Near-Optimal List Replicability from Littlestone Dimension

Ari Blondal, Pooya Hatami, Chavdar Lalov, Sivan Tretiak

We study the list replicability number in the realizable PAC learning framework and prove that every concept class $H$ over a countable domain with Littlestone dimension $d$ and dual VC dimension $v^*$ satisfies \[ \mathrm{LR}(H)\leq 2^{O(d\min\{\log d,v^*\})}. \] This matches the worst-case lower bound of $2^{Ω(d)}$ up to a logarithmic factor in the exponent and exponentially improves the previous best upper bound $\mathrm{LR}(H)\le 2^{2^{O(d)}}$ of Bun, Livni, and Moran [FOCS 2020]. Unlike earlier bounds of $2^{O_ε(d^2)}$ for fixed accuracy $ε>0$ implied by the work of Ghazi, Golowich, Kumar, and Manurangsi [STOC 2021] and Lyu [STOC 2026], our bound is independent of $ε$. The $2^{O(d\log d)}$ bound follows from a new inductive framework for bounding list replicability that builds on the recent notion of irreducibility from differentially private learning. The $2^{O(dv^*)}$ bound follows from a simple globally stable learner that takes the majority vote of multiple empirical risk minimizers, with stability established using tools from stable regularity.