Neural Networks Trained by Weight Permutation are Universal Approximators

TL;DR

This paper proves that neural networks trained by weight permutation can universally approximate functions, offering a new perspective on neural network training and learning behavior.

Contribution

It provides the first theoretical guarantee for permutation-based training methods, demonstrating their ability to approximate continuous functions.

Findings

Permutation training guides ReLU networks to approximate functions.

Numerical results validate efficiency across various initializations.

Permutation training reveals insights into network learning dynamics.

Abstract

The universal approximation property is fundamental to the success of neural networks, and has traditionally been achieved by training networks without any constraints on their parameters. However, recent experimental research proposed a novel permutation-based training method, which exhibited a desired classification performance without modifying the exact weight values. In this paper, we provide a theoretical guarantee of this permutation training method by proving its ability to guide a ReLU network to approximate one-dimensional continuous functions. Our numerical results further validate this method's efficiency in regression tasks with various initializations. The notable observations during weight permutation suggest that permutation training can provide an innovative tool for describing network learning behavior.