We use and adapt the Borsuk-Ulam Theorem from topology to derive limitations on list-replicable and globally stable learning algorithms. We further demonstrate the applicability of our methods in combinatorics and topology. We show that, besides trivial cases, both list-replicable and globally stable learning are impossible in the agnostic PAC setting. This is in contrast with the realizable case where it is known that any class with a finite Littlestone dimension can be learned by such algorithms. In the realizable PAC setting, we sharpen previous impossibility results and broaden their scope. Specifically, we establish optimal bounds for list replicability and global stability numbers in finite classes. This provides an exponential improvement over previous works and implies an exponential separation from the Littlestone dimension. We further introduce lower bounds for weak learners, i.e., learners that are only marginally better than random guessing. Lower bounds from previous works apply only to stronger learners. To offer a broader and more comprehensive view of our topological approach, we prove a local variant of the Borsuk-Ulam theorem in topology and a result in combinatorics concerning Kneser colorings. In combinatorics, we prove that if $c$ is a coloring of all non-empty subsets of $[n]$ such that disjoint sets have different colors, then there is a chain of subsets that receives at least $1+ \lfloor n/2\rfloor$ colors (this bound is sharp). In topology, we prove e.g. that for any open antipodal-free cover of the $d$-dimensional sphere, there is a point $x$ that belongs to at least $t=\lceil\frac{d+3}{2}\rceil$ sets.

翻译：我们运用并适配拓扑学中的Borsuk-Ulam定理，推导出列表可复制与全局稳定学习算法的局限性，并进一步展示该方法在组合学与拓扑学中的适用性。研究表明，除平凡情形外，在不可知PAC（agnostic PAC）设定下，列表可复制与全局稳定学习均不可实现。这与可实现情形形成对比——已知在该情形下，任何具有有限Littlestone维度的类别均可通过此类算法学习。在可实现PAC设定中，我们进一步强化了先前的不可能性结论并拓展其适用范围：具体而言，为有限类别中的列表可复制性与全局稳定性指数建立了最优界，该结果相比先前工作呈指数级改进，并暗示其与Littlestone维度存在指数级分离。我们进一步引入了弱学习器（即仅比随机猜测略优的学习器）的下界，而先前工作的下界仅适用于更强学习器。为提供本拓扑方法更全面深入的视角，我们证明了拓扑学中Borsuk-Ulam定理的局部变体，以及组合学中关于Kneser着色的一个结果。在组合学中，我们证明：若$c$是$[n]$所有非空子集的一种着色，满足不相交集合具有不同颜色，则存在一条包含至少$1+\lfloor n/2\rfloor$种颜色的子集链（该界是紧的）。在拓扑学中，我们证明：对于$d$维球面的任意开无对径覆盖，存在一个点$x$至少属于$t=\lceil\frac{d+3}{2}\rceil$个集合。