table summary of definitions for supervised learning (xyf ( xt or tn or tn loss( , `(gggg ` ( ` (gggt or gt ggt or gt fixed explanatory (featurevector random explanatory (featurevector fixed (real-valuedresponse random response joint pdf of and yevaluated at (xyconditional pdf of given xevaluated at fixed training data {(xi yi ) nrandom training data {(xi yi ) nmatrix of explanatory variableswith rows > and dim(xfeature columnsone of the features may be the constant vector of response variables ( yn )prediction (guessfunction loss incurred when predicting response with risk for prediction function gthat ise loss(yg( )optimal prediction functionthat isargming `(goptimal prediction function in function class gthat isargmingg `(gtraining loss for prediction function gthat isthe sample average estimate of `(gbased on fixed training sample the same as ` ( )but now for random training sample the learnerargmingg ` (gthat isthe optimal prediction function based on fixed training set and function class we suppress the superscript if the function class is implicit the learnerwhere we have replaced with random training set to compare the predictive performance of various learners in the function class gas measured by the test losswe can use the same fixed training set and test set for all learners when there is an abundance of datathe "overalldata set is usually (randomlydivided into training and test setas depicted in figure we then use the training data to construct various learners ggt ggt and use the test data to select the best (with the smallest test lossamong these learners in this context the test set is called the validation set once the best learner has been chosena third "testset can be used to assess the predictive performance of the best learner the trainingvalidationand test sets can again be obtained from the overall data set via random allocation when the overall data set is of modest sizeit is customary to perform the validation phase (model selectionon the training set onlyusing cross-validation this is the topic of section validation set