# Instrument Space Selection for Kernel Maximum Moment Restriction

arXiv: https://arxiv.org/abs/2106.03340 · HTML: https://mlsanity.com/paper/2106.03340/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 82 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 19 of 19)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Generalized Random Forests](https://mlsanity.com/paper/1610.01271/) (2016)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Breaking the Curse of Horizon: Infinite-Horizon Off-Policy Estimation](https://mlsanity.com/paper/1810.12429/) (2018)
- [Minimax Weight and Q-Function Learning for Off-Policy Evaluation](https://mlsanity.com/paper/1910.12809/) (2019)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [Deep Generalized Method of Moments for Instrumental Variable Analysis](https://mlsanity.com/paper/1905.12495/) (2019)
- [Minimax Estimation of Conditional Moment Models](https://mlsanity.com/paper/2006.07201/) (2020)
- [Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction](https://mlsanity.com/paper/2105.04544/) (2021)
- [A Linear-Time Kernel Goodness-of-Fit Test](https://mlsanity.com/paper/1705.07673/) (2017)
- [Generalized Optimal Matching Methods for Causal Inference](https://mlsanity.com/paper/1612.08321/) (2016)
- [A Kernel Loss for Solving the Bellman Equation](https://mlsanity.com/paper/1905.10506/) (2019)
- [Instrumental Variable Value Iteration for Causal Offline Reinforcement Learning](https://mlsanity.com/paper/2102.09907/) (2021)
- [Adversarial Generalized Method of Moments](https://mlsanity.com/paper/1803.07164/) (2018)
- [Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach](https://mlsanity.com/paper/2007.01290/) (2020)
- [Valid Causal Inference with (Some) Invalid Instruments](https://mlsanity.com/paper/2006.11386/) (2020)
- [Kernel Conditional Moment Test via Maximum Moment Restriction](https://mlsanity.com/paper/2002.09225/) (2020)
- [Ivy: Instrumental Variable Synthesis for Causal Inference](https://mlsanity.com/paper/2004.05316/) (2020)
- [An Adaptive Test of Independence with Analytic Kernel Embeddings](https://mlsanity.com/paper/1610.04782/) (2016)

## Built on by (most cited first, 2 of 2)

- [Fast Instrument Learning with Faster Rates](https://mlsanity.com/paper/2205.10772/) (2022)
- [RFFNet: Large-Scale Interpretable Kernel Methods via Random Fourier Features](https://mlsanity.com/paper/2211.06410/) (2022)

## Abstract (arXiv)

Kernel maximum moment restriction (KMMR) recently emerges as a popular framework for instrumental variable (IV) based conditional moment restriction (CMR) models with important applications in conditional moment (CM) testing and parameter estimation for IV regression and proximal causal learning. The effectiveness of this framework, however, depends critically on the choice of a reproducing kernel Hilbert space (RKHS) chosen as a space of instruments. In this work, we presents a systematic way to select the instrument space for parameter estimation based on a principle of the least identifiable instrument space (LIIS) that identifies model parameters with the least space complexity. Our selection criterion combines two distinct objectives to determine such an optimal space: (i) a test criterion to check identifiability; (ii) an information criterion based on the effective dimension of RKHSs as a complexity measure. We analyze the consistency of our method in determining the LIIS, and demonstrate its effectiveness for parameter estimation via simulations.