  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.
