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Kernel Treatment Effects with Adaptively Collected Data

Houssam Zenati, Bariscan Bozkurt, Arthur Gretton

TL;DR

This work develops the first kernel-based framework for distributional causal inference under adaptive data collection. It introduces kernel treatment effects (KTE) defined via counterfactual mean embeddings in an RKHS and derives a variance-stabilized, doubly robust estimator that admits a Hilbert-space martingale CLT under adaptivity. A sample-split stabilized test is proposed to yield valid Gaussian limits for testing equality of counterfactual distributions, with a practical, fold-based procedure for variance estimation and cross-fitting to handle nuisance components. Through synthetic, IHDP, and dSprite experiments, the method achieves calibrated type-I error and high power for both mean shifts and higher-moment distributional changes, outperforming mean-focused adaptive baselines. The results suggest a general strategy for distributional inference under adaptivity using variance-stabilized martingale arguments for RKHS-valued estimands.

Abstract

Adaptive experiments improve efficiency by adjusting treatment assignments based on past outcomes, but this adaptivity breaks the i.i.d. assumptions that underpins classical asymptotics. At the same time, many questions of interest are distributional, extending beyond average effects. Kernel treatment effects (KTE) provide a flexible framework by representing counterfactual outcome distributions in an RKHS and comparing them via kernel distances. We present the first kernel-based framework for distributional inference under adaptive data collection. Our method combines doubly robust scores with variance stabilization to ensure asymptotic normality via a Hilbert-space martingale CLT, and introduces a sample-fitted stabilized test with valid type-I error. Experiments show it is well calibrated and effective for both mean shifts and higher-moment differences, outperforming adaptive baselines limited to scalar effects.

Kernel Treatment Effects with Adaptively Collected Data

TL;DR

This work develops the first kernel-based framework for distributional causal inference under adaptive data collection. It introduces kernel treatment effects (KTE) defined via counterfactual mean embeddings in an RKHS and derives a variance-stabilized, doubly robust estimator that admits a Hilbert-space martingale CLT under adaptivity. A sample-split stabilized test is proposed to yield valid Gaussian limits for testing equality of counterfactual distributions, with a practical, fold-based procedure for variance estimation and cross-fitting to handle nuisance components. Through synthetic, IHDP, and dSprite experiments, the method achieves calibrated type-I error and high power for both mean shifts and higher-moment distributional changes, outperforming mean-focused adaptive baselines. The results suggest a general strategy for distributional inference under adaptivity using variance-stabilized martingale arguments for RKHS-valued estimands.

Abstract

Adaptive experiments improve efficiency by adjusting treatment assignments based on past outcomes, but this adaptivity breaks the i.i.d. assumptions that underpins classical asymptotics. At the same time, many questions of interest are distributional, extending beyond average effects. Kernel treatment effects (KTE) provide a flexible framework by representing counterfactual outcome distributions in an RKHS and comparing them via kernel distances. We present the first kernel-based framework for distributional inference under adaptive data collection. Our method combines doubly robust scores with variance stabilization to ensure asymptotic normality via a Hilbert-space martingale CLT, and introduces a sample-fitted stabilized test with valid type-I error. Experiments show it is well calibrated and effective for both mean shifts and higher-moment differences, outperforming adaptive baselines limited to scalar effects.

Paper Structure

This paper contains 91 sections, 11 theorems, 246 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem statement
  4. Adaptive data collection setting
  5. Target Parameter
  6. Failure of standard asymptotic normality under adaptivity
  7. Variance-stabilized estimator
  8. Stabilized estimator with plug-in weights
  9. Asymptotic normality guarantees
  10. Conditional Variance Estimation
  11. KTE Estimation: A Sample-Split Test
  12. Numerical Simulations
  13. Synthetic data.
  14. IHDP dataset
  15. dSprite dataset
  16. ...and 76 more sections

Key Result

Theorem 4.5

Under Assumptions assum:selection_observables, assumption:efficiency_bound, assumption:variance_estimation, assumption:bounded_kernel, and assum:nuisance_regularity, where $\mathcal{N}(0,\Gamma)$ is the centered Gaussian measure on $\mathcal{H}_{\mathcal{Y}}$ with covariance $\Gamma$ (see Appendix app:background, Theorem. thm:bosq_mclt) and $\Gamma=\lim_{T\to\infty}\frac{1}{T}\sum_{t=1}^T \mathop

Figures (11)

  • Figure 1: Histogram of the miscalibrated DR-xKTE statistic over $500$ runs ($T=700$, $d=5$, $t_0=15$, $\varepsilon=10^{-3}$) with true adaptive propensities $\pi_t(1 \mid X_t)$.
  • Figure 2: Illustration of $200$ simulations of VS-DR-KTE under the null in the adaptive setting with $T=1000$: (A) Histogram with KDE and standard normal pdf, (B) Normal Q-Q plot, (C) False positives against sample sizes. The results show approximate Gaussian behaviour and controlled type-I error.
  • Figure 3: True positive rates ($200$ simulations, Scenarios II–IV). Mean-focused baselines (CADR/AW-AIPW) achieve matching performance on II; VS-DR-KTE shows markedly higher power on III–IV (higher-moment shifts).
  • Figure 4: Observational samples from the dSprite data in Scenario IV
  • Figure 5: Counterfactual pairs from the dSprite data in Scenario IV
  • ...and 6 more figures

Theorems & Definitions (33)

  • Example 3.2: Contextual extension of bibaut2025demystifying, Section 3.1.1
  • Theorem 4.5: Asymptotic normality of the stabilized RKHS estimator
  • proof : Sketch of proof
  • Proposition 5.1: Pathwise consistency of the plug-in conditional variance weights
  • Theorem 6.1: Asymptotic normality of the cross-fitted stabilized test
  • proof : Sketch of proof
  • Remark 6.2: Consistency of the sample–split KTE
  • Lemma 10.1: Covariance perturbation inequality
  • proof
  • Remark 10.2: Variance identity
  • ...and 23 more