Before revision:  We prove valid inference after first-step estimation with deep learning, a result new to the literature. We provide new rates of convergence for deep feedforward neural nets and, because our rates are sufficiently fast (in some cases minimax optimal) , obtain valid semiparametric inference .
After revision:  We establish novel rates of convergence for deep feedforward neural nets and, because our rates are sufficiently fast (in some cases minimax optimal) , obtain valid semiparametric inference .