原文地址：https://en.wikipedia.org/wiki/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
) and squared error loss (blue) as a function of {\displaystyle y-f(x)}
The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by[1]
-
{\displaystyle 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 {\displaystyle |a|=\delta }
. The variable a often refers to the residuals, that is to the difference between the observed and predicted values {\displaystyle a=y-f(x)}
, so the former can be expanded to[2]
-
{\displaystyle 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, {\displaystyle L(a)=a^{2}}
, and the absolute loss, {\displaystyle 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 {\displaystyle 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 convex in a uniform neighborhood of its minimum {\displaystyle a=0}
, at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points {\displaystyle a=-\delta }
and {\displaystyle 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, and ensures that derivatives are continuous for all degrees. It is defined as[3][4]
-
{\displaystyle L_{\delta }(a)=\delta ^{2}({\sqrt {1+(a/\delta )^{2}}}-1).}
As such, this function approximates {\displaystyle a^{2}/2}
for small values of {\displaystyle a}, and approximates a straight line with slope {\displaystyle \delta }
for large values of {\displaystyle a}.
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 {\displaystyle f(x)}
(a real-valued classifier score) and a true binary class label {\displaystyle y\in \{+1,-1\}}
, the modified Huber loss is defined as[6]
-
{\displaystyle 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 {\displaystyle \max(0,1-y\,f(x))}
is the hinge loss used by support vector machines; the quadratically smoothed hinge loss is a generalization of {\displaystyle L}
.[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.
- 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.