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Sigmoid-FTRL: Design-Based Adaptive Neyman Allocation for AIPW Estimators

Fangyi Chen, Shu Ge, Jian Qian, Christopher Harshaw

TL;DR

This work tackles adaptive Neyman Allocation for design-based AIPW estimators with deterministic potential outcomes. It introduces Sigmoid-FTRL, a two-regret online design that uses a sigmoidal transformation to convert a non-convex Neyman regret problem into convex components, achieving the optimal $T^{-1/2}R^2$ rate and matching a corresponding lower bound. The authors establish a central limit theorem and a conservative variance estimator to enable Wald-type confidence intervals, and they show non-superefficiency under the design-based framework while characterizing the exact asymptotic variance. The paper highlights a fundamental distinction between design-based and super-population settings in adaptive allocation and provides a practical, provably efficient design for AIPW inference that scales with covariate norms and sample size.

Abstract

We consider the problem of Adaptive Neyman Allocation for the class of AIPW estimators in a design-based setting, where potential outcomes and covariates are deterministic. As each subject arrives, an adaptive procedure must select both a treatment assignment probability and a linear predictor to be used in the AIPW estimator. Our goal is to construct an adaptive procedure that minimizes the Neyman Regret, which is the difference between the variance of the adaptive procedure and an oracle variance which uses the optimal non-adaptive choice of assignment probability and linear predictors. While previous work has drawn insightful connections between Neyman Regret and online convex optimization for the Horvitz--Thompson estimator, one of the central challenges for AIPW estimator is that the underlying optimization is non-convex. In this paper, we propose Sigmoid-FTRL, an adaptive experimental design which addresses the non-convexity via simultaneous minimization of two convex regrets. We prove that under standard regularity conditions, the Neyman Regret of Sigmoid-FTRL converges at a $T^{-1/2} R^2$ rate, where $T$ is the number of subjects in the experiment and $R$ is the maximum norm of covariate vectors. Moreover, we show that no adaptive design can improve upon the $T^{-1/2}$ rate under our regularity conditions. Finally, we establish a central limit theorem and a consistently conservative variance estimator which facilitate the construction of asymptotically valid Wald-type confidence intervals.

Sigmoid-FTRL: Design-Based Adaptive Neyman Allocation for AIPW Estimators

TL;DR

This work tackles adaptive Neyman Allocation for design-based AIPW estimators with deterministic potential outcomes. It introduces Sigmoid-FTRL, a two-regret online design that uses a sigmoidal transformation to convert a non-convex Neyman regret problem into convex components, achieving the optimal rate and matching a corresponding lower bound. The authors establish a central limit theorem and a conservative variance estimator to enable Wald-type confidence intervals, and they show non-superefficiency under the design-based framework while characterizing the exact asymptotic variance. The paper highlights a fundamental distinction between design-based and super-population settings in adaptive allocation and provides a practical, provably efficient design for AIPW inference that scales with covariate norms and sample size.

Abstract

We consider the problem of Adaptive Neyman Allocation for the class of AIPW estimators in a design-based setting, where potential outcomes and covariates are deterministic. As each subject arrives, an adaptive procedure must select both a treatment assignment probability and a linear predictor to be used in the AIPW estimator. Our goal is to construct an adaptive procedure that minimizes the Neyman Regret, which is the difference between the variance of the adaptive procedure and an oracle variance which uses the optimal non-adaptive choice of assignment probability and linear predictors. While previous work has drawn insightful connections between Neyman Regret and online convex optimization for the Horvitz--Thompson estimator, one of the central challenges for AIPW estimator is that the underlying optimization is non-convex. In this paper, we propose Sigmoid-FTRL, an adaptive experimental design which addresses the non-convexity via simultaneous minimization of two convex regrets. We prove that under standard regularity conditions, the Neyman Regret of Sigmoid-FTRL converges at a rate, where is the number of subjects in the experiment and is the maximum norm of covariate vectors. Moreover, we show that no adaptive design can improve upon the rate under our regularity conditions. Finally, we establish a central limit theorem and a consistently conservative variance estimator which facilitate the construction of asymptotically valid Wald-type confidence intervals.

Paper Structure

This paper contains 43 sections, 77 theorems, 359 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Design-based Adaptive Experiments
  4. Design-Based Potential Outcomes Framework
  5. Technical Assumptions
  6. Triangular Array Asymptotics
  7. Adaptive AIPW Estimators
  8. Adaptive Neyman Allocation
  9. Formulation of the Neyman Regret
  10. Online Optimization and Technical Challenges
  11. Decomposition of the Neyman Regret
  12. The Sigmoid-FTRL Design
  13. Formal Description of The Sigmoid-FTRL Design
  14. Neyman Regret Guarantee
  15. Probability Regret
  16. ...and 28 more sections

Key Result

Proposition 2.1

If $p_t \in (0,1)$ for all $t \in [T]$ a.s. then the adaptive AIPW estimator is unbiased: $\E{ \hat{\tau} } = \tau$.

Theorems & Definitions (149)

  • Proposition 2.1: AIPW Bias
  • Proposition 2.2: AIPW Variance
  • Proposition 3.1
  • proof : Proof Sketch
  • Definition 1
  • Theorem 3.2: Lower Bound
  • Lemma 3.3
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 139 more