Convergence of a Multi-Agent Projected Stochastic Gradient Algorithm for Non-Convex Optimization (1107.2526v3)

Abstract: We introduce a new framework for the convergence analysis of a class of distributed constrained non-convex optimization algorithms in multi-agent systems. The aim is to search for local minimizers of a non-convex objective function which is supposed to be a sum of local utility functions of the agents. The algorithm under study consists of two steps: a local stochastic gradient descent at each agent and a gossip step that drives the network of agents to a consensus. Under the assumption of decreasing stepsize, it is proved that consensus is asymptotically achieved in the network and that the algorithm converges to the set of Karush-Kuhn-Tucker points. As an important feature, the algorithm does not require the double-stochasticity of the gossip matrices. It is in particular suitable for use in a natural broadcast scenario for which no feedback messages between agents are required. It is proved that our result also holds if the number of communications in the network per unit of time vanishes at moderate speed as time increases, allowing for potential savings of the network's energy. Applications to power allocation in wireless ad-hoc networks are discussed. Finally, we provide numerical results which sustain our claims.

• The paper shows the convergence of a multi-agent projected stochastic gradient algorithm to KKT points under mild assumptions.
• It employs a dual-step process combining local stochastic gradient descent and gossip-based consensus that accommodates non-double stochastic matrices.
• The approach has practical implications, including energy savings and effective power allocation in wireless ad-hoc networks.

Convergence of a Multi-Agent Projected Stochastic Gradient Algorithm for Non-Convex Optimization

This paper explores a distributed algorithm designed to achieve consensus in multi-agent systems for solving non-convex optimization problems subject to constraints. The authors investigate a multi-agent projected stochastic gradient (SG) method which they propose for minimizing a non-convex objective function, represented as a sum of local utility functions of the agents. The presented algorithm consists of two principal steps: a local stochastic gradient descent at each agent and a gossip-based communication step to drive consensus across the network.

The paper makes several contributions to the field of distributed optimization. Primarily, it proves the convergence of the proposed algorithm to the set of Karush-Kuhn-Tucker (KKT) points under mild assumptions, including the requirement that the matrix sequence utilized in the gossip steps need not be double-stochastic. The latter is particularly pivotal as it broadens applicability to natural broadcast scenarios absent of feedback between agents. Remarkably, the paper shows that the algorithm’s convergence remains robust even when network communication frequency decreases over time, allowing for potential energy savings in resource-constrained networks.

Theoretical Analysis and Results

The authors ground their convergence analysis in the framework of perturbed differential inclusions, a mathematical tool robust to discontinuities in differential dynamics, which are expected in non-convex scenarios. Noteworthy is their departure from conventional convexity assumptions, instead emphasizing the algorithm's alignment with a differential variational inequality leading to convergence to KKT points.

An important component of the authors’ theoretical findings is the allowance for non-double stochasticity in gossip matrices, which has often posed significant practical difficulties. By allowing stochastic matrix entries with assured row-stochastics, the framework can accommodate one-way broadcasting, expediting implementation across networks where typical feedback protocols are prohibitive.

Practical Implications

In terms of applications, the authors apply their theoretical findings to the problem of power allocation in wireless ad-hoc networks. Interestingly, the algorithm has shown effective solutions in non-convex settings where conventional centralized and convex-based methods may falter. Numerical simulations back up the theoretical claims, depicting convergence in settings of both fixed and stochastic channels within communication networks.

Future Directions

The trajectory of this research opens ripe areas for further exploration. Particularly intriguing is the extension to deeper non-convex formulations and the exploration of broader application domains, such as large-scale machine learning problems, where decentralized processing could be beneficial. Additionally, examining message-passing techniques that further leverage the communication savings foretold by the algorithm's implementation in resource-constricted scenarios could prove valuable.

In conclusion, this paper provides a significant theoretical contribution to multi-agent optimization, especially in non-convex settings. The relaxed requirements on gossip matrix properties and energy-efficient considerations make it a practical choice for advancing distributed consensus methodologies for various real-world applications. Given these foundations, following through on proposed investigations could yield fertile results contributing to both theoretical and operational advancements in distributed optimization domains.