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Efficient Inference after Directionally Stable Adaptive Experiments

Zikai Shen, Houssam Zenati, Nathan Kallus, Arthur Gretton, Koulik Khamaru, Aurélien Bibaut

Abstract

We study inference on scalar-valued pathwise differentiable targets after adaptive data collection, such as a bandit algorithm. We introduce a novel target-specific condition, directional stability, which is strictly weaker than previously imposed target-agnostic stability conditions. Under directional stability, we show that estimators that would have been efficient under i.i.d. data remain asymptotically normal and semiparametrically efficient when computed from adaptively collected trajectories. The canonical gradient has a martingale form, and directional stability guarantees stabilization of its predictable quadratic variation, enabling high-dimensional asymptotic normality. We characterize efficiency using a convolution theorem for the adaptive-data setting, and give a condition under which the one-step estimator attains the efficiency bound. We verify directional stability for LinUCB, yielding the first semiparametric efficiency guarantee for a regular scalar target under LinUCB sampling.

Efficient Inference after Directionally Stable Adaptive Experiments

Abstract

We study inference on scalar-valued pathwise differentiable targets after adaptive data collection, such as a bandit algorithm. We introduce a novel target-specific condition, directional stability, which is strictly weaker than previously imposed target-agnostic stability conditions. Under directional stability, we show that estimators that would have been efficient under i.i.d. data remain asymptotically normal and semiparametrically efficient when computed from adaptively collected trajectories. The canonical gradient has a martingale form, and directional stability guarantees stabilization of its predictable quadratic variation, enabling high-dimensional asymptotic normality. We characterize efficiency using a convolution theorem for the adaptive-data setting, and give a condition under which the one-step estimator attains the efficiency bound. We verify directional stability for LinUCB, yielding the first semiparametric efficiency guarantee for a regular scalar target under LinUCB sampling.
Paper Structure (49 sections, 8 theorems, 146 equations)

This paper contains 49 sections, 8 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Main message: in directionally stable designs, i.i.d.-efficient estimators remain efficient.
  3. Relation to prior work.
  4. Application to LinUCB.
  5. Data and Statistical Models
  6. Data
  7. Statistical models.
  8. Sequence of Target Estimands
  9. Notation.
  10. Sequence of Target Parameters.
  11. Riesz representation of the trajectory-level target.
  12. Canonical Gradient
  13. Directional stability
  14. One-Step Estimator
  15. Asymptotic Analysis
  16. ...and 34 more sections

Key Result

Theorem 1

Under assumptions eq: characteristic_feature-ass: homoschedastic, $\Psi_T$ is pathwise differentiable at $P^{(T)}$ if, and only if, has bounded $L^2(P^{(T)})$ norm (i.e. $\|D^{\ast}_T\|^2_{L^{2}(P^{(T)})} = \frac{\sigma^2}{T}\nu_T^{\top}\Bar{\Sigma}_T^{\dag}\nu_T < \infty$), in which case it is its canonical gradient.

Theorems & Definitions (26)

  • Remark 1: Why do we consider high dimensional setting?
  • Theorem 1: Canonical gradient
  • Definition 1: Directional stability
  • Remark 2
  • Remark 3: Equivalence to Undersmoothed Plug-in Estimator
  • Theorem 2: von Mises expansion and asymptotic normality
  • proof : Proof sketch of Theorem \ref{['thm:vme-and-an']}
  • Proposition 1
  • Proposition 2
  • Remark 4
  • ...and 16 more