# 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

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]}

## See also

## References

- ↑ Huber, Peter J. (1964). "Robust Estimation of a Location Parameter".
*Annals of Statistics*.**53**(1): 73–101. doi:10.1214/aoms/1177703732. JSTOR 2238020. - ↑ Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome (2009).
*The Elements of Statistical Learning*. p. 349. Compared to Hastie*et al.*, the loss is scaled by a factor of ½, to be consistent with Huber's original definition given earlier. - ↑ Charbonnier, P.; Blanc-Feraud, L.; Aubert, G.; Barlaud, M. (1997). "Deterministic edge-preserving regularization in computed imaging".
*IEEE Trans. Image Processing*.**6**(2): 298–311. doi:10.1109/83.551699. - ↑ Hartley, R.; Zisserman, A. (2003).
*Multiple View Geometry in Computer Vision*(2nd ed.). Cambridge University Press. p. 619. ISBN 0-521-54051-8. - ↑ Lange, K. (1990). "Convergence of Image Reconstruction Algorithms with Gibbs Smoothing".
*IEEE Trans. Medical Imaging*.**9**(4): 439–446. doi:10.1109/42.61759. - 1 2 Zhang, Tong (2004).
*Solving large scale linear prediction problems using stochastic gradient descent algorithms*. ICML. - ↑ Friedman, J. H. (2001). "Greedy Function Approximation: A Gradient Boosting Machine".
*Annals of Statistics*.**26**(5): 1189–1232. doi:10.1214/aos/1013203451. JSTOR 2699986.