  This paper gives a consistent, asymptotically normal estimator of the
expected value function when the state space is high-dimensional and the
first-stage nuisance functions are estimated by modern machine learning tools.
First, we show that value function is orthogonal to the conditional choice
probability, therefore, this nuisance function needs to be estimated only at
$n^{-1/4}$ rate. Second, we give a correction term for the transition density
of the state variable. The resulting orthogonal moment is robust to
misspecification of the transition density and does not require this nuisance
function to be consistently estimated. Third, we generalize this result by
considering the weighted expected value. In this case, the orthogonal moment is
doubly robust in the transition density and additional second-stage nuisance
functions entering the correction term. We complete the asymptotic theory by
providing bounds on second-order asymptotic terms.
