Editing Loss function (section)

===Quadratic loss function===

The use of a [[quadratic function|quadratic]] loss function is common, for example when using [[least squares]] techniques. It is often more mathematically tractable than other loss functions because of the properties of [[variance]]s, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target.  If the target is ''t'', then a quadratic loss function is
:<math>\lambda(x) = C (t-x)^2 \; </math>
for some constant ''C''; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the '''squared error loss''' ('''SEL''').<ref name="ttf2001" />

Many common [[statistic]]s, including [[t-test]]s, [[Regression analysis|regression]] models, [[design of experiments]], and much else, use [[least squares]] methods applied using [[linear regression]] theory, which is based on the quadratic loss function.

The quadratic loss function is also used in [[Linear-quadratic regulator|linear-quadratic optimal control problems]]. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a [[quadratic form]] in the deviations of the variables of interest from their desired values; this approach is [[closed-form expression|tractable]] because it results in linear [[first-order condition]]s. In the context of [[stochastic control]], the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the [[Huber loss|Huber]], Log-Cosh and SMAE losses are used when the data has many large outliers.
[[File:Fitting a straight line to a data with outliers.png|thumb|Effect of using different loss functions, when the data has outliers]]