- We provide an extensive theoretical analysis of the properties of the considered objective and prove that maximizing this objective in any tree node simultaneously encourages balanced partition of the data in that node and improves the purity of the class distributions at its children nodes.
- We show a formal relation of this objective to some more standard entropy-based objectives, i.e., Shannon entropy, Gini-entropy and its modified variant, for which online optimization schemes in the context of multiclass classification are largely unknown. In particular we show that i) the improvement in the value of entropy resulting from performing the node split is lower-bounded by an expression that increases with the value of the objective and thus ii) the considered objective can be used as a surrogate function for indirectly optimizing any of the three considered entropy-based criteria.
- We present three boosting theorems for each of the three entropy criteria, which provide the number of iterations needed to reduce each of them below an arbitrary threshold. Their weak hypothesis assumptions rely on the considered objective function.
- We establish the error bound that relates maximizing the objective function with reducing the multi-class classification error.
- Finally, in the Appendix A we establish an empirical connection between the multiclass classification error and the entropy criteria and show that Gini-entropy most closely resembles the behavior of the test error in practice.
2. Related Work
3. Theoretical Properties of the Objective Function
4. Main Theoretical Results
- Shannon entropy :
- Gini-entropy :
- Modified Gini-entropy :
5.1. Properties of the Entropy-Based Criteria
5.1.1. Bounds on the Entropy-Based Criteria
5.1.2. Strong Concativity Properties of the Entropy-Based Criteria
5.2. Proof of Lemma 4 and Theorems 1–3
5.3. Proof of Theorem 4
Conflicts of Interest
Appendix A. Extreme Multiclass Classification Criteria
Appendix A.1. Numerical Experiments
Appendix A.2. Additional Proofs
- Let . ThenThus which, when solved, yields the lemma.
- Let (thus ). Note that can be written asThus as before we obtain which, when solved, yields the lemma. □
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