Optimal Rates for Regularized Conditional Mean Embedding Learning
Authors: Zhu Li, Dimitri Meunier, Mattes Mollenhauer, Arthur Gretton
NeurIPS 2022 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | We address the consistency of a kernel ridge regression estimate of the conditional mean embedding (CME), which is an embedding of the conditional distribution of Y given X into a target reproducing kernel Hilbert space HY . The CME allows us to take conditional expectations of target RKHS functions, and has been employed in nonparametric causal and Bayesian inference. We address the misspecified setting, where the target CME is in the space of Hilbert-Schmidt operators acting from an input interpolation space between HX and L2, to HY . This space of operators is shown to be isomorphic to a newly defined vector-valued interpolation space. Using this isomorphism, we derive a novel and adaptive statistical learning rate for the empirical CME estimator under the misspecified setting. Our analysis reveals that our rates match the optimal O(log n/n) rates without assuming HY to be finite dimensional. We further establish a lower bound on the learning rate, which shows that the obtained upper bound is optimal. |
| Researcher Affiliation | Academia | Zhu Li Gatsby Computational Neuroscience Unit University College London zhu.li@ucl.ac.uk Dimitri Meunier Gatsby Computational Neuroscience Unit University College London dimitri.meunier.21@ucl.ac.uk Mattes Mollenhauer Department of Mathematics and Computer Science Freie Universität Berlin mattes.mollenhauer@fu-berlin.de Arthur Gretton Gatsby Computational Neuroscience Unit University College London arthur.gretton@gmail.com |
| Pseudocode | No | The paper contains mathematical formulations and theoretical results (Theorems), but does not include any explicitly labeled pseudocode or algorithm blocks. |
| Open Source Code | No | The Ethics Review section states: "Did you include the code, data, and instructions needed to reproduce the main experimental results? [N/A]" indicating no code is provided. |
| Open Datasets | No | The paper is theoretical and does not conduct empirical studies with datasets. The Ethics Review section states: "Did you include the code, data, and instructions needed to reproduce the main experimental results? [N/A]" |
| Dataset Splits | No | The paper is theoretical and does not conduct empirical studies with datasets. The Ethics Review section states: "Did you include the code, data, and instructions needed to reproduce the main experimental results? [N/A]" |
| Hardware Specification | No | The paper is theoretical and does not describe any experimental setup or the hardware used to run experiments. |
| Software Dependencies | No | The paper is theoretical and does not describe any specific software dependencies with version numbers for experimental reproducibility. |
| Experiment Setup | No | The paper is theoretical and focuses on mathematical derivations and proofs. It does not describe any experimental setup details, hyperparameters, or training configurations. |