Sliced $L^p$ Distributional Balancing

Abstract: A popular class of causal inference methods addresses confounding through weighting, which reweights treated and control groups to balance their covariate distributions without using outcome information, thereby preserving a design-based perspective. In this paper, we propose sliced $L<sup>p$ distributional balancing (SLDB), a family indexed by $p\in[1,\infty)$ that measures imbalance by averaging squared $L<sup>p$ distances between the cumulative distribution functions of one-dimensional linear projections. The Cramér--Wold device ensures that this criterion identifies equality of multivariate distributions, while projection reduces its computation to sorting-based one-dimensional operations. Because our method lies outside the maximum mean discrepancy (MMD) framework underlying many existing distributional balancing methods, their theoretical and computational tools do not directly apply. We therefore develop a computationally efficient projected subgradient descent algorithm for estimating balancing weights, offering improved computational complexity over MMD-based methods. Furthermore, we establish a novel theoretical framework for SLDB-based causal effect estimation and prove, under suitable conditions, $\sqrt{n}$-consistency and asymptotic normality, with the asymptotic variance attaining the semiparametric efficiency bound. Finally, we develop inferential procedures that do not require augmentation with an outcome model, thereby retaining the design-based principle. Simulation studies and a real-world application demonstrate that SLDB performs competitively with existing methods.