mlsanityData to 14 Oct 2026

The Variational Method of Moments

arXiv 2012.09422 ↗First posted 17 Dec 2020cs.LG econ.EM math.STcited by 23

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Abstract (arXiv)

The conditional moment problem is a powerful formulation for describing structural causal parameters in terms of observables, a prominent example being instrumental variable regression. A standard approach reduces the problem to a finite set of marginal moment conditions and applies the optimally weighted generalized method of moments (OWGMM), but this requires we know a finite set of identifying moments, can still be inefficient even if identifying, or can be theoretically efficient but practically unwieldy if we use a growing sieve of moment conditions. Motivated by a variational minimax reformulation of OWGMM, we define a very general class of estimators for the conditional moment problem, which we term the variational method of moments (VMM) and which naturally enables controlling infinitely-many moments. We provide a detailed theoretical analysis of multiple VMM estimators, including ones based on kernel methods and neural nets, and provide conditions under which these are consistent, asymptotically normal, and semiparametrically efficient in the full conditional moment model. We additionally provide algorithms for valid statistical inference based on the same kind of variational reformulations, both for kernel- and neural-net-based varieties. Finally, we demonstrate the strong performance of our proposed estimation and inference algorithms in a detailed series of synthetic experiments.

Key facts. Cited by 23 papers in the map (data through 14 Oct 2026). First posted on arXiv 17 Dec 2020.

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Citing papers per year: 2021: 6; 2022: 6; 2023: 5; 2024: 3; 2025: 1; 2026: 2.

23 papers.

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  1. 2021Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approachcited by 44
  2. 2021Proximal Reinforcement Learning: Efficient Off-Policy Evaluation in Partially Observed Markov Decision Processescited by 41
  3. 2021A Minimax Learning Approach to Off-Policy Evaluation in Confounded Partially Observable Markov Decision Processescited by 36
  4. 2022Long-term Causal Inference Under Persistent Confounding via Data Combinationcited by 27
  5. 2022Robust and Agnostic Learning of Conditional Distributional Treatment Effectscited by 18
  6. 2022Inference on Strongly Identified Functionals of Weakly Identified Functionscited by 12
  7. 2022Instrumental Variables in Causal Inference and Machine Learning: A Surveycited by 10
  8. 2023Falsification of Internal and External Validity in Observational Studies via Conditional Moment Restrictionscited by 10
  9. 2023Source Condition Double Robust Inference on Functionals of Inverse Problemscited by 9
  10. 2022Functional Generalized Empirical Likelihood Estimation for Conditional Moment Restrictionscited by 8
  11. 2021Learning Causal Models from Conditional Moment Restrictions by Importance Weightingcited by 7
  12. 2024Regularized DeepIV with Model Selectioncited by 7
11 more
  1. 2023Estimation Beyond Data Reweighting: Kernel Method of Momentscited by 5
  2. 2021Towards Principled Causal Effect Estimation by Deep Identifiable Modelscited by 4
  3. 2021Quasi-Bayesian Dual Instrumental Variable Regressioncited by 3
  4. 2024Causal Inference with Complex Treatments: A Surveycited by 3
  5. 2022Adversarial Estimatorscited by 2
  6. 2023Personalized Pricing with Invalid Instrumental Variables: Identification, Estimation, and Policy Learningcited by 1
  7. 2023Heterogeneous Transfer Learning for Building High-Dimensional Generalized Linear Models with Disparate Datasetscited by 1
  8. 2024Geometry-Aware Instrumental Variable Regressioncited by 1
  9. 2025Detecting clinician implicit biases in diagnoses using proximal causal inferencecited by 1
  10. 2026Adaptive Estimation and Inference in Conditional Moment Models via the Discrepancy Principlecited by 0
  11. 2026Conformal Prediction for Nonparametric Instrumental Regressioncited by 0