mlsanityData to 14 Oct 2026

Optimal Rates for Regularized Conditional Mean Embedding Learning

arXiv 2208.01711 ↗First posted 2 Aug 2022stat.ML cs.LGcited by 50

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Abstract (arXiv)

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.

Key facts. Cited by 50 papers in the map (data through 14 Oct 2026). Builds on 14 of its 48 references that are in the map. First posted on arXiv 2 Aug 2022.

Builds on

14 of 48 references.

References · 14

  1. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 364
  2. 2018Gaussian Processes and Kernel Methods: A Review on Connections and Equivalencescited by 192
  3. 2019Kernel Instrumental Variable Regressioncited by 118
  4. 2017Sobolev Norm Learning Rates for Regularized Least-Squares Algorithmcited by 114
  5. 2020A Measure-Theoretic Approach to Kernel Conditional Mean Embeddingscited by 86
  6. 2017On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operatorcited by 73
  7. 2016Optimal Rates For Regularization Of Statistical Inverse Learning Problemscited by 68
  8. 2018Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spacescited by 59
  9. 2016A Consistent Regularization Approach for Structured Predictioncited by 33
  10. 2019A Rigorous Theory of Conditional Mean Embeddingscited by 31
  11. 2018Causal Inference via Kernel Deviance Measurescited by 22
  12. 2020Nonparametric approximation of conditional expectation operatorscited by 21
2 more
  1. 2020The linear conditional expectation in Hilbert spacecited by 8
  2. 2021Sobolev Norm Learning Rates for Conditional Mean Embeddingscited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 3; 2023: 13; 2024: 15; 2025: 14; 2026: 5.

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  1. 2023Sharp Spectral Rates for Koopman Operator Learningcited by 36
  2. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 28
  3. 2023On the Optimality of Misspecified Spectral Algorithmscited by 28
  4. 2023Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithmcited by 22
  5. 2023On the Optimality of Misspecified Kernel Ridge Regressioncited by 20
  6. 2022Efficient Conditionally Invariant Representation Learningcited by 19
  7. 2023Learning invariant representations of time-homogeneous stochastic dynamical systemscited by 17
  8. 2023Estimating Koopman operators with sketching to provably learn large scale dynamical systemscited by 15
  9. 2024Practical Kernel Tests of Conditional Independencecited by 15
  10. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  11. 2024Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithmscited by 12
  12. 2024Spectral Representation for Causal Estimation with Hidden Confounderscited by 12
38 more
  1. 2023An Efficient Doubly-Robust Test for the Kernel Treatment Effectcited by 10
  2. 2023Consistent Long-Term Forecasting of Ergodic Dynamical Systemscited by 10
  3. 2024Data-Driven Distributionally Robust Safety Verification Using Barrier Certificates and Conditional Mean Embeddingscited by 9
  4. 2024Learning the Infinitesimal Generator of Stochastic Diffusion Processescited by 9
  5. 2024Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimalcited by 8
  6. 2024Nonparametric Control Koopman Operatorscited by 7
  7. 2024Operator World Models for Reinforcement Learningcited by 7
  8. 2024Doubly Robust Conditional Independence Testing with Generative Neural Networkscited by 7
  9. 2024Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroupscited by 7
  10. 2024An Overview of Causal Inference using Kernel Embeddingscited by 6
  11. 2025Koopman-Equivariant Gaussian Processescited by 6
  12. 2023Kernel Single Proxy Control for Deterministic Confoundingcited by 5
  13. 2024On the Consistency of Kernel Methods with Dependent Observationscited by 4
  14. 2025Learning Curves of Stochastic Gradient Descent in Kernel Regressioncited by 4
  15. 2026One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operatorscited by 4
  16. 2025Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regressioncited by 3
  17. 2025Regularized least squares learning with heavy-tailed noise is minimax optimalcited by 3
  18. 2025Doubly-Robust Estimation of Counterfactual Policy Mean Embeddingscited by 3
  19. 2025Spectral Algorithms in Misspecified Regression: Convergence under Covariate Shiftcited by 3
  20. 2024Spectrally Transformed Kernel Regressioncited by 2
  21. 2024Nonparametric Sparse Online Learning of the Koopman Operatorcited by 2
  22. 2025All Models Are Miscalibrated, But Some Less So: Comparing Calibration with Conditional Mean Operatorscited by 2
  23. 2025Kernel Treatment Effects with Adaptively Collected Datacited by 2
  24. 2025LUCID: Learning-Enabled Uncertainty-Aware Certification of Stochastic Dynamical Systemscited by 2
  25. 2023Physics-Informed Kernel Embeddings: Integrating Prior System Knowledge with Data-Driven Controlcited by 1
  26. 2023Conditional expectation using compactification operatorscited by 1
  27. 2024Kernel-Based Optimal Control: An Infinitesimal Generator Approachcited by 1
  28. 2025Density Ratio-Free Doubly Robust Proxy Causal Learningcited by 1
  29. 2025kNNSampler: Stochastic Imputations for Recovering Missing Value Distributionscited by 1
  30. 2026Estimating Aleatoric Uncertainty in the Causal Treatment Effectcited by 1
  31. 2023Returning The Favour: When Regression Benefits From Probabilistic Causal Knowledgecited by 0
  32. 2023Propagating Kernel Ambiguity Sets in Nonlinear Data-driven Dynamics Modelscited by 0
  33. 2025Nonparametric Sparse Online Learning of the Koopman Operatorcited by 0
  34. 2025Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learningcited by 0
  35. 2025Kernel conditional tests from learning-theoretic boundscited by 0
  36. 2026Kernel-Based Learning of Safety Barrierscited by 0
  37. 2026Safety Certification is Classificationcited by 0
  38. 2026Sequential Kernel-based Conditional Independence Testing via Adaptive Bettingcited by 0