Before revision:  We  provide new rates of convergence for deep feedforward neural nets and, because our rates are sufficiently fast (in some cases minimax optimal), prove that semiparametric inferenceis valid using deep nets for first-step estimation .
After 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), prove that semiparametric inferenceis valid using deep nets for first-step estimation .