Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression (2501.04898v1)

Abstract: We provide a convergence analysis of deep feature instrumental variable (DFIV) regression (Xu et al., 2021), a nonparametric approach to IV regression using data-adaptive features learned by deep neural networks in two stages. We prove that the DFIV algorithm achieves the minimax optimal learning rate when the target structural function lies in a Besov space. This is shown under standard nonparametric IV assumptions, and an additional smoothness assumption on the regularity of the conditional distribution of the covariate given the instrument, which controls the difficulty of Stage 1. We further demonstrate that DFIV, as a data-adaptive algorithm, is superior to fixed-feature (kernel or sieve) IV methods in two ways. First, when the target function possesses low spatial homogeneity (i.e., it has both smooth and spiky/discontinuous regions), DFIV still achieves the optimal rate, while fixed-feature methods are shown to be strictly suboptimal. Second, comparing with kernel-based two-stage regression estimators, DFIV is provably more data efficient in the Stage 1 samples.

• The paper establishes that DFIV attains minimax optimal rates for nonparametric IV regression when the structural function lies within a Besov space with defined smoothness.
• The paper demonstrates that DFIV adaptively captures both smooth and spiky data behaviors, outperforming traditional 2SLS and kernel-based methods.
• The paper highlights DFIV’s sample efficiency by requiring fewer first-stage samples and introduces novel empirical process techniques for precise error bounds.

Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression

Instrumental Variable (IV) regression is a statistical method used widely in econometrics and causal inference to estimate causal relationships when controlled experimentation is not feasible. Traditional implementations, such as Two-Stage Least Squares (2SLS), rely on fixed-feature sets, which can be suboptimal in capturing the complexities of real-world data. This paper introduces Deep Feature Instrumental Variable (DFIV) regression to address these limitations by utilizing deep neural networks (DNNs) to learn adaptively selected feature representations.

Summary of Contributions

1. Minimax Optimality: The paper establishes theoretically that DFIV attains the minimax optimal rates for nonparametric IV regression when the structural function resides within a Besov space with certain smoothness parameters. This assertion provides a sound statistical basis for employing DNNs in IV regression, showing that their adaptive capacities can be harnessed without sacrificing performance guarantees.
2. Superior Adaptivity: The analysis explores scenarios where the structural function possesses regions of both smooth and spiky behavior, and demonstrates that DFIV can adapt optimally to such spatial inhomogeneities. This adaptivity is crucial for real-world applications where data distributions are rarely homogeneous across the feature space.
3. Sample Efficiency: A critical insight from the paper is that DFIV requires fewer samples in the first-stage regression compared to existing kernel IV methods. This sample efficiency is particularly beneficial when data collection is costly or logistically challenging.
4. Empirical Process Techniques: The authors develop a novel theoretical framework leveraging empirical process theory, bounding complexities of DNN classes, and introducing a dynamic cover technique, to analyze the estimation error. This approach enables them to provide precise error bounds and insights into the behavior of neural networks in the IV setting.
5. Regularization and Smoothness Control: To ensure practical utility and theoretical guarantees, DFIV incorporates smoothness regularization. This not only facilitates better empirical performance but also maintains the estimator within a desirable function class, ensuring the smoothness required for optimal performance.

Theoretical Insights and Implications

• Link and Smoothness Conditions: Crucial to the analysis are conditions on the operator linking the instrumental variable to the endogenous variable. These conditions ensure the manageable complexity of both the structural function and its neural network approximation, balancing approximation and estimation errors effectively.
• Lower Bound Analysis: The paper extends classical statistical minimax theory to nonparametric IV regression settings, particularly with DNNs. It provides a robust framework showing that no estimator can surpass the derived minimax rate, solidifying the optimality claims for DFIV.

Future Directions

Future research can explore the extension of these results to broader function spaces, such as mixed or anisotropic Besov spaces, which capture more complex dependencies inherent in high-dimensional datasets. Additionally, the application of DFIV in dynamic settings, such as reinforcement learning or time-series analysis, where the relationships evolve over time, presents a promising area for further exploration.

Moreover, understanding the practical implications of these theoretical results in domains beyond econometrics, such as bioinformatics or medical diagnostics, where the causality and adaptability of the model are crucial, would be highly valuable. The integration of domain-specific knowledge with deep learning approaches like DFIV has the potential to offer more targeted insights and predictions in these complex fields.

In summary, the paper bridges a significant gap between theory and practice in nonparametric IV regression, demonstrating the value of deep learning as a flexible and powerful tool in causal inference.