Approximation with Random Shallow ReLU Networks for Model Reference Adaptive Control

Core Concepts

Random Shallow ReLU Networks can efficiently approximate functions for adaptive control applications.

Abstract

Neural networks with single hidden layers are commonly used in adaptive control. Approximation properties for control with neural networks are assumed but not proven. Lamperski and Lekang aim to show that ReLU networks with random weights achieve accurate approximations. The paper introduces a new integral representation theorem for ReLU activations and smooth functions. The results can be applied to construct neural networks for model reference adaptive control problems. Theoretical properties of neural networks with random initializations have been extensively studied. The paper addresses the gap in proving the required approximation properties for adaptive control. Theoretical challenges include quantifying the effects of smoothness in high dimensions and relaxing smoothness requirements.

Stats

"ReLU networks with randomly generated weights and biases achieve L8 error of Opm´1{2q with high probability." "The worst-case error on balls around the origin decays like Opm´1{2q, where m is the number of neurons." "The bounds simplify for the uniform distribution over Sn´1 ˆ r´R, Rs."

Quotes

"Neural networks are regularly employed in adaptive control of nonlinear systems and related methods of reinforcement learning." "The main contribution of this paper shows that two-layer neural networks with ReLU activation functions can approximate sufficiently smooth functions on bounded sets to arbitrary accuracy."

Key Insights Distilled From

Approximation with Random Shallow ReLU Networks with Applications to Model Reference Adaptive Control

by Andrew Lampe... at arxiv.org 03-27-2024

https://arxiv.org/pdf/2403.17142.pdf

Deeper Inquiries

How can the results of this study be applied to real-world adaptive control systems

The results of this study can be applied to real-world adaptive control systems by providing a theoretical foundation for using random shallow ReLU networks for function approximation. In adaptive control systems, where nonlinear dynamics are common, neural networks can be used to model unknown nonlinearities and approximate value functions. By showing that ReLU networks with randomly generated weights and biases can achieve a desired level of approximation accuracy, this study fills a crucial gap in the current use of neural networks in adaptive control. The results provide a framework for constructing neural network approximators that can be used in model reference adaptive control applications. By leveraging the approximation properties of random shallow ReLU networks, adaptive control systems can achieve better performance and accuracy in tracking and controlling nonlinear systems.

What are the potential limitations of using random shallow ReLU networks for approximation in control systems

One potential limitation of using random shallow ReLU networks for approximation in control systems is the need for a large number of neurons to achieve the desired level of accuracy. While the study shows that these networks can approximate sufficiently smooth functions on bounded sets to arbitrary accuracy, the number of neurons required may be high, especially for complex functions or high-dimensional spaces. This can lead to increased computational complexity and training time, making it challenging to implement these networks in real-time control systems. Additionally, the reliance on random initialization for weights and biases may introduce variability in the performance of the networks, leading to potential inconsistencies in control outcomes. Ensuring robustness and stability in control systems when using random shallow ReLU networks for approximation is crucial to address these limitations.

How can the concept of integral representations in neural networks be extended to other areas of machine learning and artificial intelligence

The concept of integral representations in neural networks can be extended to other areas of machine learning and artificial intelligence to enhance the understanding and interpretability of neural network models. By deriving integral representations for activation functions and smooth functions over bounded domains, researchers can gain insights into the inner workings of neural networks and how they approximate complex functions. This approach can be extended to analyze the behavior of neural networks in different tasks such as classification, regression, and reinforcement learning. By exploring integral representations in neural networks, researchers can develop new theoretical frameworks, optimization techniques, and interpretability methods for improving the performance and reliability of neural network models across various applications in machine learning and artificial intelligence.