Overfitting, regularisation and condition estimation in regression

Abstract Overfitting is a problem in regression and deep neural networks, and it is often stated that Tikhonov regularisation minimises its adverse effects, but the relationship between regularisation and overfitting has not been established. The theory of regularisation is well developed, but overfitting has a qualitative description and it is not defined mathematically. This paper addresses the relationship between overfitting, regularisation and condition estimation by considering underdetermined and overdetermined least squares ( $$\text {LS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>LS</mml:mtext> </mml:math> ) problems that arise in regression. This study is important because regularisation is not benign since its use when a condition on the decay of the singular values of the coefficient matrix in the $$\text {LS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>LS</mml:mtext> </mml:math> minimisation is not satisfied leads to a large error in the solution of the regularised $$\text {LS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>LS</mml:mtext> </mml:math> problem. Examples in which the regression curve overfits the data are shown, but regularisation must not be applied because the $$\text {LS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>LS</mml:mtext> </mml:math> problem is well conditioned. Also, an ill conditioned $$\text {LS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>LS</mml:mtext> </mml:math> problem whose solution does not display overfitting is shown, but its ill conditioned nature implies regularisation should be applied in order to obtain a numerically stable solution. It is concluded that regularisation does not solve the problem of overfitting in regression.