DNA-SE: Towards Deep Neural-Nets Assisted Semiparametric Estimation
This work addresses a computational bottleneck in semiparametric statistics, impacting fields like causal inference and missing data, though it is a novel method rather than a paradigm shift.
The paper tackles the challenge of solving Fredholm integral equations in semiparametric estimation, which is computationally intensive for multi-dimensional problems, by proposing DNA-SE, a deep neural network-based framework that improves scalability and statistical performance over traditional methods.
Semiparametric statistics play a pivotal role in a wide range of domains, including but not limited to missing data, causal inference, and transfer learning, to name a few. In many settings, semiparametric theory leads to (nearly) statistically optimal procedures that yet involve numerically solving Fredholm integral equations of the second kind. Traditional numerical methods, such as polynomial or spline approximations, are difficult to scale to multi-dimensional problems. Alternatively, statisticians may choose to approximate the original integral equations by ones with closed-form solutions, resulting in computationally more efficient, but statistically suboptimal or even incorrect procedures. To bridge this gap, we propose a novel framework by formulating the semiparametric estimation problem as a bi-level optimization problem; and then we develop a scalable algorithm called Deep Neural-Nets Assisted Semiparametric Estimation (DNA-SE) by leveraging the universal approximation property of Deep Neural-Nets (DNN) to streamline semiparametric procedures. Through extensive numerical experiments and a real data analysis, we demonstrate the numerical and statistical advantages of $\dnase$ over traditional methods. To the best of our knowledge, we are the first to bring DNN into semiparametric statistics as a numerical solver of integral equations in our proposed general framework.