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Multiclass problems are often decomposed into multiple binary problems thatare solved by individual binary classifiers whose results are integrated into afinal answer. Various methods, including all-pairs (APs), one-versus-all (OVA),and error correcting output code (ECOC), have been studied, to decomposemulticlass problems into binary problems. However, little study has been madeto optimally aggregate binary problems to determine a final answer to themulticlass problem. In this paper we present a convex optimization method foran optimal aggregation of binary classifiers to estimate class membershipprobabilities in multiclass problems. We model the class membership probabilityas a softmax function which takes a conic combination of discrepancies inducedby individual binary classifiers, as an input. With this model, we formulatethe regularized maximum likelihood estimation as a convex optimization problem,which is solved by the primal-dual interior point method. Connections of ourmethod to large margin classifiers are presented, showing that the large marginformulation can be considered as a limiting case of our convex formulation.Numerical experiments on synthetic and real-world data sets demonstrate thatour method outperforms existing aggregation methods as well as direct methods,in terms of the classification accuracy and the quality of class membershipprobability estimates.

Convex Optimization for Binary Classifier Aggregation in Multiclass Problems