  In deep learning, classification tasks are formalized as optimization
problems often solved via the minimization of the cross-entropy. However,
recent advancements in the design of objective functions allow the usage of the
$f$-divergence to generalize the formulation of the optimization problem for
classification. We adopt a Bayesian perspective and formulate the
classification task as a maximum a posteriori probability problem. We propose a
class of objective functions based on the variational representation of the
$f$-divergence. Furthermore, driven by the challenge of improving the
state-of-the-art approach, we propose a bottom-up method that leads us to the
formulation of an objective function corresponding to a novel $f$-divergence
referred to as shifted log (SL). We theoretically analyze the objective
functions proposed and numerically test them in three application scenarios:
toy examples, image datasets, and signal detection/decoding problems. The
analyzed scenarios demonstrate the effectiveness of the proposed approach and
that the SL divergence achieves the highest classification accuracy in almost
all the considered cases.
