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regularize

Find weights to minimize resubstitution error plus penalty term

Syntax

ens1 = regularize(ens)
ens1 = regularize(ens,Name,Value)

Description

ens1 = regularize(ens) finds optimal weights for learners in ens by lasso regularization. regularize returns a regression ensemble identical to ens, but with a populated Regularization property.

ens1 = regularize(ens,Name,Value) computes optimal weights with additional options specified by one or more Name,Value pair arguments. You can specify several name-value pair arguments in any order as Name1,Value1,…,NameN,ValueN.

Input Arguments

ens

A regression ensemble, created by fitrensemble.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'lambda'

Vector of nonnegative regularization parameter values for lasso. For the default setting of lambda, regularize calculates the smallest value lambda_max for which all optimal weights for learners are 0. The default value of lambda is a vector including 0 and nine exponentially-spaced numbers from lambda_max/1000 to lambda_max.

Default: [0 logspace(log10(lambda_max/1000),log10(lambda_max),9)]

'MaxIter'

Maximum number of iterations allowed, specified as a positive integer. If the algorithm executes MaxIter iterations before reaching the convergence tolerance, then the function stops iterating and returns a warning message. The function can return more than one warning when either npass or the number of lambda values is greater than 1.

Default: 1e3

'npass'

Maximal number of passes for lasso optimization, a positive integer.

Default: 10

'reltol'

Relative tolerance on the regularized loss for lasso, a numeric positive scalar.

Default: 1e-3

'verbose'

Verbosity level, either 0 or 1. When set to 1, regularize displays more information as it runs.

Default: 0

Output Arguments

ens1

A regression ensemble. Usually you set ens1 to the same name as ens.

Examples

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Regularize an ensemble of bagged trees.

Generate sample data.

rng(10,'twister') % For reproducibility
X = rand(2000,20);
Y = repmat(-1,2000,1);
Y(sum(X(:,1:5),2)>2.5) = 1;

You can create a bagged classification ensemble of 300 trees from the sample data.

bag = fitrensemble(X,Y,'Method','Bag','NumLearningCycles',300);

fitrensemble uses a default template tree object templateTree() as a weak learner when 'Method' is 'Bag'. In this example, for reproducibility, specify 'Reproducible',true when you create a tree template object, and then use the object as a weak learner.

t = templateTree('Reproducible',true); % For reproducibiliy of random predictor selections
bag = fitrensemble(X,Y,'Method','Bag','NumLearningCycles',300,'Learners',t);

Regularize the ensemble of bagged regression trees.

bag = regularize(bag,'lambda',[0.001 0.1],'verbose',1);
Starting lasso minimization for Lambda=0.001. Initial MSE=0.110607.
    Lasso minimization completed pass 1 for Lambda=0.001
        MSE = 0.0899652
        Relative change in MSE = 0.229442
        Number of learners with non-zero weights = 12
    Lasso minimization completed pass 2 for Lambda=0.001
        MSE = 0.064488
        Relative change in MSE = 0.39507
        Number of learners with non-zero weights = 43
    Lasso minimization completed pass 3 for Lambda=0.001
        MSE = 0.0608422
        Relative change in MSE = 0.0599211
        Number of learners with non-zero weights = 64
    Lasso minimization completed pass 4 for Lambda=0.001
        MSE = 0.060069
        Relative change in MSE = 0.0128723
        Number of learners with non-zero weights = 82
    Lasso minimization completed pass 5 for Lambda=0.001
        MSE = 0.0599398
        Relative change in MSE = 0.00215497
        Number of learners with non-zero weights = 96
    Lasso minimization completed pass 6 for Lambda=0.001
        MSE = 0.0599369
        Relative change in MSE = 4.80374e-05
        Number of learners with non-zero weights = 109
    Lasso minimization completed pass 7 for Lambda=0.001
        MSE = 0.0599364
        Relative change in MSE = 9.35973e-06
        Number of learners with non-zero weights = 113
    Lasso minimization completed pass 8 for Lambda=0.001
        MSE = 0.0599364
        Relative change in MSE = 1.99253e-08
        Number of learners with non-zero weights = 114
    Lasso minimization completed pass 9 for Lambda=0.001
        MSE = 0.0599364
        Relative change in MSE = 5.04823e-08
        Number of learners with non-zero weights = 113
    Completed lasso minimization for Lambda=0.001.
    Resubstitution MSE changed from 0.110607 to 0.0599364.
    Number of learners reduced from 300 to 113.
Starting lasso minimization for Lambda=0.1. Initial MSE=0.110607.
    Lasso minimization completed pass 1 for Lambda=0.1
        MSE = 0.113013
        Relative change in MSE = 0.0212927
        Number of learners with non-zero weights = 10
    Lasso minimization completed pass 2 for Lambda=0.1
        MSE = 0.086583
        Relative change in MSE = 0.30526
        Number of learners with non-zero weights = 27
    Lasso minimization completed pass 3 for Lambda=0.1
        MSE = 0.080426
        Relative change in MSE = 0.0765551
        Number of learners with non-zero weights = 42
    Lasso minimization completed pass 4 for Lambda=0.1
        MSE = 0.0795375
        Relative change in MSE = 0.0111715
        Number of learners with non-zero weights = 57
    Lasso minimization completed pass 5 for Lambda=0.1
        MSE = 0.0792383
        Relative change in MSE = 0.00377496
        Number of learners with non-zero weights = 67
    Lasso minimization completed pass 6 for Lambda=0.1
        MSE = 0.0786905
        Relative change in MSE = 0.00696198
        Number of learners with non-zero weights = 75
    Lasso minimization completed pass 7 for Lambda=0.1
        MSE = 0.0787969
        Relative change in MSE = 0.00134974
        Number of learners with non-zero weights = 77
    Lasso minimization completed pass 8 for Lambda=0.1
        MSE = 0.0788049
        Relative change in MSE = 0.00010252
        Number of learners with non-zero weights = 87
    Lasso minimization completed pass 9 for Lambda=0.1
        MSE = 0.0788065
        Relative change in MSE = 1.98213e-05
        Number of learners with non-zero weights = 87
    Completed lasso minimization for Lambda=0.1.
    Resubstitution MSE changed from 0.110607 to 0.0788065.
    Number of learners reduced from 300 to 87.

regularize reports on its progress.

Inspect the resulting regularization structure.

bag.Regularization
ans = struct with fields:
               Method: 'Lasso'
       TrainedWeights: [300x2 double]
               Lambda: [1.0000e-03 0.1000]
    ResubstitutionMSE: [0.0599 0.0788]
       CombineWeights: @classreg.learning.combiner.WeightedSum

Check how many learners in the regularized ensemble have positive weights. These are the learners included in a shrunken ensemble.

sum(bag.Regularization.TrainedWeights > 0)
ans = 1×2

   113    87

Shrink the ensemble using the weights from Lambda = 0.1.

cmp = shrink(bag,'weightcolumn',2)
cmp = 
  classreg.learning.regr.CompactRegressionEnsemble
             ResponseName: 'Y'
    CategoricalPredictors: []
        ResponseTransform: 'none'
               NumTrained: 87


  Properties, Methods

The compact ensemble contains 87 members, less than 1/3 of the original 300.

More About

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