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

File:Huber loss.svg

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

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

$$L_{\delta }(y,f(x))=\{\begin{array}{cc}\frac{1}{2}{(y-f(x))}^2& \text{for }|y-f(x)|\le \delta ,\\ \delta \cdot (|y-f(x)|-\frac{1}{2}\delta ),& \text{otherwise.}\end{array}$$

The Huber loss is the convolution of the absolute value function with the rectangular function, scaled and translated. Thus it "smoothens out" the former's corner at the origin.

File:Comparison of loss functions.png

Comparison of Huber loss with other loss functions used for robust regression.

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 ${\sum }_{i=1}^{n}L(a_i)$), 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 L2 squared loss and L1 absolute loss by being strongly convex when close to the target/minimum and less steep for extreme values. The scale at which the Pseudo-Huber loss function transitions from L2 loss for values close to the minimum to L1 loss for extreme values and the steepness at extreme values can be controlled by the $\delta$ value. The Pseudo-Huber loss function ensures that derivatives are continuous for all degrees. It is defined as

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

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

$$L(y,f(x))=\{\begin{array}{cc}\max (0,1-yf(x){)}^2& \text{for }yf(x)>-1,\\ -4yf(x)& \text{otherwise.}\end{array}$$

The term $\max (0,1-yf(x))$ 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