Abstract-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 zeroone 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 KullbackLeibler (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.
Somewhat related, in REF Painsky and Wornell show that for binary classification problems the logarithmic loss dominates "universally" any other convenient (i.e., smooth, proper and convex) loss function, in the sense that by minimizing the logarithmic loss one minimizes the regret that is associated with any such measures.
On the Universality of the Logistic Loss Function
2018 IEEE International Symposium on Information Theory (ISIT)
2018 IEEE International Symposium on Information Theory (ISIT)
Computer Science Mathematics