Scale-free network optimization: foundations and algorithms

TL;DR

This paper introduces a new theoretical framework for scalable network optimization algorithms that leverage the decay of correlation in network systems, enabling dimension-free solutions beyond traditional methods.

Contribution

It develops a novel sensitivity-based correlation concept and designs algorithms exploiting exponential decay, advancing the theory of local and scalable network optimization.

Findings

Correlation decays exponentially with network distance

Algorithms exploit decay for dimension-free optimization

First to extend sensitivity analysis beyond infinitesimal perturbations

Abstract

We investigate the fundamental principles that drive the development of scalable algorithms for network optimization. Despite the significant amount of work on parallel and decentralized algorithms in the optimization community, the methods that have been proposed typically rely on strict separability assumptions for objective function and constraints. Beside sparsity, these methods typically do not exploit the strength of the interaction between variables in the system. We propose a notion of correlation in constrained optimization that is based on the sensitivity of the optimal solution upon perturbations of the constraints. We develop a general theory of sensitivity of optimizers the extends beyond the infinitesimal setting. We present instances in network optimization where the correlation decays exponentially fast with respect to the natural distance in the network, and we design algorithms that can exploit this decay to yield dimension-free optimization. Our results are the first of their kind, and open new possibilities in the theory of local algorithms.