As stated in Section A, we apply the softmax function such thatRAPsoftmax outputs a synthetic datasetdrawnfromsomeprobabilisticfamilyofdistributionsD = n σ(M)| M Rn

Pt i=1eqi(x)(eai eqi(Di 1)) which is the exactly the distribution computed byMWEM. D(x)log(D(x)) (6) The optimization problem becomesDt = argminD (X)Lmwem(D, eQt, eAt). We show the exact details ofGEM in Algorithms 2 and 3. Note that given a vector of queries Qt = hq1,...,qti,wedefinefQt() = hfq1(),...,fqt()i. B.1 Lossfunction(fork-waymarginals)anddistributionalfamily For anyz R,G(z)outputs a distribution over each attribute, which we can use to calculate the answer toaquery viafq. Empirically,wefindthatour model tends to better capture the distribution of the overall private dataset in this way (Figure 3).