We provide a novel characterization of augmented balancing weights, also
known as Automatic Debiased Machine Learning (AutoDML). These estimators
combine outcome modeling with balancing weights, which estimate inverse
propensity score weights directly. When the outcome and weighting models are
both linear in some (possibly infinite) basis, we show that the augmented
estimator is equivalent to a single linear model with coefficients that combine
the original outcome model coefficients and OLS; in many settings, the
augmented estimator collapses to OLS alone. We then extend these results to
specific choices of outcome and weighting models. We first show that the
combined estimator that uses (kernel) ridge regression for both outcome and
weighting models is equivalent to a single, undersmoothed (kernel) ridge
regression; this also holds when considering asymptotic rates. When the
weighting model is instead lasso regression, we give closed-form expressions
for special cases and demonstrate a ``double selection'' property. Finally, we
generalize these results to linear estimands via the Riesz representer. Our
framework ``opens the black box'' on these increasingly popular estimators and
provides important insights into estimation choices for augmented balancing
weights.

本文针对 Augmented Balancing Weights（即 AutoDML）进行了新颖的表征，研究了其估计器结合了结果建模和平衡权重的方法，提出了一种基于 Riesz 代表器的推广方法，并进一步扩展到线性估计。该研究为增强平衡权重提供了重要见解，进一步揭露了其具体的估算选择方式。