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.
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
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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 strongly 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. 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. This steepness can be controlled by the value. The Pseudo-Huber loss function ensures that derivatives are continuous for all degrees. It is defined as
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.
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
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