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# Universal Rates for Multiclass Learning

*Proceedings of Thirty Sixth Conference on Learning Theory*, PMLR 195:5615-5681, 2023.

#### Abstract

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.