Minimax Optimal Kernel Operator Learning via Multilevel Training
Authors: Jikai Jin, Yiping Lu, Jose Blanchet, Lexing Ying
ICLR 2023 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | In this paper, we study the statistical limit of learning a Hilbert Schmidt operator between two infinite-dimensional Sobolev reproducing kernel Hilbert spaces (RKHSs). We establish the information-theoretic lower bound in terms of the Sobolev Hilbert-Schmidt norm and show that a regularization that learns the spectral components below the bias contour and ignores the ones above the variance contour can achieve the optimal learning rate. At the same time, the spectral components between the bias and variance contours give us flexibility in designing computationally feasible machine learning algorithms. Based on this observation, we develop a multilevel kernel operator learning algorithm that is optimal when learning linear operators between infinite-dimensional function spaces. |
| Researcher Affiliation | Academia | Jikai Jin School of Mathematical Sciences Peking University Beijing, China jkjin@pku.edu.cn Yiping Lu Institute for Computational & Mathematical Engineering Stanford University Stanford, CA, US yplu@stanford.edu Jose Blanchet Management Science and Engineering Stanford University Stanford, CA, US jose.blanchet@stanford.edu Lexing Ying Department of Mathematics Stanford University Stanford, CA, US lexing@stanford.edu |
| Pseudocode | No | The paper provides mathematical formulations of the proposed estimators (e.g., equations 3, 5, 22, 26) but does not include a distinct pseudocode block or algorithm description with step-by-step instructions. |
| Open Source Code | No | The paper does not contain any statements about providing open-source code for the described methodology, nor does it include links to any code repositories. |
| Open Datasets | No | The paper is theoretical and focuses on statistical limits and learning rates. It mentions |
| Dataset Splits | No | The paper is theoretical and does not describe any empirical experiments or dataset usage, thus no training/validation/test dataset splits are provided. |
| Hardware Specification | No | The paper is theoretical and does not describe any empirical experiments or the hardware used to run them. |
| Software Dependencies | No | The paper is theoretical and does not describe any empirical experiments or specific software dependencies with version numbers. |
| Experiment Setup | No | The paper is theoretical and does not describe any empirical experiments or their setup, including hyperparameters or system-level training settings. |