Primary Submission Category: Weighting
The Role of the Simplex Constraint in Regularizing Treatment Effect Estimates
Authors: Avi Feller, David Arbour, Anup Rao, Tung Mai,
Presenting Author: David Arbour*
Many workhorse methods in causal inference are weighting estimators, with estimates that are linear in the observed outcomes. A key consideration for these methods is whether to constrain the weights to be non-negative (or on the simplex): matching and inverse propensity score weighting (IPW), for example, impose this constraint while linear regression does not. These constraints limit extrapolation but can introduce bias, especially with high-dimensional features. In this paper, we take a geometric perspective and argue that the simplex constraint acts as an implicit regularizer via sample trimming. We make this regularization explicit as a form of generalized ridge penalty, characterize the resulting behavior under a high-dimensional factor model, and consider the mis-specified setting, such as when the target is outside the convex hull of control units. We then extend results to other shape constraints on the weights, especially to popular forms of Augmented IPW. We argue that, beyond their recognized statistical properties, these estimators also have attractive geometric properties, especially in high dimensions. Finally, we show how this perspective helps explain recent results on high-dimensional causal inference, such as the lack of double-descent behavior for the synthetic control method.