimization in nonconvex settings. By combining SCA methods with consensus algorithms, NEXT opens avenues for further exploration into optimizing complex systems where traditional methods falter.

Practical Implications:

NEXT's flexibility and robustness make it an attractive choice for real-world applications in areas requiring distributed computation without centralized control. By ensuring convergence through local computation and limited communication, it holds promise for scalable implementations in large networks.

Future Research:

Potential developments could include extensions to asynchronous settings or real-time applications with stringent convergence requirements. Moreover, exploring the adaptation of NEXT to more specific domains like cybersecurity or autonomous systems could lead to innovative solutions in those fields.

Conclusion

Di Lorenzo and Scutari's "NEXT: In-Network Nonconvex Optimization" contributes a significant step forward in distributed optimization techniques applicable to nonconvex problems. With provable convergence properties and practical applicability across a spectrum of problems, NEXT positions itself as a powerful tool for contemporary multi-agent systems and networks.