  Bayes estimators are well known to provide a means to incorporate prior
knowledge that can be expressed in terms of a single prior distribution.
However, when this knowledge is too vague to express with a single prior, an
alternative approach is needed. Gamma-minimax estimators provide such an
approach. These estimators minimize the worst-case Bayes risk over a set
$\Gamma$ of prior distributions that are compatible with the available
knowledge. Traditionally, Gamma-minimaxity is defined for parametric models. In
this work, we define Gamma-minimax estimators for general models and propose
adversarial meta-learning algorithms to compute them when the set of prior
distributions is constrained by generalized moments. Accompanying convergence
guarantees are also provided. We also introduce a neural network class that
provides a rich, but finite-dimensional, class of estimators from which a
Gamma-minimax estimator can be selected. We illustrate our method in two
settings, namely entropy estimation and a prediction problem that arises in
biodiversity studies.
