We consider the problem of multiclass transductive online learning when the number of labels can be unbounded. Previous works by Ben-David et al. [1997] and Hanneke et al. [2023b] only consider the case of binary and finite label spaces, respectively. The latter work determined that their techniques fail to extend to the case of unbounded label spaces, and they pose the question of characterizing the optimal mistake bound for unbounded label spaces. We answer this question by showing that a new dimension, termed the Level-constrained Littlestone dimension, characterizes online learnability in this setting. Along the way, we show that the trichotomy of possible minimax rates of the expected number of mistakes established by Hanneke et al. [2023b] for finite label spaces in the realizable setting continues to hold even when the label space is unbounded. In particular, if the learner plays for $T \in \mathbb{N}$ rounds, its minimax expected number of mistakes can only grow like $\Theta(T)$, $\Theta(\log T)$, or $\Theta(1)$. To prove this result, we give another combinatorial dimension, termed the Level-constrained Branching dimension, and show that its finiteness characterizes constant minimax expected mistake-bounds. The trichotomy is then determined by a combination of the Level-constrained Littlestone and Branching dimensions. Quantitatively, our upper bounds improve upon existing multiclass upper bounds in Hanneke et al. [2023b] by removing the dependence on the label set size. In doing so, we explicitly construct learning algorithms that can handle extremely large or unbounded label spaces. A key and novel component of our algorithm is a new notion of shattering that exploits the sequential nature of transductive online learning. Finally, we complete our results by proving expected regret bounds in the agnostic setting, extending the result of Hanneke et al. [2023b].

翻译：本文研究了标签数量可能无界情况下的多类别转导在线学习问题。Ben-David等人[1997]和Hanneke等人[2023b]的前期工作分别仅考虑了二元标签空间和有限标签空间的情形。后者指出其技术无法推广到无界标签空间的情况，并提出了刻画无界标签空间最优错误界的问题。我们通过证明一种称为"层级约束Littlestone维数"的新维度能够刻画该场景下的在线可学习性，从而回答了这个问题。在此过程中，我们证明了Hanneke等人[2023b]在可实现场景下为有限标签空间建立的三类极小极大期望错误率，在标签空间无界时仍然成立。具体而言，若学习器进行$T \in \mathbb{N}$轮博弈，其极小极大期望错误数只能以$\Theta(T)$、$\Theta(\log T)$或$\Theta(1)$的速率增长。为证明该结论，我们提出了另一种称为"层级约束分支维数"的组合维度，并证明其有限性刻画了常数极小极大期望错误界。最终的三类划分由层级约束Littlestone维数与分支维数共同决定。在量化分析上，我们的上界通过消除对标签集规模的依赖，改进了Hanneke等人[2023b]中现有的多类别上界。为此，我们显式构建了能够处理极大或无界标签空间的学习算法。算法的关键创新组件是一种利用转导在线学习时序特性的新型"粉碎"概念。最后，我们在不可知场景下证明了期望遗憾界，从而扩展了Hanneke等人[2023b]的结果。