I have a feature selection and regression task in which a dataset *dataset* is provided with 64×3000 numeric data. I want to use the best features for best settings possible for a number of learners (i.e., linear regressor and decision tree). I use mlr package in R. If I understand the provided tutorial correctly, I have to use resampling (terms like *inner* and *outer* resampling) . Why is this type of nested resampling required while I am cross-validating the learner and settings during parameter optimization?

**Code:** (dataset is not provided)

`library('mlr') set.seed(1234, "L'Ecuyer") dataset = read.csv("dataset.csv") # shuffle columns dataset <- dataset[, c(sample(ncol(dataset) - 1), ncol(dataset))] # try to make full rank cov matrix for linear regression q <- qr(dataset) dataset <- dataset[, q$pivot[seq(q$rank)]] regr.task = makeRegrTask(id = "dataset.ig", data = dataset, target = "target") rdescCV2 = makeResampleDesc("CV", iters=2) rdescCV3 = makeResampleDesc("CV", iters=3) inner = rdescCV2 outer = rdescCV3 lrns = list( "regr.lm" , makeLearner("regr.rpart", minbucket = 1) ) measures = list(mse, rsq) for (lrn1 in lrns) { set.seed(1234, "L'Ecuyer") lrnName = ifelse(typeof(lrn1) == "list", lrn1$id, lrn1) if (typeof(lrn1) == "list") { lrnPars = lrn1$par.vals } else { lrnPars = list() } lrnName2Save = lrnName lrn = makeFilterWrapper(learner = makeLearner(cl = lrnName, par.vals = lrnPars)) ps = makeParamSet( makeDiscreteParam("fw.abs", values = seq(3, 5, 1)), makeDiscreteParam("fw.method", values = c('chi.squared', 'information.gain' ))) if ("minsplit" %in% names(getParamSet(lrn)$pars)) ps$pars$minsplit = makeIntegerParam("minsplit", lower = 2L, upper = 3L) # try to find the best feature set and setting for each learner res = makeTuneWrapper( lrn, resampling = inner, measures = measures, par.set = ps, control = makeTuneControlGrid(), show.info = FALSE ) # same indices for each learner set.seed(1234, "L'Ecuyer") r = resample(res, regr.task, outer, models = TRUE, measures = measures) res2 = lapply(r$models, getTuneResult) opt.paths = lapply(res2, function(x) as.data.frame(x$opt.path)) optimalFeatures[[lrnName2Save]] = lapply(r$models, function(x) getFilteredFeatures(x$learner.model$next.model)) print(res2) print(opt.paths) print(optimalFeatures[[lrnName2Save]]) } `

**Output:**

`[Resample] cross-validation iter 1: mse.test.mean=4.95e+03,rsq.test.mean=0.222 [Resample] cross-validation iter 2: mse.test.mean=4.03e+03,rsq.test.mean=0.497 [Resample] cross-validation iter 3: mse.test.mean= 961,rsq.test.mean=0.765 [Resample] Aggr. Result: mse.test.mean=3.31e+03,rsq.test.mean=0.495 [[1]] Tune result: Op. pars: fw.abs=4; fw.method=information.gain mse.test.mean=2.73e+03,rsq.test.mean=0.568 [[2]] Tune result: Op. pars: fw.abs=5; fw.method=information.gain mse.test.mean=2.9e+03,rsq.test.mean=0.383 [[3]] Tune result: Op. pars: fw.abs=5; fw.method=chi.squared mse.test.mean=6.64e+03,rsq.test.mean=-0.0448 [[1]] fw.abs fw.method mse.test.mean rsq.test.mean dob eol error.message exec.time 1 3 chi.squared 4711.697 0.3248701 1 NA <NA> 0.36 2 4 chi.squared 2891.273 0.5480474 2 NA <NA> 0.33 3 5 chi.squared 2861.078 0.5526319 3 NA <NA> 0.31 4 3 information.gain 2726.971 0.5631411 4 NA <NA> 0.43 5 4 information.gain 2726.018 0.5678868 5 NA <NA> 0.38 6 5 information.gain 2970.028 0.5395522 6 NA <NA> 0.39 [[2]] fw.abs fw.method mse.test.mean rsq.test.mean dob eol error.message exec.time 1 3 chi.squared 5357.465 -0.2319388 1 NA <NA> 0.34 2 4 chi.squared 3747.050 0.2437902 2 NA <NA> 0.35 3 5 chi.squared 2897.023 0.3831484 3 NA <NA> 0.31 4 3 information.gain 5357.465 -0.2319388 4 NA <NA> 0.41 5 4 information.gain 3747.050 0.2437902 5 NA <NA> 0.42 6 5 information.gain 2897.023 0.3831484 6 NA <NA> 0.43 [[3]] fw.abs fw.method mse.test.mean rsq.test.mean dob eol error.message exec.time 1 3 chi.squared 7593.989 -0.10557789 1 NA <NA> 0.37 2 4 chi.squared 6786.384 -0.02621949 2 NA <NA> 0.33 3 5 chi.squared 6637.264 -0.04484878 3 NA <NA> 0.32 4 3 information.gain 7593.989 -0.10557789 4 NA <NA> 0.40 5 4 information.gain 6786.384 -0.02621949 5 NA <NA> 0.39 6 5 information.gain 6637.264 -0.04484878 6 NA <NA> 0.41 [[1]] [1] "RDF065u_640" "RTp_1225" "L2u_940" "TIC3_182" [[2]] [1] "RTp_1225" "L2u_940" "Mor03m_813" "TIC3_182" "Mor03m_2294" [[3]] [1] "H.046_1401" "Mor21u_2280" "RDF065u_640" "RTp_1225" "CIC2_1660" [Resample] cross-validation iter 1: mse.test.mean=3.13e+03,rsq.test.mean=0.509 [Resample] cross-validation iter 2: mse.test.mean=8.04e+03,rsq.test.mean=-0.00294 [Resample] cross-validation iter 3: mse.test.mean=3.49e+03,rsq.test.mean=0.148 [Resample] Aggr. Result: mse.test.mean=4.89e+03,rsq.test.mean=0.218 [[1]] Tune result: Op. pars: fw.abs=5; fw.method=chi.squared; minsplit=3 mse.test.mean=3.15e+03,rsq.test.mean=0.443 [[2]] Tune result: Op. pars: fw.abs=5; fw.method=information.gain; minsplit=3 mse.test.mean=3.3e+03,rsq.test.mean=0.206 [[3]] Tune result: Op. pars: fw.abs=5; fw.method=chi.squared; minsplit=2 mse.test.mean=4.33e+03,rsq.test.mean=0.368 [[1]] fw.abs fw.method minsplit mse.test.mean rsq.test.mean dob eol error.message exec.time 1 3 chi.squared 2 3875.576 0.3448855 1 NA <NA> 0.35 2 4 chi.squared 2 4054.182 0.2971222 2 NA <NA> 0.33 3 5 chi.squared 2 3149.302 0.4433532 3 NA <NA> 0.34 4 3 information.gain 2 3351.588 0.4077916 4 NA <NA> 0.42 5 4 information.gain 2 3904.129 0.3151364 5 NA <NA> 0.41 6 5 information.gain 2 3649.004 0.3833628 6 NA <NA> 0.39 7 3 chi.squared 3 3875.576 0.3448855 7 NA <NA> 0.35 8 4 chi.squared 3 4054.182 0.2971222 8 NA <NA> 0.35 9 5 chi.squared 3 3149.302 0.4433532 9 NA <NA> 0.38 10 3 information.gain 3 3351.588 0.4077916 10 NA <NA> 0.40 11 4 information.gain 3 3904.129 0.3151364 11 NA <NA> 0.41 12 5 information.gain 3 3649.004 0.3833628 12 NA <NA> 0.42 [[2]] fw.abs fw.method minsplit mse.test.mean rsq.test.mean dob eol error.message exec.time 1 3 chi.squared 2 4846.020 -0.01409290 1 NA <NA> 0.32 2 4 chi.squared 2 3316.516 0.20477753 2 NA <NA> 0.32 3 5 chi.squared 2 3304.965 0.20643353 3 NA <NA> 0.36 4 3 information.gain 2 4848.166 -0.01480330 4 NA <NA> 0.43 5 4 information.gain 2 3316.516 0.20477753 5 NA <NA> 0.42 6 5 information.gain 2 3304.965 0.20643353 6 NA <NA> 0.42 7 3 chi.squared 3 4613.949 0.05281112 7 NA <NA> 0.38 8 4 chi.squared 3 3316.516 0.20477753 8 NA <NA> 0.41 9 5 chi.squared 3 3304.965 0.20643353 9 NA <NA> 0.33 10 3 information.gain 3 4795.237 -0.00721534 10 NA <NA> 0.39 11 4 information.gain 3 3316.516 0.20477753 11 NA <NA> 0.38 12 5 information.gain 3 3304.965 0.20643353 12 NA <NA> 0.36 [[3]] fw.abs fw.method minsplit mse.test.mean rsq.test.mean dob eol error.message exec.time 1 3 chi.squared 2 8346.300 -0.0896325 1 NA <NA> 0.29 2 4 chi.squared 2 10435.255 -0.3316064 2 NA <NA> 0.32 3 5 chi.squared 2 4325.461 0.3684383 3 NA <NA> 0.30 4 3 information.gain 2 8346.300 -0.0896325 4 NA <NA> 0.39 5 4 information.gain 2 10435.255 -0.3316064 5 NA <NA> 0.39 6 5 information.gain 2 4325.461 0.3684383 6 NA <NA> 0.41 7 3 chi.squared 3 8346.300 -0.0896325 7 NA <NA> 0.36 8 4 chi.squared 3 10435.255 -0.3316064 8 NA <NA> 0.34 9 5 chi.squared 3 4325.461 0.3684383 9 NA <NA> 0.34 10 3 information.gain 3 8346.300 -0.0896325 10 NA <NA> 0.41 11 4 information.gain 3 10435.255 -0.3316064 11 NA <NA> 0.44 12 5 information.gain 3 4325.461 0.3684383 12 NA <NA> 0.40 [[1]] [1] "RDF065u_640" "RTp_1225" "L2u_940" "Mor03m_813" "TIC3_182" [[2]] [1] "RTp_1225" "L2u_940" "Mor03m_813" "TIC3_182" "Mor03m_2294" [[3]] [1] "H.046_1401" "Mor21u_2280" "RDF065u_640" "RTp_1225" "CIC2_1660" `

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#### Best Answer

The purpose of a nested cross-validation is to reduce overfitting — that is, thinking a model has good generalization performance when it doesn't.

To see why this is a problem, consider the extreme case where a model simply memorizes the data (for example KNN with 1 neighbour). If you evaluate this model with the data you've trained it on (or part thereof), you'll get perfect performance, but any other data will probably give terrible results.

That's why you need separate train and test sets. But even with that, it's possible that you get an unlucky split and train and test end up being too similar, again giving a misleading impression of the real performance. It could work the other way, too, where the train and test sets are too dissimilar and no matter what you learn on the training set, you won't do well on the test set.

So you can go one step further and use a series of train and test sets — cross-validation. You take your entire data and split into n of folds, using 1 fold for testing and remaining n-1 for training, then another for testing, and so on for n rounds.

Why is nested cross-validation for things like tuning (where lots of different models are evaluated and compared) better? Consider the following thought experiment. A learner has one parameter which just adds random noise; there's no real effect. Comparing different parameterizations of this learner will result in one of those being best by pure chance, even when using a cross-validation. In a nested cross-validation, the models will be evaluated on yet another set, showing that the parameter doesn't actually do anything (or at least more likely to show that).

Neither cross-validation nor nested cross-validation are really required in any case, but they'll likely improve the generalisation performance of the end result dramatically.