Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural Networks

TL;DR

This paper provides theoretical guarantees for deep neural networks in causal inference, showing they adapt to low-dimensional structures in high-dimensional data and achieve fast convergence rates in off-policy learning.

Contribution

It establishes nonasymptotic regret bounds for deep neural networks in causal inference on low-dimensional manifolds, bridging theory and practice.

Findings

Deep neural networks adapt to low-dimensional structures.

Fast convergence rates depend on intrinsic manifold dimension.

Results cover both finite and continuous action spaces.

Abstract

Causal inference explores the causation between actions and the consequent rewards on a covariate set. Recently deep learning has achieved a remarkable performance in causal inference, but existing statistical theories cannot well explain such an empirical success, especially when the covariates are high-dimensional. Most theoretical results in causal inference are asymptotic, suffer from the curse of dimensionality, and only work for the finite-action scenario. To bridge such a gap between theory and practice, this paper studies doubly robust off-policy learning by deep neural networks. When the covariates lie on a low-dimensional manifold, we prove nonasymptotic regret bounds, which converge at a fast rate depending on the intrinsic dimension of the manifold. Our results cover both the finite- and continuous-action scenarios. Our theory shows that deep neural networks are adaptive to the low-dimensional geometric structures of the covariates, and partially explains the success of deep learning for causal inference.