  For many inference problems in statistics and econometrics, the unknown
parameter is identified by a set of moment conditions. A generic method of
solving moment conditions is the Generalized Method of Moments (GMM). However,
classical GMM estimation is potentially very sensitive to outliers. Robustified
GMM estimators have been developed in the past, but suffer from several
drawbacks: computational intractability, poor dimension-dependence, and no
quantitative recovery guarantees in the presence of a constant fraction of
outliers. In this work, we develop the first computationally efficient GMM
estimator (under intuitive assumptions) that can tolerate a constant $\epsilon$
fraction of adversarially corrupted samples, and that has an $\ell_2$ recovery
guarantee of $O(\sqrt{\epsilon})$. To achieve this, we draw upon and extend a
recent line of work on algorithmic robust statistics for related but simpler
problems such as mean estimation, linear regression and stochastic
optimization. As two examples of the generality of our algorithm, we show how
our estimation algorithm and assumptions apply to instrumental variables linear
and logistic regression. Moreover, we experimentally validate that our
estimator outperforms classical IV regression and two-stage Huber regression on
synthetic and semi-synthetic datasets with corruption.
