Differentially-Private Bayes Consistency
This resolves a fundamental theoretical gap in differentially private machine learning by enabling universal consistency, which was previously impossible in distribution-free models.
The paper constructs a universally Bayes consistent learning rule that satisfies differential privacy, showing it can learn arbitrary distributions with a single DP algorithm, unlike the distribution-free PAC model where DP learning is extremely limited. As an application, it proves VC classes can be privately learned in a semi-supervised setting with a near-optimal labeled sample complexity of Õ(d/ε).
We construct a universally Bayes consistent learning rule that satisfies differential privacy (DP). We first handle the setting of binary classification and then extend our rule to the more general setting of density estimation (with respect to the total variation metric). The existence of a universally consistent DP learner reveals a stark difference with the distribution-free PAC model. Indeed, in the latter DP learning is extremely limited: even one-dimensional linear classifiers are not privately learnable in this stringent model. Our result thus demonstrates that by allowing the learning rate to depend on the target distribution, one can circumvent the above-mentioned impossibility result and in fact, learn \emph{arbitrary} distributions by a single DP algorithm. As an application, we prove that any VC class can be privately learned in a semi-supervised setting with a near-optimal \emph{labeled} sample complexity of $\tilde{O}(d/\varepsilon)$ labeled examples (and with an unlabeled sample complexity that can depend on the target distribution).