  This paper studies debiased machine learning when nuisance parameters appear
in indicator functions. An important example is maximized average welfare under
optimal treatment assignment rules. For asymptotically valid inference for a
parameter of interest, the current literature on debiased machine learning
relies on Gateaux differentiability of the functions inside moment conditions,
which does not hold when nuisance parameters appear in indicator functions. In
this paper, we propose smoothing the indicator functions, and develop an
asymptotic distribution theory for this class of models. The asymptotic
behavior of the proposed estimator exhibits a trade-off between bias and
variance due to smoothing. We study how a parameter which controls the degree
of smoothing can be chosen optimally to minimize an upper bound of the
asymptotic mean squared error. A Monte Carlo simulation supports the asymptotic
distribution theory, and an empirical example illustrates the implementation of
the method.
