Weak learners

• Models that perform only slightly better than random guessing are called weak learners (Bishop 2007).

• Weak learners low predictive accuracy may be due, to the predictor having high bias or high variance.

Trees may be weak learners

• May be sensitive to small changes in training data that lead to very different tree structure.
• High sensitivity implies predictions are highly variable
• They are greedy algorithms making locally optimal decisions at each node without considering the global optimal solution.
  • This can lead to suboptimal splits and ultimately a weaker predictive performance.

There’s room for improvement

• In many situations trees may be a good option (e.g. for simplicity and interpretability)

• But there are issues that, if solved, may improve performance.

• It is to be noted, too, that these problems are not unique to trees.

  • Other simple models such as linear regression may be considered as weakl learners in may situations.

The bias-variance trade-off

• When we try to improve weak learners we need to deal with the bias-variance trade-off.

The bias-variance trade-off cheatsheet

How to deal with such trade-off

• How can a model be made less variable or less biased without this increasing its bias or variance?

• There are distinct appraches to deal with this problem

  • Regularization,
  • Feature engineering,
  • Model selection
  • Ensemble learning

Ensemble learners

• Ensemble learning takes a distinct based on “the wisdom of crowds”.

• Predictors, also called, ensembles are built by fitting repeated (weak learners) models on the same data and combining them to form a single result.

• As a general rule, ensemble learners tend to improve the results obtained with the weak learners they are made of.

Ensemble methods

• If we rely on how they deal with the bias-variance trade-off we can consider distinct groups of ensemble methods:

  • Bagging

  • Boosting

  • Hybrid learners

Emnsemble methods cheatsheet

Bagging

• Bagging, Random forests and Random patches yield a smaller variance than simple trees by building multiple trees based on aggregating trees built on a subset of
  • individual (1),
  • features (2) or
  • both (3).

Boosting & Stacking

• Boosting or Stacking combine distinct predictors to yield a model with an increasingly smaller error, and so reduce the bias.

• They differ on if do the combination

  • sequentially (1) or

  • using a meta-model (2) .

Hybrid Techniques

• Hybrid techniques combine approaches in order to deal with both variance and bias.

• The approach should be clear from their name:

  • Gradient Boosted Trees with Bagging

  • Stacked bagging

Bagging: bootstrap aggregation

• Decision trees suffer from high variance when compared with other methods such as linear regression, especially when \(n/p\) is moderately large.

• Given that this is intrinsic to trees, Breiman (1996) sugested to build multiple trees derived from the same dataset and, somehow, average them.

Averaging decreases variance

• Bagging relies, informally, on the idea that:

  • given \(X\sim F()\), s.t. \(Var_F(X)=\sigma^2\),
  • given a s.r.s. \(X_1, ..., X_n\) from \(F\) then
  • if \(\overline{X}=\frac{1}{N}\sum_{i=1}^n X_i\) then \(var_F(\overline{X})=\sigma^2/n\).

• That is, relying on the sample mean instead of on simple observations, decreases variance by a factor of \(n\).

• BTW this idea is still (approximately) valid for more general statistics where the CLT applies.

Back to bagging

• Breiman (1996) combined the ideas of:
  • Averaging provides decreased variance estimates,
  • Bootstrap provides multiple (re)samples.
• He suggested: bootstrap aggregating :
  • Take resamples from the original training dataset
  • Learn the model on each bootstrapped training set to get a prediction \(\hat f^{*b}(x)\).
  • Average \(\hat f^{*b}(x)\) to obtain improved prediction/classification.

Bagging prediction/classifier

• For regression (trees) the bagged estimate is the average prediction at \(x\) from these \(B\) trees.

• For classification (trees) the bagged classifier selects the class with the most “votes” from the \(B\) trees:

Resampling based estimators

• The bootstrap was introduced as a way to provide standard error estimators.
• When used to compute \(\hat f_{bag}(x)\) or \(\hat G_{bag}(x)\), as described above, it provides direct estimators of a characteristic, not of their standard errors.
• However, the bagging process can also provide resampling based estimates of the precision of these estimators.

Out-Of-Bag observations

• Every time a resample is taken with replacement, some observations are omitted, due to the multiple occurring of others.

• These out-of-bag (OOB) observations can be used to build an estimate of prediction error.

Illustration of OOB EE

Source: https://www.baeldung.com/cs/random-forests-out-of-bag-error

Bagging in R (1.1)

• The example uses the AmesHousing dataset on house prices in Ames, IA to predict the “Sales price” of houses.

• We start by splitting the dataset in test/train subsets

• Next we build a bagged tree on the train subset and evaluate it on the test subset.

• Last we try to interpret the results using “Variable importance”

• Bagging is equivalent to RandomForest if we use all the trees so the library randomForest is used.

Bagging - Prepare data

# Prepare "clean" dataset from raw data
ames <- AmesHousing::make_ames()

# Split in test/training
set.seed(123)
train <- sample(1:nrow(ames), nrow(ames)/2)
# split <- rsample::initial_split(ames, prop = 0.7, 
#                        strata = "Sale_Price")
ames_train  <- ames[train,]
ames_test   <- ames[-train,]

Bagging - Build bag of trees

library(randomForest)
set.seed(12543)
bag.Ames <- randomForest(Sale_Price ~ ., 
                         data = ames_train, 
                         mtry = ncol(ames_train-1), 
                         ntree = 100,
                         importance = TRUE)

Bagging - Results

show(bag.Ames )

Call:
 randomForest(formula = Sale_Price ~ ., data = ames_train, mtry = ncol(ames_train -      1), ntree = 100, importance = TRUE) 
               Type of random forest: regression
                     Number of trees: 100
No. of variables tried at each split: 80

          Mean of squared residuals: 771498315
                    % Var explained: 87.89

Bagging - Prediction and accuracy (1)

Bagging - Prediction and accuracy (1)

Interpetability: The “achiles heel”

• Trees may have a straightforward interpretation,
  • Plotting the tree provides information about
    • which variables are important
    • how they act on the prediction
• Ensembles are less intuitive because
  • there is no consensus tree.
  • not clear which variables are most important.

Feature importance from Trees

• To measure feature importance, the reduction in the loss function (e.g., SSE) attributed to each variable at each split is tabulated.

• The total reduction in the loss function across all splits by a variable are summed up and used as the total feature importance.

Feature importance for Ensembles

• Variable importance measures can be extended to an ensemble simply by adding up variable importance over all trees built.

Random forests: decorrelating predictors

• Bagged trees, based on re-samples (of the same sample) tend to be highly correlated.
• Leo Breimann, again, introduced a modification to bagging, he called Random forests, that aims at decorrelating trees as follows:
  • When growing a tree from one bootstrap sample,
  • At each split use only a randomly selected subset of predictors.

Split variable randomization

• While growing a decision tree, during the bagging process,

• Random forests perform split-variable randomization:

  • each time a split is to be performed,
  • the search for the split variable is limited to a random subset of \(m_{try}\) of the original \(p\) features.

Random forests

Source: AIML.com research

How many variables per split?

• \(m\) can be chosen using cross-validation, but
• The usual recommendation for random selection of variables at each split has been studied by simulation:
  • For regression default value is \(m=p/3\)
  • For classification default value is \(m=\sqrt{p}\).
• If \(m=p\), we have bagging instead of RF.

Random forest algorithm (1)

Random Forests Algorithm, from chapter 11 in (Boehmke and Greenwell 2020)

Random forest algorithm (2)

Random Forests Algorithm, from chapter 17 in (Hastie and Efron 2016).

Out-of-the box performance

• Random forests tend to provide very good out-of-the-box performance, that is:
  • Although several hyperparameters can be tuned,
  • Default values tend to produce good results.
• Moreover, among the more popular machine learning algorithms, RFs have the least variability in their prediction accuracy when tuning (Probst, Wright, and Boulesteix 2019).

Out of the box performance

• A random forest trained with all hyperparameters set to their default values can yield an OOB RMSE that is better than many other classifiers, with or without tuning.
• This combined with good stability and ease-of-use has made it the option of choice for many problems.

Out of the box performance example

# number of features
n_features <- length(setdiff(names(ames_train), "Sale_Price"))

# train a default random forest model
ames_rf1 <- ranger(
  Sale_Price ~ ., 
  data = ames_train,
  mtry = floor(n_features / 3),
  respect.unordered.factors = "order",
  seed = 123
)

# get OOB RMSE
(default_rmse <- sqrt(ames_rf1$prediction.error))
## [1] 24859.27

1. Number of trees

2. Number of features (\(m_{try}\)).

3. Complexity of each tree.

4. Sampling scheme.

5. Splitting rule

• Default splitting rules favor features with many splits, potentially biasing towards certain variable types.
• Conditional inference trees offer an alternative to reduce bias, but may not always improve predictive accuracy and have longer training times.
• Randomized splitting rules, like extremely randomized trees (which draw split points completely randomly), improve computational efficiency but may not enhance predictive accuracy and can even reduce it.

Tuning hyperparameters

• RF are a good example of a common situachion in ML:
  • As the number of parameter increases,
  • finding their optimal values requires more effort
  • and can even become computationally unfeasible.
• As more complex algorithms with greater number of hyperparameters are introduced tuning strategies should also be considered.

Tuning strategies

Random forests in bioinformatics

• Random forests have been thoroughly used in Bioinformatics (Boulesteix et al. 2012).
• Bioinformatics data are often high dimensional with
  • dozens or (less often) hundreds of samples/individuals
  • thousands (or hundreds of thousands) of variables.

Application of Random forests

• Random forests provide robust classifiers for:
  • Distinguishing cancer from non cancer,
  • Predicting tumor type in cancer of unknown origin,
  • Selecting variables (SNPs) in Genome Wide Association Studies…
• Some variation of Random forests are used only for variable selection

Another ensemble approach

Bagging vs Boosting

Source: Ensemble Learning: Bagging & Boosting

So what is Boosting

• Idea: create a model that is better than any of its individual components by combining their strengths and compensating for their weaknesses.

• For this, multiple weak models are trained sequentially, and each new model is trained to improve the errors made by the previous model.

• The final model is a weighted combination of all the models, and the weights are determined by the accuracy of each model.

Historical background

• Introduced by Robert Schapire (Schapire 1989) and Yoav Freund in the 1990s .
• AdaBoost algorithm became the first widely-used Boosting algorithm
• It achieved significant success in various applications, including face detection and handwriting recognition.
• Since then, several other Boosting algorithms have been developed, including: Gradient Boosting, XGBoost, and LightGBM.

Advantages of Boosting

• Boosting, like other Ensemble methods, improves the accuracy of weak learners and achieve better predictive performance than individual models.

• Boosting also reduces overfitting by improving the generalization ability of models.

• Available in many flavors,

• Can be parallelized

• Strong experience in Real world applications and industry.

Limitations of Boosting

• Can be computationally expensive, especially when dealing with large datasets and complex models.

• Can be sensitive to noisy data and outliers,

• May not work well with certain types of data distributions.

• Not so good as “out-of-the-box”: Requires careful tuning of hyperparameters to achieve optimal performance, which can be time-consuming and challenging.

Adaboost

• AdaBoost (Adaptive Boosting) directly reflects what has been described above as “boosting”.

• Its main objective is to Improve classification accuracy by combining multiple “weak learners”

• In order to run AdaBoost it is required:

  • To choose a weak learner with slightly better than random guessing capability (e.g., shallow decision trees).
  • To encoding classesas +1 and -1.
  • To set **initial equal weights* for all training observations.

Running Adaboost

• Adaboost proceeds iteratively by, at each iteration:
  • Fit the weak learner using initial weights.
  • Predict and identify misclassified observations.
  • Update observation weights: decrease for correctly classified, increase for misclassified.
  • Assign a learner weight to the weak learner based on its accuracy.
• The final prediction is a weighted combination of all weak learners.

Adaboost Architecture

Source: Ensemble Learning: Bagging & Boosting

Adaboost applications

Adaboost has limitations

• Adaboost was a breakthrough algorithm that significantly improved the accuracy of ML models.
• However, Adaboost has its limitations.
  • It does not handle continuous variables very well.
  • Can be sensitive to noisy data and outliers,
  • May not perform well with complex datasets.
  • Its performance can reach a “plateau”: it will no longer improve after a certain number of iterations.

Gradient Boosting

• Developed to overcome the limitations of Adaboost.
• Takes a different approach that can be linked with Optimization by Gradient Descent.
• Several advantages over Adaboost
  • Can handle continuous variables much better,
  • It is more robust to noisy data and outliers.
  • Can handle complex datasets and
  • Can continue to improve its accuracy even after Adaboost’s performance has “plateaued”.

Gradient boosting generalizes Adaboost

• It is a generalization of the AdaBoost algorithm that allows the use of any cost function, as long as it is differentiable.

• The flexibility of this algorithm has made it possible to apply boosting to a multitude of problems (regression, multiple classification, etc.), making it one of the most successful machine learning methods.

• While there are various adaptations, the general idea behind all of them is the same: train models sequentially, with each model adjusting the residuals (errors) of the previous models.

Gradient boosting architechture

Relation with Gradient Descent

• Gradient Boosting can be seen as an extension of Gradient Descent, a popular optimization algorithm used to find the minimum of a function.
  • In Gradient Descent, the weights of the model are updated in the opposite direction of the gradient of the cost function.
  • In Gradient Boosting, the new model is trained on the negative gradient of the loss function, which is equivalent to minimizing the loss function in the direction of steepest descent.

Gradient Descent Variations

• Multiple extensions from Gradient Boosting.

• XGBoost

  • Optimized implementation that uses regularization to control overfitting and provide better accuracy.
  • Won many competitions.

• LightGBM

  • Relies on a technique to reduce the number of samples used in each iteration.
  • Faster training, good for large datasets.

Boosting applications

• Fraud Detection

• Image and Speech Recognition

• Anomaly Detection

• Medical Diagnosis

• Amazon’s recommendation engine

• Models that predict protein structures from amino acid sequences

• Pattern identification in fMRI brain scans.

Boosting application with R

• Many packages implement the many variations of boosting:
  • ada, adabag, mboost, gbm, xgboost
• An interesting option is to rely on the caret package which allows to run the distinct methods with a common interface.