Minimizing Stochastic Complexity with Ridge Regression

We derive a penalty strength criterion for ridge regression using stochastic complexity, which is a refined variant of the minimum description length principle. Since stochastic complexity does not typically account for the effect of regularization on complexity, despite its ability to simplify models, we are required to make a slight modification to the underlying coding scheme. Our scheme makes use of a weighted ensemble of regularized model fits rather than a mixture of maximum likelihood estimates. Under this modification, regularization is interpreted as reducing model complexity by constraining flexibility. In the case of ridge regression, the complexity penalty term that we derive can be expressed analytically as the log determinant of the residual operator. We demonstrate the effect of this complexity penalty by fitting a linear readout to a reservoir computer, and by performing benchmark testing on publicly available datasets.