  Many differentially private algorithms for answering database queries involve
a step that reconstructs a discrete data distribution from noisy measurements.
This provides consistent query answers and reduces error, but often requires
space that grows exponentially with dimension. Private-PGM is a recent approach
that uses graphical models to represent the data distribution, with complexity
proportional to that of exact marginal inference in a graphical model with
structure determined by the co-occurrence of variables in the noisy
measurements. Private-PGM is highly scalable for sparse measurements, but may
fail to run in high dimensions with dense measurements. We overcome the main
scalability limitation of Private-PGM through a principled approach that
relaxes consistency constraints in the estimation objective. Our new approach
works with many existing private query answering algorithms and improves
scalability or accuracy with no privacy cost.
