Sequential Kernelized Independence Testing

TL;DR

The paper tackles sequential independence testing for streaming data by introducing Sequential Kernel Independence Tests (SKIT) that combine the betting framework with kernel-based dependence measures. SKIT leverages HSIC, COCO, and KCC to build payoff functions and applies online wagering (e.g., Online Newton Step) to achieve level-$\alpha$ time-uniform false alarm control and consistency, even under non-i.i.d. drift. It also provides symmetry-based variants and minibatching to enhance practicality, including plug-in estimators for witness functions and scalable eigenproblem solutions. Through synthetic experiments and real-data applications (e.g., climate data, MNIST-inspired tasks), SKIT demonstrates adaptive stopping—requiring fewer samples on easy problems and more on hard ones—while maintaining rigorous error control and competitive power relative to batch alternatives. Overall, the framework enables scalable, anytime-valid, streaming independence testing in time-varying environments, with broad applicability to nonparametric settings.

Abstract

Independence testing is a classical statistical problem that has been extensively studied in the batch setting when one fixes the sample size before collecting data. However, practitioners often prefer procedures that adapt to the complexity of a problem at hand instead of setting sample size in advance. Ideally, such procedures should (a) stop earlier on easy tasks (and later on harder tasks), hence making better use of available resources, and (b) continuously monitor the data and efficiently incorporate statistical evidence after collecting new data, while controlling the false alarm rate. Classical batch tests are not tailored for streaming data: valid inference after data peeking requires correcting for multiple testing which results in low power. Following the principle of testing by betting, we design sequential kernelized independence tests that overcome such shortcomings. We exemplify our broad framework using bets inspired by kernelized dependence measures, e.g., the Hilbert-Schmidt independence criterion. Our test is also valid under non-i.i.d., time-varying settings. We demonstrate the power of our approaches on both simulated and real data.

Figures (16)

• Figure 1: Valid sequential independence tests for: $Y_t=X_t\beta+\varepsilon_t$, $X_t,\varepsilon_t\sim \mathcal{N}(0,1)$. Batch + $n$-step is batch HSIC with Bonferroni correction applied every $n$ steps (allowing monitoring only at those steps). Seq-MMD refers to the reduction to two-sample testing (Appendix \ref{['appsubsec:two_sample_reduction']}). Our test outperforms other tests.
• Figure 2: Under distribution drift \ref{['eq:dist_drift_example']}, SKIT controls type I error under $H_0$ and has high power under $H_1$. Batch HSIC fails to control type I error under $H_0$ (hence we do not plot its power).
• Figure 3: Rejection rate and scaled sample size used to reject the null hypothesis for synthetic data. Inspecting the rejection rate for $\beta=0$ (independence holds) confirms that the type I error is controlled. Further, we confirm that SKITs are adaptive to the complexity (smaller $\beta$ and larger $d$ correspond to harder settings).
• Figure 4: (a) SKITs with symmetry-based payoffs have high power under the Gaussian model. (b) SKIT with linear kernel has high power under the Gaussian model ($X$ and $Y$ are linearly correlated for $\beta\neq 0$), and its false alarm rate is controlled under the spherical model ($X$ and $Y$ are linearly uncorrelated but dependent).
• Figure 5: Solid lines connect cities for which the null is rejected. SKIT supports the conjecture regarding dependent temperature fluctuations in nearby locations.

Theorems & Definitions (38)

• Example 1
• Theorem 1
• Remark 1
• Example 2
• Theorem 2
• Proposition 1
• Remark 2: Minibatching
• Theorem 3
• Theorem 4
• Remark 3