This paper presents a 4-objective evolutionary multiobjective optimization study for optimizing the error rates (false positives, false negatives), reliability, and complexity of binary classifiers. The example taken is the email anti-spam filtering problem. The two major goals of the optimization is to minimize the error rates that is the false negative rate and the false positive rate. Our approach discusses three-way classification, that is the binary classifier can also not classify an instance in cases where there is not enough evidence to assign the instance to one of the two classes. In this case the instance is marked as suspicious but still presented to the user. The number of unclassified (suspicious) instances should be minimized, as long as this does not lead to errors. This will be termed the coverage objective. The set (ensemble) of rules needed for the anti-spam filter to operate in optimal conditions is addressed as a fourth objective. All objectives stated above are in general conflicting with each other and that is why we address the problem as a 4-objective (quadcriteria) optimization problem. We assess the performance of a set of state-of-the-art evolutionary multiobjective optimization algorithms. These are NSGA-II, SPEA2, and the hypervolume indicator-based SMS-EMOA. Focusing on the anti-spam filter optimization, statistical comparisons on algorithm performance are provided on several benchmarks and a range of performance indicators. Moreover, the resulting 4-D Pareto hyper-surface is discussed in the context of binary classifier optimization.

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Binary classification; Evolutionary multi-objective optimization; Parallel coordinates; Parsimony; Three-way classification;

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