Learning multi-class decision trees over arbitrary real domains

Investigate learning multi-class decision trees—decision trees in which each leaf corresponds to a distinct class label—over arbitrary finite point sets X ⊂ R^d in the membership-query active learning setting. Determine efficient algorithms and characterize the query complexity for exact or approximate learning of such multi-class decision trees over arbitrary X ⊂ R^d.

Background

The paper studies active learning of halfspaces without point synthesis and achieves optimal query bounds in several settings. It notes that for Boolean decision trees of depth at least two over arbitrary point sets X ⊂ Rd, there is an Ω(n) query lower bound, which makes direct generalization beyond depth one infeasible in this worst-case domain setting.

As an alternative generalization, the authors propose studying multi-class decision trees, where each leaf corresponds to a distinct label, over arbitrary X ⊂ Rd. They explicitly state that they are unaware of prior work on this question, highlighting it as a potentially unexplored problem and suggesting it as a direction for future paper.

References

On the other hand, an alternative generalization is the problem of learning multi-class decision trees, where each leaf must correspond to a distinct label, over arbitrary X \subset Rd. We do not know of any prior works studying this question and we believe this may be an interesting direction for future work.

— Actively Learning Halfspaces without Synthetic Data (2509.20848 - Black et al., 25 Sep 2025) in Section 1, Introduction (Main application: decision stumps, or axis-aligned halfspaces)