Multiclass Transductive Online Learning

• Main Authors: Hanneke, Steve, Raman, Vinod, Shaeiri, Amirreza, Subedi, Unique
• Format: Journal Article
• Language: English
• Published: 03-11-2024
• Subjects: Computer Science - Learning; Statistics - Machine Learning
• DOI: 10.48550/arxiv.2411.01634

We consider the problem of multiclass transductive online learning when the
number of labels can be unbounded. Previous works by Ben-David et al. [1997]
and Hanneke et al. [2023b] only consider the case of binary and finite label
spaces, respectively. The latter work determined that their techniques fail to
extend to the case of unbounded label spaces, and they pose the question of
characterizing the optimal mistake bound for unbounded label spaces. We answer
this question by showing that a new dimension, termed the Level-constrained
Littlestone dimension, characterizes online learnability in this setting. Along
the way, we show that the trichotomy of possible minimax rates of the expected
number of mistakes established by Hanneke et al. [2023b] for finite label
spaces in the realizable setting continues to hold even when the label space is
unbounded. In particular, if the learner plays for $T \in \mathbb{N}$ rounds,
its minimax expected number of mistakes can only grow like $\Theta(T)$,
$\Theta(\log T)$, or $\Theta(1)$. To prove this result, we give another
combinatorial dimension, termed the Level-constrained Branching dimension, and
show that its finiteness characterizes constant minimax expected
mistake-bounds. The trichotomy is then determined by a combination of the
Level-constrained Littlestone and Branching dimensions. Quantitatively, our
upper bounds improve upon existing multiclass upper bounds in Hanneke et al.
[2023b] by removing the dependence on the label set size. In doing so, we
explicitly construct learning algorithms that can handle extremely large or
unbounded label spaces. A key and novel component of our algorithm is a new
notion of shattering that exploits the sequential nature of transductive online
learning. Finally, we complete our results by proving expected regret bounds in
the agnostic setting, extending the result of Hanneke et al. [2023b].