🤖 AI Summary
This work addresses the potential nonexistence of a global optimum in linear ensembles of multiple binary classifiers by proposing a theoretical framework grounded in truth-table logical structuring and equivalence class partitioning, which establishes sufficient conditions for the existence of a convexified empirical risk minimizer. By introducing a multidimensional generalization of classification-calibrated loss functions and the notion of φ-frontiers, the study analyzes solution stability in relation to data quality. Under exponential (Boost) and logistic (Logit) losses, the authors derive, for the first time, explicit closed-form expressions for the optimal ensemble weights and fully characterize all solution regimes in the three-classifier setting. This approach circumvents iterative optimization, thereby substantially enhancing both the interpretability and computational efficiency of ensemble models.
📝 Abstract
This paper studies an optimal linear combination of binary classifiers based on a logical structuration of the dataset via truth tables. The given classifiers partition data into equivalence classes, allowing for a rigorous analysis of the convexified empirical risk through a multidimensional generalization of classification calibrated functions. We establish sufficient conditions for the existence and uniqueness of the (global) point of minimum of the convexified empirical risk for any list of classifiers (when the number of classifiers is large, there frequently could be no point of minimum). In the case of three classifiers, our analysis allows to list all the configurations leading to either a unique solution, infima or non-unique points of minimum. Furthermore, we derive explicit analytical formulae for optimal weights using Exponential (Boost) and Logistic (Logit) loss functions, bypassing iterative optimization.
The stability of the resulting classifier and the analysis of data quality can be evaluated through the introduction of the notion of $φ$-frontiers.