In this paper we study the problem of multiclass classification with a bounded number of different labels $k$, in the realizable setting. We extend the traditional PAC model to a) distribution-dependent learning rates, and b) learning rates under data-dependent assumptions. First, we consider the universal learning setting (Bousquet, Hanneke, Moran, van Handel and Yehudayoff, STOC '21), for which we provide a complete characterization of the achievable learning rates that holds for every fixed distribution. In particular, we show the following trichotomy: for any concept class, the optimal learning rate is either exponential, linear or arbitrarily slow. Additionally, we provide complexity measures of the underlying hypothesis class that characterize when these rates occur. Second, we consider the problem of multiclass classification with structured data (such as data lying on a low dimensional manifold or satisfying margin conditions), a setting which is captured by partial concept classes (Alon, Hanneke, Holzman and Moran, FOCS '21). Partial concepts are functions that can be undefined in certain parts of the input space. We extend the traditional PAC learnability of total concept classes to partial concept classes in the multiclass setting and investigate differences between partial and total concepts.

翻译：本文研究在可实现设定下，具有有限标签数$k$的多类分类问题。我们将传统PAC模型扩展到：a) 依赖于分布的学习速率，以及b) 数据依赖假设下的学习速率。首先，我们考虑通用学习设定（Bousquet, Hanneke, Moran, van Handel 和 Yehudayoff, STOC '21），针对该设定，我们给出了对任意固定分布均成立的可实现学习速率的完整刻画。特别地，我们展示了如下三分法：对于任何概念类，最优学习速率要么是指数级、线性级，要么是任意慢速。此外，我们提供了表征这些速率发生时底层假设类的复杂度度量。其次，我们考虑结构化数据的多类分类问题（例如位于低维流形或满足间隔条件的数据），该设定由部分概念类（Alon, Hanneke, Holzman 和 Moran, FOCS '21）所刻画。部分概念是在输入空间某些部分可能未定义的函数。我们将在多类设定中，将传统全概念类的PAC可学习性扩展到部分概念类，并研究部分概念与全概念之间的差异。