Regularized DeepIV with Model Selection (2403.04236v1)

Abstract: In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. While recent advancements in machine learning have introduced flexible methods for IV estimation, they often encounter one or more of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) requiring minimax computation oracle, which is highly unstable in practice; (3) absence of model selection procedure. In this paper, we present the first method and analysis that can avoid all three limitations, while still enabling general function approximation. Specifically, we propose a minimax-oracle-free method called Regularized DeepIV (RDIV) regression that can converge to the least-norm IV solution. Our method consists of two stages: first, we learn the conditional distribution of covariates, and by utilizing the learned distribution, we learn the estimator by minimizing a Tikhonov-regularized loss function. We further show that our method allows model selection procedures that can achieve the oracle rates in the misspecified regime. When extended to an iterative estimator, our method matches the current state-of-the-art convergence rate. Our method is a Tikhonov regularized variant of the popular DeepIV method with a non-parametric MLE first-stage estimator, and our results provide the first rigorous guarantees for this empirically used method, showcasing the importance of regularization which was absent from the original work.

• The paper presents Regularized DeepIV, which employs Tikhonov regularization in a neural network framework to overcome key challenges in IV estimation.
• It introduces an iterative estimation method with recursive penalties that adaptively enhances convergence rates across varied data-generating processes.
• The model selection algorithm robustly identifies optimal parameters, outperforming benchmarks like DeepIV and KernelIV in reducing MSE.

Fast and Adaptive Rates for Regularized DeepIV with Iterative Estimator and Model Selection

Introduction

Instrumental variable (IV) estimation is a crucial technique employed across various disciplines such as econometrics, epidemiology, and machine learning for identifying causal relationships. Traditional methods of IV estimation face challenges concerning non-uniqueness, high computational demands, and model selection difficulties. This paper introduces an innovative approach, Regularized DeepIV (RDIV), which leverages neural networks for function approximation in a two-stage method, overcoming the limitations of previous approaches. Additionally, the paper extends RDIV to an iterative version and demonstrates an effective model selection process, capable of handling a diverse set of data-generating processes (DGP).

RDIV: Addressing Key Challenges

RDIV presents a solution to three significant challenges in IV estimation: the uniqueness of the solution, computational instability, and model selection. By incorporating Tikhonov regularization, RDIV converges to the least-norm IV solution, allowing for non-unique solutions, which in practice occur due to weak instrumental variables or complex relationships between variables.

The computational efficiency of RDIV is noteworthy. Unlike traditional methods that require non-convex non-concave minimax optimization oracles, RDIV utilizes standard empirical risk minimization oracles. This architectural choice ensures stability and convergence with the standard optimization techniques used in supervised learning contexts, especially when employing neural networks for function approximation.

Model selection is a pivotal component of RDIV, enabling the evaluation of different model configurations based on their performance. Utilizing a validation dataset, RDIV can effectively determine the optimal model parameters, addressing a significant gap in previous IV estimation methods where clear model selection procedures were absent.

Iterative RDIV: Exploiting Ill-Posedness

To further leverage the smoothness properties of the solution and address the level of ill-posedness in the problem, the paper extends RDIV to an iterative version. This iteration introduces a recursive penalization strategy, where each iteration regularizes towards the prior one, allowing for an adaptive and fine-tuned approach to reaching the solution. Empirical results demonstrate that for a wide range of β-source conditions, the iterative RDIV achieves superior convergence rates, matching state-of-the-art results without necessitating computationally intensive minimax oracles.

Model Selection and Numerical Experiments

The paper extensively evaluates RDIV and its iterative variant, comparing them with benchmark methods across various DGPs. These numerical experiments underline the practical efficacy of RDIV, showing notable improvement over benchmarks such as DeepIV and KernelIV in terms of Mean Squared Error (MSE) performance.

A significant contribution of this paper is its model selection algorithm, which further refines the estimator's performance. By selecting the best model based on the validation set performance, RDIV showcases enhanced adaptability and efficiency across different experimental settings. This model selection component not only enhances RDIV's accuracy but also its applicability to real-world datasets where the ground truth DGP is unknown.

Conclusion and Future Directions

RDIV, with its iterative extension and model selection procedure, presents a robust, efficient, and adaptable framework for nonparametric IV regression. It operates effectively without the unique solution assumption, negates the need for complex computational oracles, and incorporates a clear model selection mechanism. Future work could explore extending the RDIV framework to other settings beyond IV estimation, where similar challenges of non-uniqueness, computational instability, and model selection persist. Further research could also investigate more refined iterative schemes and their theoretical properties, potentially offering even faster convergence rates and broader applicability.