# Kernel-Based Functional Balancing for Causal Inference with Compositional Treatments

arXiv: https://arxiv.org/abs/2606.17308 · HTML: https://mlsanity.com/paper/2606.17308/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 39 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [The Influence Function of Semiparametric Estimators](https://mlsanity.com/paper/1508.01378/) (2015)
- [Augmented Minimax Linear Estimation](https://mlsanity.com/paper/1712.00038/) (2017)
- [Entropy Balancing for Continuous Treatments](https://mlsanity.com/paper/2001.06281/) (2020)
- [Kernel Balancing: A flexible non-parametric weighting procedure for estimating causal effects](https://mlsanity.com/paper/1605.00155/) (2016)
- [Independence weights for causal inference with continuous treatments](https://mlsanity.com/paper/2107.07086/) (2021)
- [Flexible Functional Treatment Effect Estimation](https://mlsanity.com/paper/2309.08039/) (2023)
- [Causal Effect Estimation for Multivariate Continuous Treatments](https://mlsanity.com/paper/2205.08730/) (2022)

## Abstract (arXiv)

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 √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.