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.

翻译：我们提供了增强平衡权重（也称为自动去偏机器学习，AutoDML）的新型刻画。这类估计量将结果建模与平衡权重相结合，其中平衡权重直接估计逆倾向得分权重。当结果模型和权重模型均基于（可能无限的）基函数呈线性时，我们证明增强估计量等价于单个线性模型，其系数结合了原始结果模型系数和普通最小二乘法；在许多情形下，增强估计量简化为单独的普通最小二乘法。随后，我们将这些结果扩展到结果模型和权重模型的具体选择中。首先，我们证明使用（核）岭回归同时作为结果模型和权重模型的组合估计量等价于单个欠光滑（核）岭回归；这在考虑渐近速率时同样成立。当权重模型换为套索回归时，我们给出了特殊情形下的闭合表达式，并展示了“双重筛选”性质。最后，我们通过Riesz表示器将这些结果推广到线性估计量。我们的框架“打开了这些日益普及的估计量黑箱”，并为增强平衡权重的估计选择提供了重要见解。