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Optimize a Boosted Regression Ensemble

This example shows how to optimize hyperparameters of a boosted regression ensemble. The optimization minimizes the cross-validation loss of the model.

The problem is to model the efficiency in miles per gallon of an automobile, based on its acceleration, engine displacement, horsepower, and weight. Load the carsmall data, which contains these and other predictors.

load carsmall
X = [Acceleration Displacement Horsepower Weight];
Y = MPG;

Fit a regression ensemble to the data using the LSBoost algorithm, and using surrogate splits. Optimize the resulting model by varying the number of learning cycles, the maximum number of surrogate splits, and the learn rate. Furthermore, allow the optimization to repartition the cross-validation between every iteration.

For reproducibility, set the random seed and use the 'expected-improvement-plus' acquisition function.

rng default
Mdl = fitrensemble(X,Y,...
    'Method','LSBoost',...
    'Learner',templateTree('Surrogate','on'),...
    'OptimizeHyperparameters',{'NumLearningCycles','MaxNumSplits','LearnRate'},...
    'HyperparameterOptimizationOptions',struct('Repartition',true,...
    'AcquisitionFunctionName','expected-improvement-plus'))
|====================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   | NumLearningC-|    LearnRate | MaxNumSplits |
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    | ycles        |              |              |
|====================================================================================================================|
|    1 | Best   |      3.5891 |      8.8707 |      3.5891 |      3.5891 |          383 |      0.51519 |            4 |
|    2 | Best   |      3.4929 |      0.4762 |      3.4929 |       3.498 |           16 |      0.66503 |            7 |
|    3 | Best   |      3.1712 |     0.97701 |      3.1712 |      3.1713 |           33 |       0.2556 |           92 |
|    4 | Accept |      6.3074 |     0.38797 |      3.1712 |      3.1717 |           13 |    0.0053227 |            5 |
|    5 | Accept |      3.2808 |     0.39457 |      3.1712 |      3.1715 |           13 |      0.53319 |           99 |
|    6 | Best   |       2.974 |     0.32124 |       2.974 |      2.9768 |           10 |      0.30539 |           90 |
|    7 | Accept |      4.6086 |     0.29482 |       2.974 |      2.9757 |           10 |      0.09622 |            2 |
|    8 | Accept |      3.2302 |      0.3267 |       2.974 |      3.1035 |           10 |      0.33326 |           40 |
|    9 | Accept |      3.3755 |      3.1982 |       2.974 |       3.111 |          119 |       0.3092 |           99 |
|   10 | Accept |      2.9805 |     0.28517 |       2.974 |      3.0718 |           10 |      0.31553 |            1 |
|   11 | Accept |      3.0656 |     0.31749 |       2.974 |       3.033 |           10 |      0.31311 |           91 |
|   12 | Best   |      2.9546 |     0.30311 |      2.9546 |      2.9548 |           10 |      0.52213 |            1 |
|   13 | Accept |      3.1134 |     0.31655 |      2.9546 |      3.0514 |           10 |       0.4158 |            1 |
|   14 | Accept |       5.506 |       12.83 |      2.9546 |      3.0524 |          487 |    0.0010022 |           35 |
|   15 | Accept |       3.162 |      13.382 |      2.9546 |      3.0315 |          499 |     0.021297 |           31 |
|   16 | Accept |      5.8944 |     0.30931 |      2.9546 |      3.0558 |           10 |      0.02851 |            1 |
|   17 | Accept |      3.3265 |      13.283 |      2.9546 |      3.0028 |          499 |     0.074578 |           15 |
|   18 | Accept |      3.1752 |      13.219 |      2.9546 |      3.0574 |          494 |      0.04424 |           99 |
|   19 | Accept |      6.4219 |     0.25977 |      2.9546 |      3.0591 |           10 |    0.0010027 |           11 |
|   20 | Accept |       3.358 |      12.422 |      2.9546 |      3.0583 |          498 |    0.0043108 |           95 |
|====================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   | NumLearningC-|    LearnRate | MaxNumSplits |
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    | ycles        |              |              |
|====================================================================================================================|
|   21 | Accept |      2.9861 |      11.547 |      2.9546 |      3.0591 |          499 |    0.0092444 |            2 |
|   22 | Accept |      3.0766 |      11.332 |      2.9546 |      3.0581 |          500 |    0.0098187 |           37 |
|   23 | Accept |      3.0247 |      11.945 |      2.9546 |      3.0568 |          500 |     0.010251 |           81 |
|   24 | Accept |      3.1616 |      11.112 |      2.9546 |       3.057 |          500 |    0.0097239 |            3 |
|   25 | Accept |      4.1776 |      1.9193 |      2.9546 |      3.0575 |           85 |     0.015418 |            7 |
|   26 | Accept |      6.1517 |      1.6863 |      2.9546 |      3.0564 |           79 |    0.0019248 |            1 |
|   27 | Accept |      3.8779 |      3.8071 |      2.9546 |      3.0572 |          181 |      0.99678 |            2 |
|   28 | Accept |      3.2471 |      2.1952 |      2.9546 |      3.0449 |           90 |      0.06971 |            3 |
|   29 | Accept |      3.4336 |      4.3663 |      2.9546 |      3.0444 |          159 |      0.10695 |           27 |
|   30 | Accept |      3.5563 |      11.127 |      2.9546 |      3.0433 |          499 |      0.96204 |            3 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 198.6669 seconds.
Total objective function evaluation time: 153.2122

Best observed feasible point:
    NumLearningCycles    LearnRate    MaxNumSplits
    _________________    _________    ____________

           10             0.52213          1      

Observed objective function value = 2.9546
Estimated objective function value = 3.0975
Function evaluation time = 0.30311

Best estimated feasible point (according to models):
    NumLearningCycles    LearnRate    MaxNumSplits
    _________________    _________    ____________

           10             0.4158           1      

Estimated objective function value = 3.0433
Estimated function evaluation time = 0.29822
Mdl = 
  RegressionEnsemble
                         ResponseName: 'Y'
                CategoricalPredictors: []
                    ResponseTransform: 'none'
                      NumObservations: 94
    HyperparameterOptimizationResults: [1×1 BayesianOptimization]
                           NumTrained: 10
                               Method: 'LSBoost'
                         LearnerNames: {'Tree'}
                 ReasonForTermination: 'Terminated normally after completing the requested number of training cycles.'
                              FitInfo: [10×1 double]
                   FitInfoDescription: {2×1 cell}
                       Regularization: []


  Properties, Methods

Compare the loss to that of a boosted, unoptimized model, and to that of the default ensemble.

loss = kfoldLoss(crossval(Mdl,'kfold',10))
loss = 17.8526
Mdl2 = fitrensemble(X,Y,...
    'Method','LSBoost',...
    'Learner',templateTree('Surrogate','on'));
loss2 = kfoldLoss(crossval(Mdl2,'kfold',10))
loss2 = 31.0623
Mdl3 = fitrensemble(X,Y);
loss3 = kfoldLoss(crossval(Mdl3,'kfold',10))
loss3 = 34.0239

For a different way of optimizing this ensemble, see Optimize Regression Ensemble Using Cross-Validation.