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Kernel conditional tests from learning-theoretic bounds
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Turns time-uniform confidence bounds for vector-valued kernel ridge regression into finite-sample conditional two-sample tests of functionals of conditional distributions, with bootstrapped thresholds.
Problem
Conditional hypothesis testing at arbitrary covariates with finite-sample guarantees on the covariate rejection region, for non-iid (e.g., online, dependent) data and functionals of conditional distributions such as moments or full distributions.
Approach
Transform confidence bounds of a learning method (KRR) into a conditional test by comparing predictions of two learned models against the sum of their bounds; extend KRR confidence bounds to infinite-dimensional outputs with non-trace-class kernels via uniform-block-diagonal kernels, and embed distributions via kernel mean embeddings so that functionals reduce to conditional expectations.
Key results
- Bootstrapped thresholds (naive and wild) keep type I error at or below the prescribed level alpha while giving substantially higher power than the analytical bounds, which are very conservative.
- The local test achieves lower type II error than the aggregated baseline of Singh et al. when the functions differ only in rarely sampled covariate regions (e.g., lower type II error for small mixture weight theta).
- In process monitoring of a linear dynamical system, the test detects the dynamics change reliably, even before the sliding window is fully filled with perturbed data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use bootstrapped rather than analytical thresholds | KRR conditional test · Controlled type I and type II error experiments | type II error | Helped |
| Use analytical rather than bootstrapped thresholds (ablation) | KRR conditional test · Controlled type I and type II error experiments | type I error calibration | Hurt |
| Use the local conditional test rather than the aggregated baseline | Conditional test · Rarely sampled difference region | type II error | Helped |
| Increase the difference between conditional expectations | KRR conditional test · Empirical type II error experiments | type II error | Helped |
| Use a Gaussian output kernel instead of a linear output kernel (ablation) | KRR functional test · Different noise distributions with equal means | positive rate | Helped |
| Use a linear output kernel instead of a Gaussian output kernel (ablation) | KRR functional test · Different noise distributions with equal means | positive rate | Hurt |
| Use a richer output kernel, such as a Gaussian kernel with small bandwidth, rather than a kernel tailored to the mean (ablation) | KRR functional test · Conditional distributions differing by their means | type II error | Hurt |
| Increase the data set size (ablation) | KRR conditional test · Bootstrap validation | power | Helped |
| Decrease the regularization parameter (ablation) | KRR conditional test · Bootstrap validation | power | Helped |
| Increase the regularization parameter (ablation) | KRR conditional test · Bootstrap validation | power | Hurt |
| Increase the perturbation magnitude (ablation) | KRR process-monitoring test · Linear dynamical system change detection | detection | Helped |
| Increase the state dimension (ablation) | KRR process-monitoring test · Linear dynamical system change detection | detection performance | Hurt |
Surprising findings
- Rich kernels (e.g., Gaussian with small bandwidth) are less efficient than linear kernels at detecting mean differences, since they see finer distributional differences.
- Analytical thresholds are very conservative, so the bootstrapped thresholds are needed for practical power.
Limitations
- Analytical thresholds depend on hard-to-infer quantities (RKHS norm bound, subgaussianity operator) and are conservative.
- Bootstrap schemes are heuristic; no guarantee is given, and time-uniform guarantees are not preserved by bootstrapping.
- Type I calibration is only checked on a finite grid of covariates.
Experimental setup
Metrics: type I error type II error positive rate
Baselines: Singh et al. conditional two-sample test (kernel-based, density-estimator calibr
Builds on: Gaussian process posterior variance bounds vector-valued kernel ridge regression confidence bounds wild bootstrap
Closest papers by meaning
- Kernel Regression in Structured Non-IID Settings: Theory and Implications for Denoising Score Learning (NeurIPS 2025 · similarity 0.89)
- On the Hardness of Conditional Independence Testing In Practice (NeurIPS 2025 · similarity 0.88)
- A Comprehensive Analysis on the Learning Curve in Kernel Ridge Regression (NeurIPS 2024 · similarity 0.88)
- Towards Theoretical Understanding of Learning Large-scale Dependent Data via Random Features (ICML 2024 · similarity 0.88)
- Characterizing Overfitting in Kernel Ridgeless Regression Through the Eigenspectrum (ICML 2024 · similarity 0.87)
- On the Consistency of Kernel Methods with Dependent Observations (ICML 2024 · similarity 0.87)
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