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：我们研究多类别在线预测问题，其中预测器可以使用多个标签的列表进行预测（而非传统设置中仅使用单一标签）。我们通过$b$元Littlestone维度刻画该模型中的可学习性。该维度是经典Littlestone维度的变体，其区别在于将二叉错误树替换为$(k+1)$元错误树（其中$k$为列表中的标签数量）。在不可知论设置中，我们根据比较器类别由单标签函数或多标签函数构成以及其与算法所用列表大小的权衡，探索不同场景。我们发现某些情况下可实现负遗憾，并给出这种可能性成立的完整刻画。作为工作的一部分，我们改编了Littlestone的SOA和Rosenblatt的感知机等经典算法，使其使用标签列表进行预测。我们还建立了列表可学习类别的组合学结果，包括Sauer-Shelah-Perles引理的列表在线版本。我们以模式类别（假设类别的泛化形式，可表示自适应假设（即带记忆函数）并建模数据依赖假设如带间隔线性分类）为框架陈述所有结论。