Before revision:  We show that in the case of identification failures , a very general family of continuously-regularized estimators is consistent for the best approximation of the parameter of interest and obtain L_2 and L_\infty finite-sample risk bounds. This class includes Tikhonov, iterated Tikhonov, spectral cut-off, and Landweber-Fridman as special cases. We show that in many cases the best approximation coincides with the structural parameter and can be a useful and tractable object to infer relation between structural variables otherwise.
After revision:  We show that in the case of identification failures , a very general family of continuously-regularized estimators is consistent for the best approximation of the structural parameter in the orthogonal complement to the null space of the operator. We illustrate that in many cases the best approximation coincides with the structural parameter and can be a useful and tractable object to infer relation between structural variables otherwise.