Private sample-complexity separation for finite classes

Construct a sequence of finite concept classes whose logarithmic cardinality is superpolynomial in VC dimension while every differentially private learner requires Ω(log|Cκ|) samples to learn the class.

Background

The open problem asks whether private learning can be exponentially harder than suggested by VC dimension for a suitable sequence of finite classes. The desired separation requires both superpolynomial growth of log class size relative to VC dimension and an Ω(log class-size) private sample lower bound.

The paper proves only a conditional lower bound for classes containing disjoint-threshold minors, yielding Ω(k log* N), and explicitly leaves the requested existential construction unresolved, particularly without a small-δ restriction.

References

Does there exist a sequence of finite-size classes $C = {C_\kappa}{\kappa\in \mathbb N}$ where: (1) $\lim{\kappa\to \infty} |C_\kappa| = \infty,$ (2) $\log |C_\kappa|$ is superpolynomial in $VC(C_\kappa)$, and (3) The number of samples required to learn $C_\kappa$ under differential privacy is $\Omega(\log |C_\kappa|)$.

— VALG: An Agentic System for ML Theory Research  (2608.13060 - Zhang et al., 13 Aug 2026) in Section 4, Subsection “Does Differential Privacy Make PAC Learning Much Harder?”, Subproblem 2: Class Existence