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GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Function

Brian M Cho, Raaz Dwivedi, Nathan Kallus

TL;DR

GAAVI develops nonparametric, asymptotic anytime-valid tests for global null hypotheses on the conditional mean function (CMF) and CATE under general covariate spaces. It constructs a weighted, Gaussian-mixture martingale with a time-uniform lower bound to enable early, error-controlled rejection and inverts these tests to form function-valued confidence sequences. Theoretical guarantees include asymptotic AV type I error control, power-one under mild assumptions, and asymptotically optimal sample complexities relative to Gaussian shift models, even with nuisance estimators that do not converge. Empirical results on synthetic data and a real-world randomized trial demonstrate robust error control and competitive power under continuous monitoring, outperforming discretized or linear-projection baselines in nonlinear settings. The framework enables timely, high-confidence decisions in online experiments, algorithmic auditing, and fairness assessments while accommodating flexible, model-agnostic regression methods.

Abstract

Inference on the conditional mean function (CMF) is central to tasks from adaptive experimentation to optimal treatment assignment and algorithmic fairness auditing. In this work, we provide a novel asymptotic anytime-valid test for a CMF global null (e.g., that all conditional means are zero) and contrasts between CMFs, enabling experimenters to make high confidence decisions at any time during the experiment beyond a minimum sample size. We provide mild conditions under which our tests achieve (i) asymptotic type-I error guarantees, (i) power one, and, unlike past tests, (iii) optimal sample complexity relative to a Gaussian location testing. By inverting our tests, we show how to construct function-valued asymptotic confidence sequences for the CMF and contrasts thereof. Experiments on both synthetic and real-world data show our method is well-powered across various distributions while preserving the nominal error rate under continuous monitoring.

GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Function

TL;DR

GAAVI develops nonparametric, asymptotic anytime-valid tests for global null hypotheses on the conditional mean function (CMF) and CATE under general covariate spaces. It constructs a weighted, Gaussian-mixture martingale with a time-uniform lower bound to enable early, error-controlled rejection and inverts these tests to form function-valued confidence sequences. Theoretical guarantees include asymptotic AV type I error control, power-one under mild assumptions, and asymptotically optimal sample complexities relative to Gaussian shift models, even with nuisance estimators that do not converge. Empirical results on synthetic data and a real-world randomized trial demonstrate robust error control and competitive power under continuous monitoring, outperforming discretized or linear-projection baselines in nonlinear settings. The framework enables timely, high-confidence decisions in online experiments, algorithmic auditing, and fairness assessments while accommodating flexible, model-agnostic regression methods.

Abstract

Inference on the conditional mean function (CMF) is central to tasks from adaptive experimentation to optimal treatment assignment and algorithmic fairness auditing. In this work, we provide a novel asymptotic anytime-valid test for a CMF global null (e.g., that all conditional means are zero) and contrasts between CMFs, enabling experimenters to make high confidence decisions at any time during the experiment beyond a minimum sample size. We provide mild conditions under which our tests achieve (i) asymptotic type-I error guarantees, (i) power one, and, unlike past tests, (iii) optimal sample complexity relative to a Gaussian location testing. By inverting our tests, we show how to construct function-valued asymptotic confidence sequences for the CMF and contrasts thereof. Experiments on both synthetic and real-world data show our method is well-powered across various distributions while preserving the nominal error rate under continuous monitoring.
Paper Structure (51 sections, 8 theorems, 98 equations, 7 figures, 1 table)

This paper contains 51 sections, 8 theorems, 98 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Setup and Preliminaries
  4. Problem Statement
  5. Testing the Conditional Mean Function
  6. Predictable Weights for GAAVI
  7. Variance Estimation for Sequential Testing
  8. Generalization to Confidence Sequences
  9. Theoretical Guarantees
  10. Tests of Power One
  11. Achieving Optimal Sample Complexities
  12. Experiments
  13. Setup.
  14. Discussion of Results.
  15. Conclusions and Future Directions
  16. ...and 36 more sections

Key Result

Theorem 4.2

Let Assumptions assump:pos_var- assump:strict_pos and assump:bounded_nuisances hold. Then, for fixed error tolerance $\alpha \in (0,1)$, $\rho > 0$, conditional variance threshold $l > 0$ (as defined in Equation eq:cond_var_eq), and weight bounds $\epsilon_t = \Omega(t^{-\gamma})$ for $\gamma \in for all distributions $P \in \mathcal{H}(f)$ and $f \in L_\infty(B)$. Likewise, under the same cond

Figures (7)

  • Figure 1: Visualization of our method for one instance with $t_0 = 200$ (grey region). The red dotted line indicates time of rejection.
  • Figure 2: Scatterplot and conditional means (red lines) of $Y$ with respect to latent variable $\zeta$ for each of our synthetic examples.
  • Figure 3: Cumulative density functions for the first time of rejection $N_f$. Shaded region denotes pointwise 95% confidence interval.
  • Figure 4: Plot of $\rho_{\text{1-sided}}$ for Differing values of $t^*$, $\alpha = 0.1$.
  • Figure 5: Enlarged version of Figure \ref{['fig:synthetic_examples']} in Section \ref{['sec:experiments']}. Scatterplot and conditional means (red lines) of $Y$ with respect to latent variable $\zeta$ for each of our synthetic examples. Blue points denote 10,000 samples for each distribution.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 2.4: Asymptotic Anytime Validity
  • Remark 3.1: On Thresholds and Variance Weighing
  • Remark 3.2: Comparisons with Sample Variance
  • Remark 3.3: Choice of Regression Functions
  • Theorem 4.2: Asymptotic Anytime Validity of $\xi_t$
  • Theorem 4.4: Test of Power 1
  • Theorem 4.7: Sample Complexities Bounds
  • Remark 4.8: Optimality Bound for Gaussian Testing
  • Definition 2.3: Anytime Valid Testing
  • Definition 2.4: Test of Power One
  • ...and 5 more