We study universal rates for multiclass classification, establishing the optimal rates (up to log factors) for all hypothesis classes. This generalizes previous results on binary classification (Bousquet, Hanneke, Moran, van Handel, and Yehudayoff, 2021), and resolves an open question studied by Kalavasis, Velegkas, and Karbasi (2022) who handled the multiclass setting with a bounded number of class labels. In contrast, our result applies for any countable label space. Even for finite label space, our proofs provide a more precise bounds on the learning curves, as they do not depend on the number of labels. Specifically, we show that any class admits exponential rates if and only if it has no infinite Littlestone tree, and admits (near-)linear rates if and only if it has no infinite Daniely-Shalev-Shwartz-Littleston (DSL) tree, and otherwise requires arbitrarily slow rates. DSL trees are a new structure we define in this work, in which each node of the tree is given by a pseudo-cube of possible classifications of a given set of points. Pseudo-cubes are a structure, rooted in the work of Daniely and Shalev-Shwartz (2014), and recently shown by Brukhim, Carmon, Dinur, Moran, and Yehudayoff (2022) to characterize PAC learnability (i.e., uniform rates) for multiclass classification. We also resolve an open question of Kalavasis, Velegkas, and Karbasi (2022) regarding the equivalence of classes having infinite Graph-Littlestone (GL) trees versus infinite Natarajan-Littlestone (NL) trees, showing that they are indeed equivalent.

翻译：我们研究多类别分类的普适速率，为所有假设类建立了最优速率（对数因子内）。这推广了先前关于二分类的结果（Bousquet, Hanneke, Moran, van Handel 和 Yehudayoff, 2021），并解决了 Kalavasis、Velegkas 和 Karbasi（2022）研究的一个开放问题，他们的工作处理了标签类别数量有界的多类别场景。相比之下，我们的结果适用于任意可数标签空间。即使对于有限标签空间，我们的证明也提供了学习曲线更精确的界，因为它们不依赖于标签数量。具体而言，我们证明：任何类若且唯若无无限Littlestone树时具有指数级速率，若且唯若无无限Daniely-Shalev-Shwartz-Littlestone（DSL）树时具有（近）线性速率，否则需要任意慢的速率。DSL 树是本文定义的新结构，其中每个节点由给定点集可能分类的伪立方体给出。伪立方体是一种源于 Daniely 和 Shalev-Shwartz（2014）工作的结构，且最近由 Brukhim、Carmon、Dinur、Moran 和 Yehudayoff（2022）证明其刻画了多类别分类的PAC可学习性（即一致速率）。我们还解决了 Kalavasis、Velegkas 和 Karbasi（2022）的一个开放问题，即关于具有无限图-Littlestone（GL）树的类与具有无限Natarajan-Littlestone（NL）树的类之间的等价性，证明它们确实是等价的。