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Disease registries, surveillance data, and other datasets with extremely large sample sizes become increasingly available in providing population-based information on disease incidence, survival probability, or other important public health characteristics. Such information can be leveraged in studies that collect detailed measurements but with smaller sample sizes. In contrast to recent proposals that formulate additional information as constraints in optimization problems, we develop a general framework to construct simple estimators that update the usual regression estimators with some functionals of data that incorporate the additional information. We consider general settings that incorporate nuisance parameters in the auxiliary information, non-i.i.d. data such as those from case-control studies, and semiparametric models with infinite-dimensional parameters common in survival analysis. Details of several important data and sampling settings are provided with numerical examples.