We study multiclass online prediction where the learner can predict using a list of multiple labels (as opposed to just one label in the traditional setting). We characterize learnability in this model using the $b$-ary Littlestone dimension. This dimension is a variation of the classical Littlestone dimension with the difference that binary mistake trees are replaced with $(k+1)$-ary mistake trees, where $k$ is the number of labels in the list. In the agnostic setting, we explore different scenarios depending on whether the comparator class consists of single-labeled or multi-labeled functions and its tradeoff with the size of the lists the algorithm uses. We find that it is possible to achieve negative regret in some cases and provide a complete characterization of when this is possible. As part of our work, we adapt classical algorithms such as Littlestone's SOA and Rosenblatt's Perceptron to predict using lists of labels. We also establish combinatorial results for list-learnable classes, including an list online version of the Sauer-Shelah-Perles Lemma. We state our results within the framework of pattern classes -- a generalization of hypothesis classes which can represent adaptive hypotheses (i.e. functions with memory), and model data-dependent assumptions such as linear classification with margin.

翻译：我们研究多类在线预测问题，其中学习器可以使用多个标签的列表进行预测（而非传统设置中仅预测单个标签）。我们利用 b 元 Littlestone 维数刻画该模型中的可学习性。该维数是经典 Littlestone 维数的变体，区别在于将二元错误树替换为 (k+1) 元错误树，其中 k 为列表中的标签数。在不可知设置中，我们根据比较器类别是单标签函数还是多标签函数，以及其与算法所用列表大小的权衡，探索了不同场景。我们发现在某些情况下可以实现负遗憾值，并对此类情况给出了完整刻画。作为工作的一部分，我们改编了 Littlestone SOA 和 Rosenblatt 感知机等经典算法，使其利用标签列表进行预测。我们还为列表可学习类别建立了组合数学结果，包括 Sauer-Shelah-Perles 引理的列表在线版本。我们在模式类别框架下陈述结果——该框架是假设类别的推广，能表示自适应假设（即具有记忆的函数），并对数据依赖性假设（如带间隔的线性分类）进行建模。