Kernel-Based Functional Balancing for Causal Inference with Compositional Treatments

2026-06-15

AI Summary

This study addresses causal effect estimation under compositional treatments in the simplex space by proposing the Augmented Weighting Estimator (AWE). AWE constructs weights by directly minimizing worst-case balance error in a reproducing kernel Hilbert space (RKHS) defined over the joint space of treatments and covariates, and augments these weights with kernel ridge regression and the marginal covariate distribution. The method achieves √n-consistency and asymptotic normality without explicitly modeling the treatment assignment mechanism or requiring consistent weight estimation. To the best of our knowledge, this work establishes the first asymptotic theory for causal inference with compositional treatments based on functional balancing. The estimator is efficiently implemented via kernel methods, low-rank approximations, and convex optimization. Empirical results demonstrate that AWE exhibits superior performance in both simulated and real-world data, achieving √n convergence rates and strong robustness.

Abstract

We study causal effect estimation with compositional treatments, where the exposure lies on a simplex and the estimand is defined over compositions rather than scalar or binary values. By considering a projection of the average potential outcome onto the treatment space, a kernel-based covariate functional balancing approach is adopted for weight construction. The weights are obtained by directly minimizing a worst-case balancing error over a reproducing kernel Hilbert space (RKHS) defined on the joint space of treatments and covariates, instead of being estimated under a treatment assignment model. Building on these weights, an augmented weighted estimator (AWE) is proposed, where the outcome function is estimated via kernel ridge regression and combined with a marginal augmentation over the covariate distribution. Despite the complex structure of the resulting objective, a finite-dimensional convex optimization problem is formulated via a representer theorem and a low-rank approximation. The proposed estimator achieves $\sqrt{n}$-consistency without requiring consistent estimation or smoothness of the weights. An asymptotic normality result is established around a sample-specific target. Empirical performance is demonstrated through simulation studies and a real data application.

Problem

• compositional treatments
• causal inference
• kernel methods
• functional balancing
• average potential outcome

Innovation

• compositional treatments
• kernel-based balancing
• reproducing kernel Hilbert space
• augmented weighted estimator
• causal inference