The Dimension of Self-Directed Learning
This work addresses a foundational problem in online learning theory by providing a precise characterization of self-directed learning, which is incremental but offers new insights into learnability gaps.
The paper tackles the problem of characterizing self-directed learning complexity by introducing a dimension called SDdim, which exactly determines the mistake-bound for any concept class in both binary and multi-class settings, with applications to examples like axis-aligned rectangles and linear separators.
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.