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Regularization aims to improve prediction performance of a given statisticalmodeling approach by moving to a second approach which achieves worse trainingerror but is expected to have fewer degrees of freedom, i.e., better agreementbetween training and prediction error. We show here, however, that thisexpected behavior does not hold in general. In fact, counter examples are giventhat show regularization can increase the degrees of freedom in simplesituations, including lasso and ridge regression, which are the most commonregularization approaches in use. In such situations, the regularizationincreases both training error and degrees of freedom, and is thus inherentlywithout merit. On the other hand, two important regularization scenarios aredescribed where the expected reduction in degrees of freedom is indeedguaranteed: (a) all symmetric linear smoothers, and (b) linear regressionversus convex constrained linear regression (as in the constrained variant ofridge regression and lasso).

When Does More Regularization Imply Fewer Degrees of Freedom? Sufficient Conditions and Counter Examples from Lasso and Ridge Regression