Actually, All Learnable Distribution Classes Are Privately Learnable

We show that every class of distributions that is learnable up to arbitrarily small target error in total variation distance with finitely many samples is also learnable under ($\epsilon ,\delta$)-differential privacy up to arbitrarily small target error with finitely many samples. This proves a conjecture of Ashtiani. Previous work by Bun, Kamath, Mouzakis, and Singhal [arXiv:2402.00267] gave a weak refutation of this conjecture, showing that this implication is not true when learnability is defined with respect to a fixed target error. In contrast, we show that, under the standard definition of learnability, the conjecture is true.