Gradient boosting

Part of a series on
Machine learning and data mining
hide Problems Classification Clustering Regression Anomaly detection Data Cleaning AutoML Association rules Reinforcement learning Structured prediction Feature engineering Feature learning Online learning Semi-supervised learning Unsupervised learning Learning to rank Grammar induction
hide Supervised learning (classification • regression) Decision trees Ensembles Bagging Boosting Random forest k-NN Linear regression Naive Bayes Artificial neural networks Logistic regression Perceptron Relevance vector machine (RVM) Support vector machine (SVM)
hide Clustering BIRCH CURE Hierarchical k-means Expectation–maximization (EM) DBSCAN OPTICS Mean shift
hide Dimensionality reduction Factor analysis CCA ICA LDA NMF PCA PGD t-SNE
hide Structured prediction Graphical models Bayes net Conditional random field Hidden Markov
hide Anomaly detection k-NN Local outlier factor
hide Artificial neural network Autoencoder Cognitive computing Deep learning DeepDream Multilayer perceptron RNN LSTM GRU ESN Restricted Boltzmann machine GAN SOM Convolutional neural network U-Net Transformer Spiking neural network Memtransistor Electrochemical RAM (ECRAM)
hide Reinforcement learning Q-learning SARSA Temporal difference (TD)
hide Theory Bias–variance tradeoff Computational learning theory Empirical risk minimization Occam learning PAC learning Statistical learning VC theory
hide Machine-learning venues NeurIPS ICML ML JMLR ArXiv:cs.LG
hide Related articles Glossary of artificial intelligence List of datasets for machine-learning research Outline of machine learning
v t

Gradient boosting is a machine learning technique for regression, classification and other tasks, which produces a prediction model in the form of an ensemble of weak prediction models, typically decision trees.^[1][2] When a decision tree is the weak learner, the resulting algorithm is called gradient boosted trees, which usually outperforms random forest.^[1][2][3] It builds the model in a stage-wise fashion like other boosting methods do, and it generalizes them by allowing optimization of an arbitrary differentiable loss function.

History[]

The idea of gradient boosting originated in the observation by Leo Breiman that boosting can be interpreted as an optimization algorithm on a suitable cost function.^[4] Explicit regression gradient boosting algorithms were subsequently developed by Jerome H. Friedman,^[5][6] simultaneously with the more general functional gradient boosting perspective of Llew Mason, Jonathan Baxter, Peter Bartlett and Marcus Frean.^[7][8] The latter two papers introduced the view of boosting algorithms as iterative functional gradient descent algorithms. That is, algorithms that optimize a cost function over function space by iteratively choosing a function (weak hypothesis) that points in the negative gradient direction. This functional gradient view of boosting has led to the development of boosting algorithms in many areas of machine learning and statistics beyond regression and classification.

Informal introduction[]

(This section follows the exposition of gradient boosting by Li.^[9])

Like other boosting methods, gradient boosting combines weak "learners" into a single strong learner in an iterative fashion. It is easiest to explain in the least-squares regression setting, where the goal is to "teach" a model ${\displaystyle F}$ to predict values of the form ${\displaystyle {\hat {y}}=F(x)}$ by minimizing the mean squared error ${\displaystyle {\tfrac {1}{n}}\sum _{i}({\hat {y}}_{i}-y_{i})^{2}}$, where ${\displaystyle i}$ indexes over some training set of size ${\displaystyle n}$ of actual values of the output variable ${\displaystyle y}$:

• ${\displaystyle {\hat {y}}_{i}=}$ the predicted value ${\displaystyle F(x)}$
• ${\displaystyle y_{i}=}$ the observed value
• ${\displaystyle n}$ the number of samples in ${\displaystyle y}$

Now, let us consider a gradient boosting algorithm with ${\displaystyle M}$ stages. At each stage ${\displaystyle m}$ (${\displaystyle 1\leq m\leq M}$) of gradient boosting, suppose some imperfect model ${\displaystyle F_{m}}$ (for low ${\displaystyle m}$, this model may simply return ${\displaystyle {\hat {y}}_{i}={\bar {y}}}$, where the RHS is the mean of ${\displaystyle y}$). In order to improve ${\displaystyle F_{m}}$, our algorithm should add some new estimator, ${\displaystyle h_{m}(x)}$. Thus,

${\displaystyle F_{m+1}(x)=F_{m}(x)+h_{m}(x)=y}$

or, equivalently,

${\displaystyle h_{m}(x)=y-F_{m}(x)}$.

Therefore, gradient boosting will fit h to the residual ${\displaystyle y-F_{m}(x)}$. As in other boosting variants, each ${\displaystyle F_{m+1}}$ attempts to correct the errors of its predecessor ${\displaystyle F_{m}}$. A generalization of this idea to loss functions other than squared error, and to classification and ranking problems, follows from the observation that residuals ${\displaystyle h_{m}(x)}$ for a given model are the negative gradients of the mean squared error (MSE) loss function (with respect to ${\displaystyle F(x)}$):

${\displaystyle L_{\rm {MSE}}={\frac {1}{2}}\left(y-F(x)\right)^{2}}$

${\displaystyle h_{m}(x)=-{\frac {\partial L_{\rm {MSE}}}{\partial F}}=y-F(x)}$.

So, gradient boosting could be specialized to a gradient descent algorithm, and generalizing it entails "plugging in" a different loss and its gradient.

Algorithm[]

In many supervised learning problems there is an output variable y and a vector of input variables x, related to each other with some probabilistic distribution. The goal is to find some function ${\displaystyle {\hat {F}}(x)}$ that best approximates the output variable from the values of input variables. This is formalized by introducing some loss function ${\displaystyle L(y,F(x))}$ and minimizing it:

${\displaystyle {\hat {F}}={\underset {F}{\arg \min }}\,\mathbb {E} _{x,y}[L(y,F(x))]}$.

The gradient boosting method assumes a real-valued y and seeks an approximation ${\displaystyle {\hat {F}}(x)}$ in the form of a weighted sum of functions ${\displaystyle h_{i}(x)}$ from some class ${\displaystyle {\mathcal {H}}}$, called base (or weak) learners:

${\displaystyle {\hat {F}}(x)=\sum _{i=1}^{M}\gamma _{i}h_{i}(x)+{\mbox{const}}}$.

We are usually given a training set ${\displaystyle \{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}}$ of known sample values of x and corresponding values of y. In accordance with the empirical risk minimization principle, the method tries to find an approximation ${\displaystyle {\hat {F}}(x)}$ that minimizes the average value of the loss function on the training set, i.e., minimizes the empirical risk. It does so by starting with a model, consisting of a constant function ${\displaystyle F_{0}(x)}$, and incrementally expands it in a greedy fashion:

${\displaystyle F_{0}(x)={\underset {\gamma }{\arg \min }}{\sum _{i=1}^{n}{L(y_{i},\gamma )}}}$,

${\displaystyle F_{m}(x)=F_{m-1}(x)+{\underset {h_{m}\in {\mathcal {H}}}{\operatorname {arg\,min} }}\left[{\sum _{i=1}^{n}{L(y_{i},F_{m-1}(x_{i})+h_{m}(x_{i}))}}\right]}$,

where ${\displaystyle h_{m}\in {\mathcal {H}}}$ is a base learner function.

Unfortunately, choosing the best function h at each step for an arbitrary loss function L is a computationally infeasible optimization problem in general. Therefore, we restrict our approach to a simplified version of the problem.

The idea is to apply a steepest descent step to this minimization problem (functional gradient descent).

The basic idea behind the steepest descend is to find a local minimum of the loss function by iterating on the ${\displaystyle F_{m}(x)}$. In fact it can be proved that the maximum descend (strongest negative derivative) of the loss function to the local minimum along the ${\displaystyle F_{m}(x)}$ is the function itself subtracted by the loss function gradient itself^{[citation needed]}.^[10] Hence:

${\displaystyle F_{m}(x)=F_{m-1}(x)-\gamma \sum _{i=1}^{n}{\nabla _{F_{m-1}}L(y_{i},F_{m-1}(x_{i}))}}$

Where ${\displaystyle \gamma >0}$. This implies: ${\displaystyle L(y_{i},F_{m}(x_{i}))\leq L(y_{i},F_{m-1}(x_{i}))}$.

In order to increase further the descend optimization we can optimize ${\displaystyle \gamma }$ by finding the ${\displaystyle \gamma }$ value for which the Loss Function has a minimum:

${\displaystyle \gamma _{m}={\underset {\gamma }{\arg \min }}{\sum _{i=1}^{n}{L\left(y_{i},F_{m})\right)}}={\underset {\gamma }{\arg \min }}{\sum _{i=1}^{n}{L\left(y_{i},F_{m-1}(x_{i})-\gamma \nabla _{F_{m-1}}L(y_{i},F_{m-1}(x_{i}))\right)}},}$

If we considered the continuous case, i.e. where ${\displaystyle {\mathcal {H}}}$ is the set of arbitrary differentiable functions on ${\displaystyle \mathbb {R} }$, we would update the model in accordance with the following equations

${\displaystyle F_{m}(x)=F_{m-1}(x)-\gamma _{m}\sum _{i=1}^{n}{\nabla _{F_{m-1}}L(y_{i},F_{m-1}(x_{i}))}}$

Where:

$${\displaystyle \gamma _{m}={\underset {\gamma }{\arg \min }}{\sum _{i=1}^{n}{L\left(y_{i},F_{m-1}(x_{i})-\gamma \nabla _{F_{m-1}}L(y_{i},F_{m-1}(x_{i}))\right)}},}$$

where the derivatives are taken with respect to the functions ${\displaystyle F_{i}}$ for ${\displaystyle i\in \{1,..,m\}}$, and ${\displaystyle \gamma _{m}}$ is the step length. In the discrete case however, i.e. when the set ${\displaystyle {\mathcal {H}}}$ is finite, we choose the candidate function h closest to the gradient of L for which the coefficient γ may then be calculated with the aid of line search on the above equations. Note that this approach is a heuristic and therefore doesn't yield an exact solution to the given problem, but rather an approximation. In pseudocode, the generic gradient boosting method is:^[5][2]

Gradient tree boosting[]

Gradient boosting is typically used with decision trees (especially CART trees) of a fixed size as base learners. For this special case, Friedman proposes a modification to gradient boosting method which improves the quality of fit of each base learner.

Generic gradient boosting at the m-th step would fit a decision tree ${\displaystyle h_{m}(x)}$ to pseudo-residuals. Let ${\displaystyle J_{m}}$ be the number of its leaves. The tree partitions the input space into ${\displaystyle J_{m}}$ disjoint regions ${\displaystyle R_{1m},\ldots ,R_{J_{m}m}}$ and predicts a constant value in each region. Using the indicator notation, the output of ${\displaystyle h_{m}(x)}$ for input x can be written as the sum:

${\displaystyle h_{m}(x)=\sum _{j=1}^{J_{m}}b_{jm}\mathbf {1} _{R_{jm}}(x),}$

where ${\displaystyle b_{jm}}$ is the value predicted in the region ${\displaystyle R_{jm}}$.^[11]

Then the coefficients ${\displaystyle b_{jm}}$ are multiplied by some value ${\displaystyle \gamma _{m}}$, chosen using line search so as to minimize the loss function, and the model is updated as follows:

${\displaystyle F_{m}(x)=F_{m-1}(x)+\gamma _{m}h_{m}(x),\quad \gamma _{m}={\underset {\gamma }{\operatorname {arg\,min} }}\sum _{i=1}^{n}L(y_{i},F_{m-1}(x_{i})+\gamma h_{m}(x_{i})).}$

Friedman proposes to modify this algorithm so that it chooses a separate optimal value ${\displaystyle \gamma _{jm}}$ for each of the tree's regions, instead of a single ${\displaystyle \gamma _{m}}$ for the whole tree. He calls the modified algorithm "TreeBoost". The coefficients ${\displaystyle b_{jm}}$ from the tree-fitting procedure can be then simply discarded and the model update rule becomes:

${\displaystyle F_{m}(x)=F_{m-1}(x)+\sum _{j=1}^{J_{m}}\gamma _{jm}\mathbf {1} _{R_{jm}}(x),\quad \gamma _{jm}={\underset {\gamma }{\operatorname {arg\,min} }}\sum _{x_{i}\in R_{jm}}L(y_{i},F_{m-1}(x_{i})+\gamma ).}$

Size of trees[]

${\displaystyle J}$, the number of terminal nodes in trees, is the method's parameter which can be adjusted for a data set at hand. It controls the maximum allowed level of interaction between variables in the model. With ${\displaystyle J=2}$ (decision stumps), no interaction between variables is allowed. With ${\displaystyle J=3}$ the model may include effects of the interaction between up to two variables, and so on.

Hastie et al.^[2] comment that typically ${\displaystyle 4\leq J\leq 8}$ work well for boosting and results are fairly insensitive to the choice of ${\displaystyle J}$ in this range, ${\displaystyle J=2}$ is insufficient for many applications, and ${\displaystyle J>10}$ is unlikely to be required.

Regularization[]

Fitting the training set too closely can lead to degradation of the model's generalization ability. Several so-called regularization techniques reduce this overfitting effect by constraining the fitting procedure.

One natural regularization parameter is the number of gradient boosting iterations M (i.e. the number of trees in the model when the base learner is a decision tree). Increasing M reduces the error on training set, but setting it too high may lead to overfitting. An optimal value of M is often selected by monitoring prediction error on a separate validation data set. Besides controlling M, several other regularization techniques are used.

Another regularization parameter is the depth of the trees. The higher this value the more likely the model will overfit the training data.

Shrinkage[]

An important part of gradient boosting method is regularization by shrinkage which consists in modifying the update rule as follows:

${\displaystyle F_{m}(x)=F_{m-1}(x)+\nu \cdot \gamma _{m}h_{m}(x),\quad 0<\nu \leq 1,}$

where parameter ${\displaystyle \nu }$ is called the "learning rate".

Empirically it has been found that using small learning rates (such as ${\displaystyle \nu <0.1}$) yields dramatic improvements in models' generalization ability over gradient boosting without shrinking (${\displaystyle \nu =1}$).^[2] However, it comes at the price of increasing computational time both during training and querying: lower learning rate requires more iterations.

Stochastic gradient boosting[]

Soon after the introduction of gradient boosting, Friedman proposed a minor modification to the algorithm, motivated by Breiman's bootstrap aggregation ("bagging") method.^[6] Specifically, he proposed that at each iteration of the algorithm, a base learner should be fit on a subsample of the training set drawn at random without replacement.^[12] Friedman observed a substantial improvement in gradient boosting's accuracy with this modification.

Subsample size is some constant fraction ${\displaystyle f}$ of the size of the training set. When ${\displaystyle f=1}$, the algorithm is deterministic and identical to the one described above. Smaller values of ${\displaystyle f}$ introduce randomness into the algorithm and help prevent overfitting, acting as a kind of regularization. The algorithm also becomes faster, because regression trees have to be fit to smaller datasets at each iteration. Friedman^[6] obtained that ${\displaystyle 0.5\leq f\leq 0.8}$ leads to good results for small and moderate sized training sets. Therefore, ${\displaystyle f}$ is typically set to 0.5, meaning that one half of the training set is used to build each base learner.

Also, like in bagging, subsampling allows one to define an out-of-bag error of the prediction performance improvement by evaluating predictions on those observations which were not used in the building of the next base learner. Out-of-bag estimates help avoid the need for an independent validation dataset, but often underestimate actual performance improvement and the optimal number of iterations.^[13][14]

Number of observations in leaves[]

Gradient tree boosting implementations often also use regularization by limiting the minimum number of observations in trees' terminal nodes. It is used in the tree building process by ignoring any splits that lead to nodes containing fewer than this number of training set instances.

Imposing this limit helps to reduce variance in predictions at leaves.

Penalize complexity of tree[]

Another useful regularization techniques for gradient boosted trees is to penalize model complexity of the learned model.^[15] The model complexity can be defined as the proportional number of leaves in the learned trees. The joint optimization of loss and model complexity corresponds to a post-pruning algorithm to remove branches that fail to reduce the loss by a threshold. Other kinds of regularization such as an ${\displaystyle \ell _{2}}$ penalty on the leaf values can also be added to avoid overfitting.

Usage[]

Gradient boosting can be used in the field of learning to rank. The commercial web search engines Yahoo^[16] and Yandex^[17] use variants of gradient boosting in their machine-learned ranking engines. Gradient boosting is also utilized in High Energy Physics in data analysis. At the Large Hadron Collider (LHC), variants of gradient boosting Deep Neural Networks (DNN) were successful in reproducing the results of non-machine learning methods of analysis on datasets used to discover the Higgs boson.^[18]

Names[]

The method goes by a variety of names. Friedman introduced his regression technique as a "Gradient Boosting Machine" (GBM).^[5] Mason, Baxter et al. described the generalized abstract class of algorithms as "functional gradient boosting".^[7][8] Friedman et al. describe an advancement of gradient boosted models as Multiple Additive Regression Trees (MART);^[19] Elith et al. describe that approach as "Boosted Regression Trees" (BRT).^[20]

A popular open-source implementation for R calls it a "Generalized Boosting Model",^[13] however packages expanding this work use BRT.^[21] Yet another name is TreeNet, after an early commercial implementation from Salford System's Dan Steinberg, one of researchers who pioneered the use of tree-based methods.^[22] XGBoost is another popular modern implementation of the method with some extensions, like second-order optimization.

Disadvantages[]

While boosting can increase the accuracy of a base learner, such as a decision tree or linear regression, it sacrifices intelligibility and interpretability.^[1][23] Furthermore, its implementation may be more difficult due to the higher computational demand.

See also[]

• AdaBoost
• Random forest
• Catboost
• LightGBM
• XGBoost
• Decision tree learning

References[]

Further reading[]

• Boehmke, Bradley; Greenwell, Brandon (2019). "Gradient Boosting". Hands-On Machine Learning with R. Chapman & Hall. pp. 221–245. ISBN 978-1-138-49568-5.

External links[]

• How to explain gradient boosting
• Gradient Boosted Regression Trees
• LightGBM