Simplicial covering dimension of extremal concept classes

Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension. We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.

Borsuk-Ulam and Replicable Learning of Large-Margin Halfspaces

Recent advances in learning theory have established that, for total concepts, list replicability, global stability, differentially private (DP) learnability, and shared-randomness replicability coincide precisely with the finiteness of the Littlestone dimension. Does the same hold for partial concept classes? We answer this question by studying the large-margin half-spaces class, which has bounded Littlestone dimension and is purely DP-learnable and shared-randomness replicable even in high dimensions. We prove that the list replicability number of $\gamma$-margin half-spaces satisfies \[ \frac{d}{2} + 1 \le \mathrm{LR}(H_{\gamma}^d) \le d, \] which increases with the dimension $d$. This reveals a surprising separation for partial concepts: list replicability and global stability do not follow from bounded Littlestone dimension, DP-learnability, or shared-randomness replicability. By applying our main theorem, we also answer the following open problems. - We prove that any disambiguation of an infinite-dimensional large-margin half-space to a total concept class has unbounded Littlestone dimension, answering an open question of Alon et al. (FOCS '21). - We prove that the maximum list-replicability number of any *finite* set of points and homogeneous half-spaces in $d$-dimensional Euclidean space is $d$, resolving a problem of Chase et al. (FOCS '23). - We prove that any disambiguation of the Gap Hamming Distance problem in the large gap regime has unbounded public-coin randomized communication complexity. This answers an open problem of Fang et al. (STOC '25). We prove the lower bound via a topological argument involving the local Borsuk-Ulam theorem of Chase et al. (STOC '24). For the upper bound, we design a learning rule that relies on certain triangulations of the cross-polytope and recent results on the generalization properties of SVM.