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，其中学习者可以使用多个标签的列表进行预测（区别于传统设置中仅使用单个标签）。我们通过$b$元Littlestone维度刻画了该模型下的可学习性。该维度是经典Littlestone维度的变体，区别在于将二元错误树替换为$(k+1)$元错误树，其中$k$是列表中的标签数量。在不可知论设置中，我们探究了不同场景，具体取决于比较器类别由单标签或多标签函数构成，以及其与算法使用的列表规模之间的权衡。我们发现某些情况下可实现负遗憾，并给出了此类情况的完整刻画。作为研究的一部分，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结果，包括Sauer-Shelah-Perles引理的在线列表版本。我们在模式类别框架下阐述了所有结果——模式类别是假设类别的泛化，能够表示自适应假设（即具有记忆的函数），并建模依赖数据的假设（例如带间隔的线性分类）。