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

Technical Summary: GAAVI (Global Asymptotic Anytime Valid Inference)

GAAVI is a novel statistical framework designed for performing asymptotic anytime-valid inference on the Conditional Mean Function (CMF) and the Conditional Average Treatment Effect (CATE). It addresses the challenges of continuous monitoring in complex, high-dimensional, or continuous covariate spaces.

1. The Problem: Sequential Inference in Continuous Spaces

In modern data science (e.g., clinical trials, algorithmic fairness, and online platforms), researchers often need to monitor outcomes as data arrives. Traditional inference methods face two major limitations:

• Fixed Sample Size Constraints: Most tests require a pre-specified sample size, making them unsuitable for "early stopping" or real-time decision-making.
• The Curse of Dimensionality/Continuity: Existing sequential tests for CMFs often rely on discretization (binning) of the covariate space or parametric projections (e.g., linear regression). These methods lose power when the true relationship is non-linear or when the covariate space is high-dimensional, as they only test a simplified version of the function.

The goal of GAAVI is to provide a nonparametric test that can be applied at any time $t$ (after a minimum "burn-in" period $t_0$) while maintaining rigorous error control.

2. Methodology: Weighted Martingales and Predictable Weights

The core of GAAVI is the construction of a weighted martingale process that tracks the "drift" of the conditional mean.

• The Test Statistic: The authors construct a process $\psi_t(f)$ by taking a weighted sum of the differences between observed outcomes (adjusted by a regression estimate) and a hypothesized null function $f(x)$.
• Predictable Weights: Unlike standard tests that use fixed weights, GAAVI uses predictable weights $w_i(X_i, f)$. These weights are designed to be "directionally aware": they use current estimates of the CMF ($\hat{\tau}_i$) and conditional variance ($\hat{v}_i$) to assign more weight to observations that suggest the null hypothesis is false. This maximizes the "drift" of the martingale when the null is violated.
• Asymptotic Anytime Validity: Instead of requiring exact, non-asymptotic error control (which is mathematically restrictive in nonparametric settings), GAAVI uses asymptotic anytime-validity. This means that for a sufficiently large burn-in time $t_0$, the probability of a Type-I error (false rejection) is bounded by $\alpha$ across all future time steps.
• Gaussian Mixture Martingale: To derive the rejection boundaries, the authors leverage bounds based on Gaussian mixture martingales, which provide tight, time-uniform lower bounds for the average conditional mean.

3. Key Contributions

1. Nonparametric Flexibility: The framework accommodates any flexible, machine-learning-based regressor (e.g., Neural Networks, Random Forests, k-NN) to estimate the CMF and conditional variance.
2. Robust Error Control: The authors prove that the test maintains asymptotic Type-I error guarantees even if the nuisance parameters (the regression models) do not converge, provided they remain bounded.
3. Optimal Sample Complexity: They prove that under consistency of the regressors, the test achieves the optimal sample complexity relative to a parametric Gaussian location testing model.
4. Function-Valued Confidence Sequences: By inverting the test, the method constructs confidence sequences for the entire CMF/CATE function, allowing for continuous monitoring of the function's shape.

4. Key Results and Theoretical Guarantees

• Theorem 4.2 (Validity): Establishes that the test $\xi_t$ is an $\alpha$-level asymptotic anytime-valid test.
• Theorem 4.4 (Power): Proves the test has power one, meaning it is guaranteed to reject the null hypothesis in finite time if the true CMF differs from the null.
• Theorem 4.7 (Optimality): Shows that the expected number of samples required to reject a false null ($N_f$) scales with $\log(1/\alpha)$ at a rate that matches the theoretical lower bound for Gaussian models.
• Empirical Performance: Experiments on synthetic data (step, bump, and sine functions) and real-world data (French jobs RCT) demonstrate that GAAVI outperforms linear F-tests and binning-based methods, particularly when the signal is non-linear or localized.

5. Significance

GAAVI bridges the gap between high-performance machine learning and rigorous sequential statistics. It allows practitioners to use "black-box" models (like Deep Learning) to perform hypothesis testing and confidence interval construction in real-time. This is critical for:

• Safety-critical applications: Stopping clinical trials early if a subgroup is experiencing adverse effects.
• Algorithmic Auditing: Monitoring fairness metrics in real-time to halt biased model deployments.
• Adaptive Experimentation: Making high-confidence decisions in online A/B testing without the risks associated with "peeking" at data.