# One-Step Estimation of Differentiable Hilbert-Valued Parameters

arXiv: https://arxiv.org/abs/2303.16711 · HTML: https://mlsanity.com/paper/2303.16711/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 41 references that are in the map.
- First posted on arXiv 29 Mar 2023.

## Builds on (most cited first, 9 of 9)

- [Quasi-Oracle Estimation of Heterogeneous Treatment Effects](https://mlsanity.com/paper/1712.04912/) (2017)
- [ranger: A Fast Implementation of Random Forests for High Dimensional Data in C++ and R](https://mlsanity.com/paper/1508.04409/) (2015)
- [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)
- [A general framework for inference on algorithm-agnostic variable importance](https://mlsanity.com/paper/2004.03683/) (2020)
- [Semiparametric counterfactual density estimation](https://mlsanity.com/paper/2102.12034/) (2021)
- [De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers](https://mlsanity.com/paper/1802.08667/) (2018)
- [A Simple and General Debiased Machine Learning Theorem with Finite Sample Guarantees](https://mlsanity.com/paper/2105.15197/) (2021)
- [An Omnibus Nonparametric Test of Equality in Distribution for Unknown Functions](https://mlsanity.com/paper/1510.04195/) (2015)

## Built on by (most cited first, 11 of 11)

- [Simplifying debiased inference via automatic differentiation and probabilistic programming](https://mlsanity.com/paper/2405.08675/) (2024)
- [DoubleGen: Debiased Generative Modeling of Counterfactuals](https://mlsanity.com/paper/2509.16842/) (2025)
- [Automatic Debiased Machine Learning for Smooth Functionals of Nonparametric M-Estimands](https://mlsanity.com/paper/2501.11868/) (2025)
- [Doubly-Robust Estimation of Counterfactual Policy Mean Embeddings](https://mlsanity.com/paper/2506.02793/) (2025)
- [Conditional Distributional Treatment Effects: Doubly Robust Estimation and Testing](https://mlsanity.com/paper/2603.16829/) (2026)
- [Kernel Treatment Effects with Adaptively Collected Data](https://mlsanity.com/paper/2510.10245/) (2025)
- [Inference on Variable Importance for Treatment Effect Heterogeneity: Shapley Values and Beyond](https://mlsanity.com/paper/2510.18843/) (2025)
- [GDR-learners: Orthogonal Learning of Generative Models for Potential Outcomes](https://mlsanity.com/paper/2509.22953/) (2025)
- [Toward an Asymptotic Efficiency Theory on Regular Parameter Manifolds](https://mlsanity.com/paper/2510.13703/) (2025)
- [Set-Valued Policy Learning](https://mlsanity.com/paper/2605.19830/) (2026)
- [Semiparametric Efficient Bilevel Gradient Estimation](https://mlsanity.com/paper/2605.21341/) (2026)

## Abstract (arXiv)

We present estimators for smooth Hilbert-valued parameters, where smoothness is characterized by a pathwise differentiability condition. When the parameter space is a reproducing kernel Hilbert space, we provide a means to obtain efficient, root-n rate estimators and corresponding confidence sets. These estimators correspond to generalizations of cross-fitted one-step estimators based on Hilbert-valued efficient influence functions. We give theoretical guarantees even when arbitrary estimators of nuisance functions are used, including those based on machine learning techniques. We show that these results naturally extend to Hilbert spaces that lack a reproducing kernel, as long as the parameter has an efficient influence function. However, we also uncover the unfortunate fact that, when there is no reproducing kernel, many interesting parameters fail to have an efficient influence function, even though they are pathwise differentiable. To handle these cases, we propose a regularized one-step estimator and associated confidence sets. We also show that pathwise differentiability, which is a central requirement of our approach, holds in many cases. Specifically, we provide multiple examples of pathwise differentiable parameters and develop corresponding estimators and confidence sets. Among these examples, four are particularly relevant to ongoing research by the causal inference community: the counterfactual density function, dose-response function, conditional average treatment effect function, and counterfactual kernel mean embedding.