# VCNet and Functional Targeted Regularization For Learning Causal Effects of Continuous Treatments

arXiv: https://arxiv.org/abs/2103.07861 · HTML: https://mlsanity.com/paper/2103.07861/

## Key facts

- Cited by 61 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 53 references that are in the map.
- First posted on arXiv 14 Mar 2021.

## Builds on (most cited first, 16 of 16)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Estimation and Inference of Heterogeneous Treatment Effects using Random Forests](https://mlsanity.com/paper/1510.04342/) (2015)
- [Causal Effect Inference with Deep Latent-Variable Models](https://mlsanity.com/paper/1705.08821/) (2017)
- [Learning Representations for Counterfactual Inference](https://mlsanity.com/paper/1605.03661/) (2016)
- [Adapting Neural Networks for the Estimation of Treatment Effects](https://mlsanity.com/paper/1906.02120/) (2019)
- [Bayesian Inference of Individualized Treatment Effects using Multi-task Gaussian Processes](https://mlsanity.com/paper/1704.02801/) (2017)
- [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)
- [Double/Debiased/Neyman Machine Learning of Treatment Effects](https://mlsanity.com/paper/1701.08687/) (2017)
- [Nonparametric methods for doubly robust estimation of continuous treatment effects](https://mlsanity.com/paper/1507.00747/) (2015)
- [Deep Neural Networks for Estimation and Inference](https://mlsanity.com/paper/1809.09953/) (2018)
- [Conditional Density Estimation with Bayesian Normalising Flows](https://mlsanity.com/paper/1802.04908/) (2018)
- [Conditional Density Estimation with Neural Networks: Best Practices and Benchmarks](https://mlsanity.com/paper/1903.00954/) (2019)
- [The Kernel Mixture Network: A Nonparametric Method for Conditional Density Estimation of Continuous Random Variables](https://mlsanity.com/paper/1705.07111/) (2017)

## Built on by (most cited first, 30 of 61)

- [Causal machine learning for predicting treatment outcomes](https://mlsanity.com/paper/2410.08770/) (2024)
- [Causal Effect Inference for Structured Treatments](https://mlsanity.com/paper/2106.01939/) (2021)
- [Exploring Transformer Backbones for Heterogeneous Treatment Effect Estimation](https://mlsanity.com/paper/2202.01336/) (2022)
- [Normalizing Flows for Interventional Density Estimation](https://mlsanity.com/paper/2209.06203/) (2022)
- [Scalable Sensitivity and Uncertainty Analysis for Causal-Effect Estimates of Continuous-Valued Interventions](https://mlsanity.com/paper/2204.10022/) (2022)
- [Doubly Robust Causal Effect Estimation under Networked Interference via Targeted Learning](https://mlsanity.com/paper/2405.03342/) (2024)
- [Conformal Prediction for Causal Effects of Continuous Treatments](https://mlsanity.com/paper/2407.03094/) (2024)
- [Domain-Specific Risk Minimization for Out-of-Distribution Generalization](https://mlsanity.com/paper/2208.08661/) (2022)
- [Reliable Off-Policy Learning for Dosage Combinations](https://mlsanity.com/paper/2305.19742/) (2023)
- [Causal Effect Estimation: Recent Advances, Challenges, and Opportunities](https://mlsanity.com/paper/2302.00848/) (2023)
- [Deep Causal Learning: Representation, Discovery and Inference](https://mlsanity.com/paper/2211.03374/) (2022)
- [Comparison of meta-learners for estimating multi-valued treatment heterogeneous effects](https://mlsanity.com/paper/2205.14714/) (2022)
- [A Survey of Deep Causal Models and Their Industrial Applications](https://mlsanity.com/paper/2209.08860/) (2022)
- [Counterfactual Prediction Under Outcome Measurement Error](https://mlsanity.com/paper/2302.11121/) (2023)
- [Adversarially Balanced Representation for Continuous Treatment Effect Estimation](https://mlsanity.com/paper/2312.10570/) (2023)
- [End-to-End Balancing for Causal Continuous Treatment-Effect Estimation](https://mlsanity.com/paper/2107.13068/) (2021)
- [Generalization bounds and algorithms for estimating conditional average treatment effect of dosage](https://mlsanity.com/paper/2205.14692/) (2022)
- [Ensembled Prediction Intervals for Causal Outcomes Under Hidden Confounding](https://mlsanity.com/paper/2306.09520/) (2023)
- [Continuous Treatment Effect Estimation Using Gradient Interpolation and Kernel Smoothing](https://mlsanity.com/paper/2401.15447/) (2024)
- [Causal Machine Learning for Cost-Effective Allocation of Development Aid](https://mlsanity.com/paper/2401.16986/) (2024)
- [Contrastive Balancing Representation Learning for Heterogeneous Dose-Response Curves Estimation](https://mlsanity.com/paper/2403.14232/) (2024)
- [Deep Autoregressive Models as Causal Inference Engines](https://mlsanity.com/paper/2409.18581/) (2024)
- [Learning Instrumental Variable from Data Fusion for Treatment Effect Estimation](https://mlsanity.com/paper/2208.10912/) (2022)
- [Time Series Causal Link Estimation under Hidden Confounding using Knockoff Interventions](https://mlsanity.com/paper/2209.11497/) (2022)
- [Differentially Private Learners for Heterogeneous Treatment Effects](https://mlsanity.com/paper/2503.03486/) (2025)
- [Projection Pursuit Density Ratio Estimation](https://mlsanity.com/paper/2506.00866/) (2025)
- [Heterogeneous Treatment Effect with Trained Kernels of the Nadaraya-Watson Regression](https://mlsanity.com/paper/2207.09139/) (2022)
- [Uplift modeling with continuous treatments: A predict-then-optimize approach](https://mlsanity.com/paper/2412.09232/) (2024)
- [Causal Foundation Models with Continuous Treatments](https://mlsanity.com/paper/2605.15133/) (2026)
- [GCF: Generalized Causal Forest for Heterogeneous Treatment Effect Estimation in Online Marketplace](https://mlsanity.com/paper/2203.10975/) (2022)

## Abstract (arXiv)

Motivated by the rising abundance of observational data with continuous treatments, we investigate the problem of estimating the average dose-response curve (ADRF). Available parametric methods are limited in their model space, and previous attempts in leveraging neural network to enhance model expressiveness relied on partitioning continuous treatment into blocks and using separate heads for each block; this however produces in practice discontinuous ADRFs. Therefore, the question of how to adapt the structure and training of neural network to estimate ADRFs remains open. This paper makes two important contributions. First, we propose a novel varying coefficient neural network (VCNet) that improves model expressiveness while preserving continuity of the estimated ADRF. Second, to improve finite sample performance, we generalize targeted regularization to obtain a doubly robust estimator of the whole ADRF curve.