A comprehensive theoretical framework for the optimization of neural
  networks classification performance with respect to weighted metrics
Francesco Marchetti Sabrina Guastavino Cristina Campi Federico Benvenuto Michele Piana
In many contexts, customized and weighted classification scores are designed
in order to evaluate the goodness of the predictions carried out by neural
networks. However, there exists a discrepancy between the maximization of such
scores and the minimization of the loss function in the training phase. In this
paper, we provide a complete theoretical setting that formalizes weighted
classification metrics and then allows the construction of losses that drive
the model to optimize these metrics of interest. After a detailed theoretical
analysis, we show that our framework includes as particular instances
well-established approaches such as classical cost-sensitive learning, weighted
cross entropy loss functions and value-weighted skill scores.