We study multiclass classification in the agnostic adversarial online learning setting. As our main result, we prove that any multiclass concept class is agnostically learnable if and only if its Littlestone dimension is finite. This solves an open problem studied by Daniely, Sabato, Ben-David, and Shalev-Shwartz (2011,2015) who handled the case when the number of classes (or labels) is bounded. We also prove a separation between online learnability and online uniform convergence by exhibiting an easy-to-learn class whose sequential Rademacher complexity is unbounded. Our learning algorithm uses the multiplicative weights algorithm, with a set of experts defined by executions of the Standard Optimal Algorithm on subsequences of size Littlestone dimension. We argue that the best expert has regret at most Littlestone dimension relative to the best concept in the class. This differs from the well-known covering technique of Ben-David, P\'{a}l, and Shalev-Shwartz (2009) for binary classification, where the best expert has regret zero.

翻译：我们在对抗型在线学习设定下研究多类别分类问题。作为主要结论，我们证明任意多类别概念类在对抗型设定下是可学习的当且仅当其Littlestone维度有限。这解决了Daniely、Sabato、Ben-David和Shalev-Shwartz（2011,2015）研究的开放问题，他们仅处理了类别（或标签）数目有界的情况。我们还通过构造一个易于学习但其序列Rademacher复杂度无界的类，证明了在线可学习性与在线一致收敛之间的分离性。我们的学习算法采用乘法权重算法，其专家集由在大小为Littlestone维度的子序列上运行标准最优算法（Standard Optimal Algorithm）所定义。我们论证最优专家相对于该类中最优概念的遗憾至多为Littlestone维度。这与Ben-David、Pál和Shalev-Shwartz（2009）针对二分类问题的著名覆盖技术不同，在该技术中最优专家的遗憾为零。