  A loss function measures the discrepancy between the true values
(observations) and their estimated fits, for a given instance of data. A loss
function is said to be proper (unbiased, Fisher consistent) if the fits are
defined over a unit simplex, and the minimizer of the expected loss is the true
underlying probability of the data. Typical examples are the zero-one loss, the
quadratic loss and the Bernoulli log-likelihood loss (log-loss). In this work
we show that for binary classification problems, the divergence associated with
smooth, proper and convex loss functions is bounded from above by the
Kullback-Leibler (KL) divergence, up to a multiplicative normalization
constant. It implies that by minimizing the log-loss (associated with the KL
divergence), we minimize an upper bound to any choice of loss functions from
this set. This property justifies the broad use of log-loss in regression,
decision trees, deep neural networks and many other applications. In addition,
we show that the KL divergence bounds from above any separable Bregman
divergence that is convex in its second argument (up to a multiplicative
normalization constant). This result introduces a new set of divergence
inequalities, similar to the well-known Pinsker inequality.
