Distributed Nonconvex Constrained Optimization over Time-Varying Digraphs (1809.01106v1)

Abstract: This paper considers nonconvex distributed constrained optimization over networks, modeled as directed (possibly time-varying) graphs. We introduce the first algorithmic framework for the minimization of the sum of a smooth nonconvex (nonseparable) function--the agent's sum-utility--plus a Difference-of-Convex (DC) function (with nonsmooth convex part). This general formulation arises in many applications, from statistical machine learning to engineering. The proposed distributed method combines successive convex approximation techniques with a judiciously designed perturbed push-sum consensus mechanism that aims to track locally the gradient of the (smooth part of the) sum-utility. Sublinear convergence rate is proved when a fixed step-size (possibly different among the agents) is employed whereas asymptotic convergence to stationary solutions is proved using a diminishing step-size. Numerical results show that our algorithms compare favorably with current schemes on both convex and nonconvex problems.

• The paper introduces a novel algorithm tackling distributed nonconvex constrained optimization over time-varying directed graphs.
• The algorithm combines Successive Convex Approximation with a perturbed consensus mechanism for gradient tracking on dynamic networks.
• Numerical results on sparse regression and PCA show the algorithm's practical effectiveness on dynamic networks, applicable in systems like mobile and sensor networks.

An In-Depth Analysis of "Distributed Nonconvex Constrained Optimization over Time-Varying Digraphs"

The paper introduces a novel algorithmic framework designed to tackle distributed nonconvex constrained optimization problems across networks modeled as directed and potentially time-varying graphs. Traditional methods in distributed optimization primarily focus on convex problems and often require strong assumptions on network connectivity and symmetry, such as undirected graphs and doubly-stochastic weight matrices. This research fills a significant gap by accommodating nonconvex scenarios with minimal constraints on network topology, thus broadening the applicability of distributed optimization in real-world scenarios where these ideal conditions are not met.

Core Contributions and Technical Approach

The principal contribution lies in the development of an algorithm that leverages a combination of Successive Convex Approximation (SCA) techniques and a perturbation-based Putsh-Sum consensus mechanism. This hybrid approach facilitates the minimization of a composite objective function characterized by a smooth nonconvex component in addition to a difference-of-convex (DC) function. Such a formulation naturally emerges in numerous applications spanning statistical machine learning and network engineering.

Algorithmic Design

1. Local Surrogate Problem: Each agent solves a locally convexified problem, deriving from a smooth approximation of its individual nonconvex objective component and the gradient information shared across neighboring agents. This surrogate problem blends the local objective function with constraints derived from differential approximations of shared terms and graph-induced communication limits.
2. Consensus and Gradient Tracking: The algorithm tackles the inherent communication challenges of dynamic and directed graphs through a perturbed consensus mechanism. This key innovation permits each agent to asymptotically track the global gradient information, crucial for convergence in a nonconvex setting.
3. Convergence and Complexity: The research rigorously establishes asymptotic convergence to stationary points under a flexible yet precise set of assumptions, which include both constant and vanishing step-size strategies. The provided iteration complexity is sublinear, marking a significant achievement in distributed nonconvex optimization theory.

By demonstrating the provable convergence of the solution under the constraints of a time-varying network topology, the work extends the theoretical foundation of distributed optimization. The framework is equipped to handle real-world issues such as nonconvexity, communication delays, and the decentralized nature of agent networks.

Numerical Insights and Practical Implications

The paper evaluates the proposed algorithm on problems like sparse linear regression and distributed principal component analysis (PCA), leveraging both synthetic and real-world data exemplars. The positive numerical results, highlighting efficiency and robustness across various network settings, underscore the practical potency of this approach in handling high-dimensional, nonconvex optimization problems.

This framework has broad implications, especially in environments where direct communication between all nodes is infeasible, such as mobile and sensor networks. In these scenarios, the ability to effectively utilize a dynamic graph structure without requiring static and undirected assumptions is invaluable.

Future Directions and Theoretical Implications

Future works could enhance scalability and tackle even more complex instances of distributed nonconvex problems, possibly integrating stochastic components into the current deterministic framework. Additionally, exploring the trade-offs in convergence rates relative to network changes and communication bandwidth could provide further optimization.

The research not only pushes the boundary of distributed optimization into more practical realms but also lays the groundwork for subsequent explorations in networked systems where constraints, variability, and nonconvexity are the norms rather than exceptions. This work is poised to significantly influence future developments in distributed machine learning, communications networks, and beyond, where decision-making processes are decentralized and occur over vast, fluctuating topologies.