The Iterative Construction Mechanism abstraction (together with the perception based database update algorithm) was formalized by Gupta et al. [39], generalizing the median mechanism of Roth and Roughgarden [74] (initially presented as an online algorithm), the online private multiplicative weights mechanism of Hardt and Rothblum [44], and its offline variant of Gupta et al. [38]; see also Hardt et al. [41]. All these algorithm can be seen to be instantiations. The connection between query release and agnostic learning was observed in [38]. The observation that the median mechanism, when analyzed using the composition theorems of Dwork et al. [32] for