# Approximate Gradient Descent Convergence Dynamics for Adaptive Control on Heterogeneous Networks

## TL;DR

This paper studies the convergence of adaptive control algorithms on complex networks, revealing limitations of standard methods and proposing rescaling techniques that improve system optimality, supported by theoretical analysis and extensive simulations.

## Contribution

It introduces a rescaling approach to enhance backpressure algorithms for heterogeneous networks, with detailed convergence analysis and validation on synthetic and real-world network models.

## Key findings

- Rescaling improves system optimality by a factor of up to O(k).
- Theoretical convergence properties are established for generalized backpressure algorithms.
- Simulations confirm the theoretical results on synthetic and Manhattan grid networks.

## Abstract

Adaptive control is a classical control method for complex cyber-physical systems, including transportation networks. In this work, we analyze the convergence properties of such methods on exemplar graphs, both theoretically and numerically. We first illustrate a limitation of the standard backpressure algorithm for scheduling optimization, and prove that a re-scaling of the model state can lead to an improvement in the overall system optimality by a factor of at most $\mathcal{O}(k)$ depending on the network parameters, where $k$ characterizes the network heterogeneity. We exhaustively describe the associated transient and steady-state regimes, and derive convergence properties within this generalized class of backpressure algorithms. Extensive simulations are conducted on both a synthetic network and on a more realistic large-scale network modeled on the Manhattan grid on which theoretical results are verified.