Borsuk-Ulam and Replicable Learning of Large-Margin Halfspaces
Ari Blondal, Hamed Hatami, Pooya Hatami, Chavdar Lalov, Sivan Tretiak
TL;DR
The paper analyzes list replicability for large-margin half-spaces in high dimensions, proving that the list replicability number LR grows with dimension as $(d/2)+1 \le LR_\\epsilon(\\mathcal{H}^d_\\gamma) \le d$. The lower bound uses a local Borsuk–Ulam argument, while the upper bound constructs a list-replicable learning rule based on hard-SVM and a rounding net. These results imply separations between list replicability and other stability notions for partial classes, and yield lower bounds on the Littlestone dimension and randomized communication for disambiguations of both large-margin half-spaces and Gap Hamming Distance. The work also establishes a first super-constant separation between randomized and pseudo-deterministic communication, and clarifies the finitary behavior of list replicability for finite hyperplane arrangements. Overall, it reveals a nuanced landscape for partial concepts where LR can diverge from classical measures and highlights topological methods as a powerful tool in learning theory and communication complexity.
Abstract
We prove that the list replicability number of $d$-dimensional $γ$-margin half-spaces satisfies \[ \frac{d}{2}+1 \le \mathrm{LR}(H^d_γ) \le d, \] which grows with dimension. This resolves several open problems: $\bullet$ Every disambiguation of infinite-dimensional large-margin half-spaces to a total concept class has unbounded Littlestone dimension, answering an open question of Alon, Hanneke, Holzman, and Moran (FOCS '21). $\bullet$ Every disambiguation of the Gap Hamming Distance problem in the large gap regime has unbounded public-coin randomized communication complexity. This answers an open question of Fang, Göös, Harms, and Hatami (STOC '25). $\bullet$ There is a separation of $O(1)$ vs $ω(1)$ between randomized and pseudo-deterministic communication complexity. $\bullet$ 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, Moran, and Yehudayoff (FOCS '23). $\bullet$ There exists a partial concept class with Littlestone dimension $1$ such that all its disambiguations have infinite Littlestone dimension. This resolves a problem of Cheung, H. Hatami, P. Hatami, and Hosseini (ICALP '23). Our lower bound follows from a topological argument based on a local Borsuk-Ulam theorem. For the upper bound, we construct a list-replicable learning rule using the generalization properties of SVMs.