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This study investigates the optimal linear combination of binary classifiers by structuring the dataset using truth tables, which facilitates a detailed examination of the convexified empirical risk. The authors establish sufficient conditions for the existence and uniqueness of the global minimum of this risk, particularly when dealing with a large number of classifiers. For three classifiers, they enumerate configurations that lead to unique or non-unique minima and provide explicit analytical formulas for optimal weights using Exponential and Logistic loss functions, avoiding the need for iterative optimization.
Identifying unique solutions in classifier combinations could revolutionize how we approach multi-class classification problems.
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 $\phi$-frontiers.