thuswhen using squared-error lossthe generalization risk for linear class can be decomposed as`(ggt (ggt (xy) ` (gg (xg( )) (ggt (xgg ( )) { { approximation error ( statistical error note that in this decomposition the statistical error is the only term that depends on the training set example (polynomial regression (cont )we continue example here is the class of linear functions of [ uu - ]and (xxbconditional on we have that (xe( )with (xn( `)where ` ( ( )) is the irreducible error we wish to understand how the approximation and statistical errors behave as we change the complexity parameter firstwe consider the approximation error any function can be written as (xh(ub - [ uu - band so (xis distributed as [ uu - ]bwhere ( similarlyg(xis distributed as [ uu ]bit follows that an expression for the approximation error  is [ uu - [ uu bdu to minimize this errorwe set the gradient with respect to to zero and obtain the linear equations - [ uu [ uu du [ uu - [ uu ]bu du [ uu - [ uu bu - du let hp [ uu - ][ uu - du be the hilbert matrixwhich has (ij)-th entry given by uij- du /( where is the thenthe above system of linear equations can be written as hb upper left sub-block of hep and max{ the solutionwhich we denote by is ] [ ( bp [ [ - - ]  hencethe approximation error gg (xg(xis given by  [ uu - [ uu bdu ( hilbert matrix