  Local differential privacy is a powerful method for privacy-preserving data
collection. In this paper, we develop a framework for training Generative
Adversarial Networks (GANs) on differentially privatized data. We show that
entropic regularization of optimal transport - a popular regularization method
in the literature that has often been leveraged for its computational benefits
- enables the generator to learn the raw (unprivatized) data distribution even
though it only has access to privatized samples. We prove that at the same time
this leads to fast statistical convergence at the parametric rate. This shows
that entropic regularization of optimal transport uniquely enables the
mitigation of both the effects of privatization noise and the curse of
dimensionality in statistical convergence. We provide experimental evidence to
support the efficacy of our framework in practice.
