Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression

TL;DR

This paper analyzes the convergence and optimality of deep feature instrumental variable (DFIV) regression, demonstrating its advantages over fixed-feature methods in adaptivity, efficiency, and handling complex target functions.

Contribution

It provides a theoretical convergence analysis of DFIV, establishing its minimax optimal rate and superiority over fixed-feature IV methods under certain conditions.

Findings

DFIV achieves minimax optimal learning rate for target functions in Besov spaces.

DFIV outperforms fixed-feature methods on functions with low spatial homogeneity.

DFIV is more data-efficient than kernel-based two-stage regression estimators.

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.