Iteratively reweighted least squares

The method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems with objective functions of the form of a p-norm:

$${\displaystyle \mathop {\operatorname {arg\,min} } _{\boldsymbol {\beta }}\sum _{i=1}^{n}{\big |}y_{i}-f_{i}({\boldsymbol {\beta }}){\big |}^{p},}$$

by an iterative method in which each step involves solving a weighted least squares problem of the form:^[1]

$${\displaystyle {\boldsymbol {\beta }}^{(t+1)}={\underset {\boldsymbol {\beta }}{\operatorname {arg\,min} }}\sum _{i=1}^{n}w_{i}({\boldsymbol {\beta }}^{(t)}){\big |}y_{i}-f_{i}({\boldsymbol {\beta }}){\big |}^{2}.}$$

IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set, for example, by minimizing the least absolute errors rather than the least square errors.

One of the advantages of IRLS over linear programming and convex programming is that it can be used with Gauss–Newton and Levenberg–Marquardt numerical algorithms.