Core Concepts

The authors propose an adversarial framework to estimate the Riesz representer, a key component in the asymptotic variance of semiparametrically estimated linear functionals, using general function spaces. They provide nonasymptotic mean square rates for the Riesz representer estimator in terms of the critical radius, allowing for flexible function approximation beyond sparse linear or RKHS spaces.

Abstract

The paper proposes an adversarial framework to estimate the Riesz representer, a key component in the asymptotic variance of semiparametrically estimated linear functionals. The authors prove a nonasymptotic mean square rate for their Riesz representer estimator in terms of the critical radius, which quantifies the complexity of the function space used for estimation.
Key highlights:
The adversarial estimator can use general function spaces beyond sparse linear or RKHS, including neural networks and random forests.
The mean square rate is expressed in terms of the critical radius, which is a well-studied quantity in statistical learning theory. This allows the authors to leverage known critical radius results for various function spaces.
The fast estimation rate for the Riesz representer is shown to be compatible with targeted and debiased machine learning approaches for semiparametric inference, allowing for misspecification.
The authors also provide results for semiparametric inference without sample splitting, based on either estimator stability or complexity-rate robustness.
Computational aspects are analyzed for random forests and RKHS, bridging theory and practice.
The flexible Riesz representer estimator is used to extend an influential empirical study on the heterogeneous effects of matching grants, providing new insights.

Stats

The functions in the function class F and the composite functions m ◦ F have uniformly bounded ranges in [-b, b].
The critical radii of the function classes FB and m ◦ FB are upper bounded by δn.

Quotes

"Many parameters traditionally studied in statistics and econometrics are functionals, i.e. scalar summaries, of an underlying regression function."
"Estimating the Riesz representer of a linear functional is a critical building block in a variety of tasks."
"Our first contribution is a general Riesz estimator over nonlinear function spaces with fast estimation rates."

Key Insights Distilled From

by Victor Chern... at **arxiv.org** 04-29-2024

Deeper Inquiries

The adversarial estimation framework can be extended to handle more general types of functionals beyond linear and mean square continuous ones by considering a broader class of function spaces and incorporating different regularization techniques. One approach is to allow for non-linear function spaces, such as neural networks, random forests, or reproducing kernel Hilbert spaces, which can approximate a wider range of functions. By adapting the adversarial framework to these function spaces, it becomes possible to estimate more complex functionals that may not have a simple linear or mean square continuous form. Additionally, incorporating different regularization terms, such as L1 or L2 norms, can help in handling more general functionals by controlling the complexity of the estimators and preventing overfitting.

While the critical radius approach is a powerful tool for obtaining fast rates for Riesz representer estimation, it does have some limitations. One limitation is that it relies on the assumption of boundedness of the function classes, which may not always hold in practice. Additionally, the critical radius may not always provide the most accurate measure of complexity for certain function spaces, especially in high-dimensional or non-linear settings. Alternative complexity measures, such as Rademacher complexities or covering numbers, could be used to complement the critical radius approach and provide a more comprehensive understanding of the complexity of the function spaces. By combining different complexity measures, it may be possible to obtain more robust and accurate estimates of the Riesz representer with fast rates.

The insights from this work on semiparametric inference can indeed be applied to other areas of machine learning beyond econometrics. In causal inference, for example, the framework of adversarial estimation can help in estimating causal effects and treatment effects in a more robust and efficient manner. By incorporating adversarial techniques and considering the stability of estimators, it is possible to improve the accuracy and reliability of causal inference models. Similarly, in reinforcement learning, the principles of semiparametric inference can be utilized to estimate value functions and policy parameters in a more stable and efficient way. By leveraging the insights from this work, researchers in various fields of machine learning can enhance the performance and reliability of their models.

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