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 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 .