Adversarial Estimators

We develop an asymptotic theory of adversarial estimators ('A-estimators'). They generalize maximum-likelihood-type estimators ('M-estimators') as their 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.