# Huber loss

In statistics, the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the squared error loss. A variant for classification is also sometimes used.

## Definition

Huber loss (green, ) and squared error loss (blue) as a function of

The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by[1]

This function is quadratic for small values of a, and linear for large values, with equal values and slopes of the different sections at the two points where . The variable a often refers to the residuals, that is to the difference between the observed and predicted values , so the former can be expanded to[2]

## Motivation

Two very commonly used loss functions are the squared loss, , and the absolute loss, . The squared loss function results in an arithmetic mean-unbiased estimator, and the absolute-value loss function results in a median-unbiased estimator (in the one-dimensional case, and a geometric median-unbiased estimator for the multi-dimensional case). The squared loss has the disadvantage that it has the tendency to be dominated by outliers—when summing over a set of 's (as in ), the sample mean is influenced too much by a few particularly large a-values when the distribution is heavy tailed: in terms of estimation theory, the asymptotic relative efficiency of the mean is poor for heavy-tailed distributions.

As defined above, the Huber loss function is convex in a uniform neighborhood of its minimum , at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points and . These properties allow it to combine much of the sensitivity of the mean-unbiased, minimum-variance estimator of the mean (using the quadratic loss function) and the robustness of the median-unbiased estimator (using the absolute value function).

## Pseudo-Huber loss function

The Pseudo-Huber loss function can be used as a smooth approximation of the Huber loss function, and ensures that derivatives are continuous for all degrees. It is defined as[3][4]

As such, this function approximates for small values of , and approximates a straight line with slope for large values of .

While the above is the most common form, other smooth approximations of the Huber loss function also exist.[5]

## Variant for classification

For classification purposes, a variant of the Huber loss called modified Huber is sometimes used. Given a prediction (a real-valued classifier score) and a true binary class label , the modified Huber loss is defined as[6]

The term is the hinge loss used by support vector machines; the quadratically smoothed hinge loss is a generalization of .[6]

## Applications

The Huber loss function is used in robust statistics, M-estimation and additive modelling.[7]