Source Condition Double Robust Inference on Functionals of Inverse
  Problems
Andrew Bennett Nathan Kallus Xiaojie Mao Whitney Newey Vasilis Syrgkanis Masatoshi Uehara
We consider estimation of parameters defined as linear functionals of
solutions to linear inverse problems. Any such parameter admits a doubly robust
representation that depends on the solution to a dual linear inverse problem,
where the dual solution can be thought as a generalization of the inverse
propensity function. We provide the first source condition double robust
inference method that ensures asymptotic normality around the parameter of
interest as long as either the primal or the dual inverse problem is
sufficiently well-posed, without knowledge of which inverse problem is the more
well-posed one. Our result is enabled by novel guarantees for iterated Tikhonov
regularized adversarial estimators for linear inverse problems, over general
hypothesis spaces, which are developments of independent interest.