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We show that every approximately differentially private learning algorithm(possibly improper) for a class $H$ with Littlestone dimension~$d$ requires$\Omega\bigl(\log^*(d)\bigr)$ examples. As a corollary it follows that theclass of thresholds over $\mathbb{N}$ can not be learned in a private manner;this resolves open question due to [Bun et al., 2015, Feldman and Xiao, 2015].We leave as an open question whether every class with a finite Littlestonedimension can be learned by an approximately differentially private algorithm.

Private PAC learning implies finite Littlestone dimension