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.

翻译：我们考虑定义为线性逆问题解的线性泛函的参数估计。任何此类参数都拥有一个双稳健表示，该表示依赖于对偶线性逆问题的解，其中对偶解可被视为逆倾向函数的推广。我们提出了首个源条件双稳健推断方法，该方法能够确保只要原始逆问题或对偶逆问题中任意一个是适定的（无需事先知晓哪个逆问题更为适定），待估参数即可实现渐近正态性。这一成果得益于我们为线性逆问题在一般假设空间上建立的迭代Tikhonov正则化对抗估计器的新颖理论保证——这些保证本身即具有独立的研究价值。