Efficient Adaptive Experimentation with Noncompliance

TL;DR

This work develops AMRIV, an Adaptive Multiply Robust estimator for estimating the population ATE in sequential experiments where treatment uptake is endogenous and only a binary instrument can be assigned. The authors derive the semiparametric efficiency bound under history-dependent instrument policies, identify the variance-aware optimal allocation π^*(X), and couple online policy learning with a sequential EIF-based estimator that attains the bound while maintaining multiply robust consistency. They prove asymptotic normality and convergence rates, along with time-uniform confidence sequences for sequential stopping. Empirical studies on synthetic and semi-synthetic data show that adaptive instrument assignment paired with AMRIV improves efficiency and robustness relative to baselines, enabling more reliable and timely causal inference in adaptive IV settings.

Abstract

We study the problem of estimating the average treatment effect (ATE) in adaptive experiments where treatment can only be encouraged -- rather than directly assigned -- via a binary instrumental variable. Building on semiparametric efficiency theory, we derive the efficiency bound for ATE estimation under arbitrary, history-dependent instrument-assignment policies, and show it is minimized by a variance-aware allocation rule that balances outcome noise and compliance variability. Leveraging this insight, we introduce AMRIV -- an Adaptive, Multiply-Robust estimator for Instrumental-Variable settings with variance-optimal assignment. AMRIV pairs (i) an online policy that adaptively approximates the optimal allocation with (ii) a sequential, influence-function-based estimator that attains the semiparametric efficiency bound while retaining multiply-robust consistency. We establish asymptotic normality, explicit convergence rates, and anytime-valid asymptotic confidence sequences that enable sequential inference. Finally, we demonstrate the practical effectiveness of our approach through empirical studies, showing that adaptive instrument assignment, when combined with the AMRIV estimator, yields improved efficiency and robustness compared to existing baselines.

Figures (4)

• Figure 1: Left: Adaptive experimentation with noncompliance. Blue elements denote learned or assigned quantities ($\pi_t$, $Z_t$), orange elements represent observed variables ($X_t$, $A_t$, $Y_t$), and the dashed red arrow indicates noncompliance ($A_t(Z_t) \ne Z_t$). Right: The adaptive policy yields faster confidence-sequence contraction, enabling earlier stopping.
• Figure 2: Optimal policy $\pi^*(X)$ as a function of compliance $\delta^A(X)$.
• Figure 3: Performance of different estimators across increasing sample size $T$. (a) Efficiency: Normalized MSE versus an oracle benchmark. (b) Consistency: MSE $\pm$ standard error. (c) Coverage: Empirical coverage of 95% confidence intervals.
• Figure 4: Performance of different estimators on TripAdvisor simulated data. (a) Efficiency: Normalized MSE versus an oracle benchmark. (b) Consistency: MSE $\pm$ standard error. (c) Coverage: Empirical coverage of 95% confidence intervals.

Theorems & Definitions (12)

• Remark 1: Interpretation under violations of Assumption \ref{['assum:compliance']}
• Theorem 1: Semiparametric Efficiency Bound
• Corollary 2: Optimal Instrument Assignment
• Remark 2: Choice of nuisance and variance estimators
• Theorem 3: Asymptotic Normality of the AMRIV Estimator
• Theorem 4: Convergence Rate of the AMRIV Estimator
• Corollary 5: Multiply Robust Consistency Guarantees
• Definition 1: Asymptotic time-uniform coverage dalal2024anytime
• Theorem 6: AsympCS for AMRIV
• Remark 3: Difference in convergence rates for fixed-time and anytime-valid inference.