On the Universality of the Logistic Loss Function

TL;DR

This paper demonstrates that for binary classification, the divergence from smooth, proper, convex loss functions is bounded by the KL divergence, justifying the widespread use of log-loss across various machine learning models.

Contribution

It establishes that minimizing log-loss bounds the divergence of other loss functions, providing a theoretical foundation for its broad application.

Findings

KL divergence bounds other convex loss divergences from above

Log-loss minimizes an upper bound to various loss functions

Introduces new divergence inequalities similar to Pinsker inequality

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 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.