Minimax optimality of convolutional neural networks for infinite dimensional input-output problems and separation from kernel methods

Yuto Nishimura, Taiji Suzuki

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Keywords: neural network, nonlinear operator learning, convolutional network, estimation error analysis

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2024/AuthorGuide.

Abstract:

Recent deep learning applications, exemplified by text-to-image tasks, often involve high-dimensional inputs and outputs. While several studies have investigated the function estimation capabilities of deep learning, research on dilated convolutional neural networks (CNNs) has mainly focused on cases where input dimensions are infinite but output dimensions are one-dimensional, similar to many other studies. However, many practical deep learning tasks involve high-dimensional (or even infinite dimensional) inputs and outputs.
In this paper, we investigate the optimality of dilated CNNs for estimating a map between infinite-dimensional input and output spaces
by analyzing their approximation and estimation abilities.
For that purpose, we first show that approximation and estimation errors depend only on the smoothness and decay rate with respect to the infinity norm of the output, and their estimation accuracy actually achieve the {\it minimax optimal} rate of convergence.
Second, we demonstrate that the dilated CNNs outperform {\it any} linear estimators including kernel ridge regression and $k$-NN estimators in a minimax error sense, highlighting the usefulness of feature learning realized by deep neural networks.
Our theoretical analysis particularly explains the success of deep learning in recent high-dimensional input-output tasks.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors' identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Primary Area: learning theory

Submission Number: 7702