  We develop an asymptotic theory of adversarial estimators ('A-estimators').
They generalize maximum-likelihood-type estimators ('M-estimators') as their
average objective is maximized by some parameters and minimized by others. This
class subsumes the continuous-updating Generalized Method of Moments,
Generative Adversarial Networks and more recent proposals in machine learning
and econometrics. In these examples, researchers state which aspects of the
problem may in principle be used for estimation, and an adversary learns how to
emphasize them optimally. We derive the convergence rates of A-estimators under
pointwise and partial identification, and the normality of functionals of their
parameters. Unknown functions may be approximated via sieves such as deep
neural networks, for which we provide simplified low-level conditions. As a
corollary, we obtain the normality of neural-net M-estimators, overcoming
technical issues previously identified by the literature. Our theory yields
novel results about a variety of A-estimators, providing intuition and formal
justification for their success in recent applications.
