Understanding the self-directed learning complexity has been an important problem that has captured the attention of the online learning theory community since the early 1990s. Within this framework, the learner is allowed to adaptively choose its next data point in making predictions unlike the setting in adversarial online learning. In this paper, we study the self-directed learning complexity in both the binary and multi-class settings, and we develop a dimension, namely $SDdim$, that exactly characterizes the self-directed learning mistake-bound for any concept class. The intuition behind $SDdim$ can be understood as a two-player game called the "labelling game". Armed with this two-player game, we calculate $SDdim$ on a whole host of examples with notable results on axis-aligned rectangles, VC dimension $1$ classes, and linear separators. We demonstrate several learnability gaps with a central focus on self-directed learning and offline sequence learning models that include either the best or worst ordering. Finally, we extend our analysis to the self-directed binary agnostic setting where we derive upper and lower bounds.

翻译：理解自导向学习的复杂性自20世纪90年代初以来一直是令在线学习理论界关注的重要问题。在该框架下，学习者能够自适应地选择下一个数据点进行预测，这与对抗性在线学习设置不同。本文研究二分类和多分类场景中的自导向学习复杂性，并提出了一个称为$SDdim$的维度，该维度精确刻画了任意概念类的自导向学习错误界。$SDdim$背后的直觉可通过一个称为"标记博弈"的双人博弈来理解。借助这一双人博弈，我们在大量示例中计算了$SDdim$，其中在轴对齐矩形、VC维度为$1$的类别和线性可分器上取得了显著结果。我们证明了几种学习性差距，重点关注包含最佳或最差排序的自导向学习和离线序列学习模型。最后，我们将分析扩展到自导向二分类不可知场景，并推导出上下界。