# Localization and Approximations for Distributed Non-convex Optimization

## TL;DR

This paper extends distributed non-convex optimization methods by reducing communication complexity and relaxing gradient assumptions, demonstrating improved resource allocation in multi-cell networks with enhanced stability.

## Contribution

It generalizes existing frameworks for non-convex distributed optimization by reducing communication and relaxing gradient bounds, with practical applications in network resource allocation.

## Key findings

- Reduced communication complexity in separable variable cases.
- Relaxed gradient assumptions via proximal approximations.
- Improved stability and performance in resource allocation simulations.

## Abstract

Distributed optimization has many applications, in communication networks, sensor networks, signal processing, machine learning, and artificial intelligence. Methods for distributed convex optimization are widely investigated, while those for non-convex objectives are not well understood. One of the first non-convex distributed optimization frameworks over an arbitrary interaction graph was proposed by Di Lorenzo and Scutari [IEEE Trans. on Signal and Information Processing over Network, 2 (2016), pp. 120-136], which iteratively applies a combination of local optimization with convex approximations and local averaging. We generalize the existing results in two ways. In the case when the decision variables are separable such that there is partial dependency in the objectives, we reduce the communication complexity of the algorithm so that nodes only keep and communicate local variables instead of the whole vector of variables. In addition, we relax the assumption that the objectives' gradients are bounded and Lipschitz by means of successive proximal approximations. Having developed the methodology, we then discuss many ways to apply our algorithmic framework to resource allocation problems in multi-cellular networks, where the two generalizations are found useful and practical. Simulation results show the superiority of our resource allocation algorithms over naive single cell methods, and furthermore, our approximation framework lead to algorithms that are numerically more stable.