# Kernel conditional tests from learning-theoretic bounds

arXiv: https://arxiv.org/abs/2506.03898 · HTML: https://mlsanity.com/paper/2506.03898/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 45 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Builds on (most cited first, 21 of 21)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Nonparametric Two-Sample Testing by Betting](https://mlsanity.com/paper/2112.09162/) (2021)
- [Learning with little mixing](https://mlsanity.com/paper/2206.08269/) (2022)
- [On the Sublinear Regret of GP-UCB](https://mlsanity.com/paper/2307.07539/) (2023)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [Sequential Predictive Two-Sample and Independence Testing](https://mlsanity.com/paper/2305.00143/) (2023)
- [A Two-Sample Conditional Distribution Test Using Conformal Prediction and Weighted Rank Sum](https://mlsanity.com/paper/2010.07147/) (2020)
- [No-regret Algorithms for Multi-task Bayesian Optimization](https://mlsanity.com/paper/2008.08885/) (2020)
- [On Safety in Safe Bayesian Optimization](https://mlsanity.com/paper/2403.12948/) (2024)
- [Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testing](https://mlsanity.com/paper/2210.08149/) (2022)
- [One or Two Things We know about Concept Drift -- A Survey on Monitoring Evolving Environments](https://mlsanity.com/paper/2310.15826/) (2023)
- [A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)](https://mlsanity.com/paper/2407.16550/) (2024)
- [Kernel Ridge Regression Inference](https://mlsanity.com/paper/2302.06578/) (2023)
- [On the Consistency of Kernel Methods with Dependent Observations](https://mlsanity.com/paper/2406.06101/) (2024)
- [General Frameworks for Conditional Two-Sample Testing](https://mlsanity.com/paper/2410.16636/) (2024)
- [A Bayesian nonparametric test for conditional independence](https://mlsanity.com/paper/1910.11219/) (2019)

## Abstract (arXiv)

We propose a framework for hypothesis testing on conditional probability distributions, which we then use to construct statistical tests of functionals of conditional distributions. These tests identify the inputs where the functionals differ with high probability, and include tests of conditional moments or two-sample tests. Our key idea is to transform confidence bounds of a learning method into a test of conditional expectations. We instantiate this principle for kernel ridge regression (KRR) with subgaussian noise. An intermediate data embedding then enables more general tests -- including conditional two-sample tests -- via kernel mean embeddings of distributions. To have guarantees in this setting, we generalize existing pointwise-in-time or time-uniform confidence bounds for KRR to previously-inaccessible yet essential cases such as infinite-dimensional outputs with non-trace-class kernels. These bounds also circumvent the need for independent data, allowing for instance online sampling. To make our tests readily applicable in practice, we introduce bootstrapping schemes leveraging the parametric form of testing thresholds identified in theory to avoid tuning inaccessible parameters. We illustrate the tests on examples, including one in process monitoring and comparison of dynamical systems. Overall, our results establish a comprehensive foundation for conditional testing on functionals, from theoretical guarantees to an algorithmic implementation, and advance the state of the art on confidence bounds for vector-valued least squares estimation.