# Weighted Spline-Expanded Networks with Distributional Balancing for Continuous Treatment Effects

arXiv: https://arxiv.org/abs/2609.33740 · HTML: https://mlsanity.com/paper/2609.33740/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 62 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [Adapting Neural Networks for the Estimation of Treatment Effects](https://mlsanity.com/paper/1906.02120/) (2019)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [Learning Counterfactual Representations for Estimating Individual Dose-Response Curves](https://mlsanity.com/paper/1902.00981/) (2019)
- [Estimating the Effects of Continuous-valued Interventions using Generative Adversarial Networks](https://mlsanity.com/paper/2002.12326/) (2020)
- [Nonparametric methods for doubly robust estimation of continuous treatment effects](https://mlsanity.com/paper/1507.00747/) (2015)
- [VCNet and Functional Targeted Regularization For Learning Causal Effects of Continuous Treatments](https://mlsanity.com/paper/2103.07861/) (2021)
- [Policy Evaluation and Optimization with Continuous Treatments](https://mlsanity.com/paper/1802.06037/) (2018)
- [Semi-Parametric Efficient Policy Learning with Continuous Actions](https://mlsanity.com/paper/1905.10116/) (2019)
- [Visually Communicating and Teaching Intuition for Influence Functions](https://mlsanity.com/paper/1810.03260/) (2018)
- [Reliable Off-Policy Learning for Dosage Combinations](https://mlsanity.com/paper/2305.19742/) (2023)
- [Deep Jump Learning for Off-Policy Evaluation in Continuous Treatment Settings](https://mlsanity.com/paper/2010.15963/) (2020)
- [Sensitivity analysis via the proportion of unmeasured confounding](https://mlsanity.com/paper/1912.02793/) (2019)
- [Adversarially Balanced Representation for Continuous Treatment Effect Estimation](https://mlsanity.com/paper/2312.10570/) (2023)
- [Energy Balancing of Covariate Distributions](https://mlsanity.com/paper/2004.13962/) (2020)
- [End-to-End Balancing for Causal Continuous Treatment-Effect Estimation](https://mlsanity.com/paper/2107.13068/) (2021)
- [Covariate balancing using the integral probability metric for causal inference](https://mlsanity.com/paper/2305.13715/) (2023)
- [Quasi-optimal Reinforcement Learning with Continuous Actions](https://mlsanity.com/paper/2301.08940/) (2023)
- [On Robustness of Individualized Decision Rules](https://mlsanity.com/paper/1903.04367/) (2019)
- [Independence weights for causal inference with continuous treatments](https://mlsanity.com/paper/2107.07086/) (2021)
- [Incremental effects for continuous exposures](https://mlsanity.com/paper/2409.11967/) (2024)
- [Falsification of Unconfoundedness by Testing Independence of Causal Mechanisms](https://mlsanity.com/paper/2502.06231/) (2025)
- [Reinforcement Learning with Continuous Actions Under Unmeasured Confounding](https://mlsanity.com/paper/2505.00304/) (2025)
- [Nonparametric Inference on Dose-Response Curves Without the Positivity Condition](https://mlsanity.com/paper/2405.09003/) (2024)

## Abstract (arXiv)

Estimating causal effects with continuous treatments in observational studies is challenging due to confounding, model misspecification, and high-dimensional covariates. We propose the Weighted Spline-Expanded Network (WSENet), an end-to-end neural framework that addresses these challenges by combining covariate balancing, structured treatment embedding, and bias-corrected outcome estimation. WSENet first applies Distance Covariate Optimal Weights to induce distributional independence between covariates and treatment without relying on parametric models. It then learns the conditional outcome via a structured network that fuses outcome-relevant representations of covariates with a spline-expanded treatment input, enabling smooth and flexible modeling of the dose-response relationship. To mitigate residual bias, we introduce Weighted Targeted Regularization, a correction technique based on efficient influence functions that yields a doubly robust estimator. Extensive evaluations on semi-synthetic and real-world datasets, including high-dimensional genomic and environmental health data, demonstrate that WSENet consistently outperforms existing baselines in both accuracy and stability.