penalty function (bb kbk = min ( bxi yi for some positive penalty constant (efind the solution the dual optimization problem ( by using sklearn' scv method note thatas the two point sets are separablethe constraint may be removedand the value of can be set to in example we used the feature map ph( [ ]to classify the points an easier way is to map the points into via the feature map ph(xkxk or any monotone function thereof translated back into this yields circular separating boundary find the radius and center of this circleusing the fact that here the sorted norms for the two groups are let { be response variable and let (xbe the regression function ( : [ xp[ xrecall that the bayes classifier is ( { ( / let { be any other classifier function belowwe denote all probabilities and expectations conditional on as [*and [*(ashow that irreducible error } [ (xyp [ (xy+| ( { (xg( )hencededuce that for learner gt constructed from training set we have [ [gt (xy ] [ (xy| ( [gt (xg( )]where the first expectation and last probability operations are with respect to (busing the previous resultdeduce that for the unconditional error (that iswe no longer condition on )we have [ (xy [gt (xy(cshow thatif gt : {ht ( / is classifier function such that as ht ( - ( ( ) ( )for some mean and variance functions (xand ( )respectivelythen sign( ( ))( ( [gt (xg ( )-ph (xwhere ph is the cdf of standard normal random variable