Early Stopping of Untrained Convolutional Neural Networks

TL;DR

This paper proves that early stopping based on the discrepancy principle ensures optimal convergence for untrained convolutional neural networks used in solving linear ill-posed problems, supported by numerical experiments.

Contribution

It provides a rigorous theoretical analysis of early stopping for untrained CNNs, showing the discrepancy principle guarantees minimax optimal convergence rates.

Findings

Discrepancy principle effectively determines early stopping.

Untrained CNNs can approximate solutions with optimal convergence.

Numerical results confirm theoretical predictions.

Abstract

In recent years, new regularization methods based on (deep) neural networks have shown very promising empirical performance for the numerical solution of ill-posed problems, e.g., in medical imaging and imaging science. Due to the nonlinearity of neural networks, these methods often lack satisfactory theoretical justification. In this work, we rigorously discuss the convergence of a successful unsupervised approach that utilizes untrained convolutional neural networks to represent solutions to linear ill-posed problems. Untrained neural networks are particularly appealing for many applications because they do not require paired training data. The regularization property of the approach relies solely on the architecture of the neural network instead. Due to the vast over-parameterization of the employed neural network, suitable early stopping is essential for the success of the method. We establish that the classical discrepancy principle is an adequate method for early stopping of two-layer untrained convolutional neural networks learned by gradient descent, and furthermore, it yields an approximation with minimax optimal convergence rates. Numerical results are also presented to illustrate the theoretical findings.