High-dimensional classification problems with Barron regular boundaries under margin conditions
This work addresses high-dimensional classification problems, such as image classification on MNIST, by providing theoretical guarantees for neural network approximation, though it appears incremental in extending existing theory to specific conditions.
The paper tackles the problem of approximating classifiers with Barron-regular decision boundaries using ReLU neural networks, achieving high polynomial approximation rates under margin conditions and demonstrating fast learning bounds close to n^{-1}.
We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong margin conditions, high-dimensional discontinuous classifiers can be approximated with a rate that is typically only achievable when approximating a low-dimensional smooth function. We demonstrate how these expression rate bounds imply fast-rate learning bounds that are close to $n^{-1}$ where $n$ is the number of samples. In addition, we carry out comprehensive numerical experimentation on binary classification problems with various margins. We study three different dimensions, with the highest dimensional problem corresponding to images from the MNIST data set.