What's the meaning of nested resampling

Question

I have a feature selection and regression task in which a dataset dataset is provided with 64x3000 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"

Lars Kotthoff · Accepted Answer · 2017-07-19 04:13:05Z

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.

So, in my example, what is the parameter optimized during the inner resampling? If the best set of features are selected in the inner loop? — remo, Jul 19 '17 at 4:34

pat-s · Accepted Answer · 2017-07-18 20:17:06Z

Nested cross-validation is a statistical concept of estimating the performance of a statistical (machine learning) model.

The outer cross-validation splits the data in train + test and estimates the model performance on each fold. To minimize variance of random partitioning within folds, this is repeated x times.

Now you could use the default parameter settings each time you train a model on the training set. However, default settings of a model are likely not optimal for each data set out in the wild. Hence, another cross-validation is performed (= the inner resampling) which has the aim to find the best parameter set of a learner for each fold. This parameter set is then applied to the actual training set of the outer resampling procedure, assuming that the model achieves better results using this non-standard parameter configuration.

This concept of finding the best parameter set is also referred to as "tuning a model". You could do a grid search and try all possible parameter configurations or you do a random search and pick randomly a parameter combination y times and see what comes out. The more random tries, the higher your probability to find the "optimal" configuration. However, it is unlikely that this is found by a random search, especially if you have numeric hyperparameters. Nevertheless, using a several hundred iterations of a random search will most likely find a better configuration than the model default.

Be aware that nested cross-validation is computational intense as you are fitting thousands of models (several hundred iterations in inner + ususally 500 - 1000 iterations in outer (folds*reps)). Therefore it is highly recommended to parallelize your code.

Here is a full parallelized example which also extracts the tuning results (see extract argument in resample). This works on a binary response task (which is not given here).

lrn_rf <- makeLearner("classif.ranger",
                      predict.type = "prob")
getHyperPars(lrn_rf)
getParamSet(lrn_rf)

filterParams(getParamSet(lrn_rf), tunable = TRUE)

## Tuning in inner resampling loop
ps <- makeParamSet(makeIntegerParam("mtry", lower = 1, upper = 11),
                   makeIntegerParam("num.trees", lower = 10, upper = 1000),
                   makeDiscreteParam("importance", values = "permutation"))

ctrl <- makeTuneControlRandom(maxit = 1000)
inner <- makeResampleDesc("CV", iters = 2)

wrapper_rf <- makeTuneWrapper(lrn_rf, resampling = inner, par.set = ps,
                              control = ctrl, show.info = T, measures = list(auc))

## Outer resampling loop
outer <- makeResampleDesc("RepCV", folds = 5, reps = 100)

library(parallelMap)
parallelStart(mode = "multicore", level = "mlr.tuneParams", cpus = 40) # only parallelize the tuning

resa_rf <- resample(wrapper_rf, task,
                    resampling = outer, extract = getTuneResult,
                    show.info = TRUE, measures = list(auc))

parallelStop()

In my example, why opt.paths include optimization of the number of features and feature selection method? As I understand, the must be optimized during the inner loop. Correct? — remo, Jul 18 '17 at 21:15
They are extracted afterwards from the output of resample (named r in your example) from all models fitted in the outer cross-validation. However, I'm not completely sure about this (!). If you want to extract from the inner loop you need to define a feature selection search strategy first (see the docs). — pat-s, Jul 18 '17 at 21:47

Lucas Roberts · Accepted Answer · 2017-07-18 18:02:02Z

According to this website: http://mlr-org.github.io/mlr-tutorial/release/html/nested_resampling/index.html the reason why you might want nested resampling is that each hyper-parameter or other component of your algorithm that is tuned according to cross validation should get a separate sample of data. This ensures that the correlation induced by cross validation is not increased in your sample. For example if you use the same sample to tune a sparsity parameter and to estimate the coefficients you might be inducing correlations that you do not want (unless you have a single oracle type procedure which does both sparsity and parameter estimation in one procedure). If you use 2 separate procedures and both require cross validation with one being nested within the other in for loops (as an example) you would assume in your outer loop the data is sampled and then in your inner for loop you would assume the sampled data is given and resample from this (outer for loop's) iteration's sample of data.

Thank you for your answer. I updated the code and output. Based on the output, why the optimal path (opt.paths) is going to optimize fw.abs and fw.method, which must be optimized within the inner loop. In fact, I can not map the output to my understanding of the whole process :( — remo, Jul 18 '17 at 18:24

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