  This paper introduces the hypervolume maximization with a single solution as
an alternative to the mean loss minimization. The relationship between the two
problems is proved through bounds on the cost function when an optimal solution
to one of the problems is evaluated on the other, with a hyperparameter to
control the similarity between the two problems. This same hyperparameter
allows higher weight to be placed on samples with higher loss when computing
the hypervolume's gradient, whose normalized version can range from the mean
loss to the max loss. An experiment on MNIST with a neural network is used to
validate the theory developed, showing that the hypervolume maximization can
behave similarly to the mean loss minimization and can also provide better
performance, resulting on a 20% reduction of the classification error on the
test set.
