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, $\delta =1$) and squared error loss (blue) as a function of $y-f(x)$

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

$L_{\delta }(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}$

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 $|a|=\delta$. The variable a often refers to the residuals, that is to the difference between the observed and predicted values $a=y-f(x)$, so the former can be expanded to

$L_{\delta }(y,f(x))={\begin{cases}{\frac {1}{2}}(y-f(x))^{2}&{\textrm {for}}|y-f(x)|\leq \delta ,\\\delta \,|y-f(x)|-{\frac {1}{2}}\delta ^{2}&{\textrm {otherwise.}}\end{cases}}$

Motivation

Two very commonly used loss functions are the squared loss, $L(a)=a^{2}$, and the absolute loss, $L(a)=|a|$. 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 $a$'s (as in ${\textstyle \sum _{i=1}^{n}L(a_{i})}$), 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 strongly convex in a uniform neighborhood of its minimum $a=0$; at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points $a=-\delta$ and $a=\delta$. 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. It combines the best properties of L1 squared loss and L2 absolute loss by being convex when close to the target/minimum and less steep for extreme values. This steepness can be controlled by the $\delta$ value. The Pseudo-Huber loss function ensures that derivatives are continuous for all degrees. It is defined as

$L_{\delta }(a)=\delta ^{2}\left({\sqrt {1+(a/\delta )^{2}}}-1\right).$

As such, this function approximates $a^{2}/2$ for small values of $a$, and approximates a straight line with slope $\delta$ for large values of $a$.

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

Variant for classification

For classification purposes, a variant of the Huber loss called modified Huber is sometimes used. Given a prediction $f(x)$ (a real-valued classifier score) and a true binary class label $y\in \{+1,-1\}$, the modified Huber loss is defined as

$L(y,f(x))={\begin{cases}\max(0,1-y\,f(x))^{2}&{\textrm {for}}\,\,y\,f(x)\geq -1,\\-4y\,f(x)&{\textrm {otherwise.}}\end{cases}}$

The term $\max(0,1-y\,f(x))$ is the hinge loss used by support vector machines; the quadratically smoothed hinge loss is a generalization of $L$.

Applications

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

See also

• Winsorizing
• Robust regression
• M-estimator
• Visual comparison of different M-estimators