  The $F_\beta$ score is a commonly used measure of classification performance,
which plays crucial roles in classification tasks with imbalanced data sets.
However, the $F_\beta$ score cannot be used as a loss function by
gradient-based learning algorithms for optimizing neural network parameters due
to its non-differentiability. On the other hand, commonly used loss functions
such as the binary cross-entropy (BCE) loss are not directly related to
performance measures such as the $F_\beta$ score, so that neural networks
optimized by using the loss functions may not yield optimal performance
measures. In this study, we investigate a relationship between classification
performance measures and loss functions in terms of the gradients with respect
to the model parameters. Then, we propose a differentiable surrogate loss
function for the optimization of the $F_\beta$ score. We show that the gradient
paths of the proposed surrogate $F_\beta$ loss function approximate the
gradient paths of the large sample limit of the $F_\beta$ score. Through
numerical experiments using ResNets and benchmark image data sets, it is
demonstrated that the proposed surrogate $F_\beta$ loss function is effective
for optimizing $F_\beta$ scores under class imbalances in binary classification
tasks compared with other loss functions.
