  Generative models trained with Differential Privacy (DP) can be used to
generate synthetic data while minimizing privacy risks. We analyze the impact
of DP on these models vis-a-vis underrepresented classes/subgroups of data,
specifically, studying: 1) the size of classes/subgroups in the synthetic data,
and 2) the accuracy of classification tasks run on them. We also evaluate the
effect of various levels of imbalance and privacy budgets. Our analysis uses
three state-of-the-art DP models (PrivBayes, DP-WGAN, and PATE-GAN) and shows
that DP yields opposite size distributions in the generated synthetic data. It
affects the gap between the majority and minority classes/subgroups; in some
cases by reducing it (a "Robin Hood" effect) and, in others, by increasing it
(a "Matthew" effect). Either way, this leads to (similar) disparate impacts on
the accuracy of classification tasks on the synthetic data, affecting
disproportionately more the underrepresented subparts of the data.
Consequently, when training models on synthetic data, one might incur the risk
of treating different subpopulations unevenly, leading to unreliable or unfair
conclusions.
