FYS-STK4155/doc/LectureNotes/_build/jupyter_execute/week47.ipynb

Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods and Summary of Course

Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University

Date: November 20-24, 2023

Plan for week 47

Active learning sessions on Tuesday and Wednesday.

• Work and Discussion of project 3

• Last weekly exercise, course feedback, to be completed by Sunday November 26

Material for the lecture on Thursday November 23, 2023.

• Thursday: Basics of decision trees, classification and regression algorithms and ensemble models

• Readings and Videos:

  • These lecture notes

  • Video of Lecture

  • Whiteboard notes

  • Video on Decision trees

  • Video on boosting methods by Hastie

  • Video on AdaBoost

  • Video on Gradient boost, part 1, parts 2-4 follow thereafter

  • Decision Trees: Geron's chapter 6 covers decision trees while ensemble models, voting and bagging are discussed in chapter 7. See also lecture from STK-IN4300, lecture 7. Chapter 9.2 of Hastie et al contains also a good discussion.

Bagging

The plain decision trees suffer from high variance. This means that if we split the training data into two parts at random, and fit a decision tree to both halves, the results that we get could be quite different. In contrast, a procedure with low variance will yield similar results if applied repeatedly to distinct data sets; linear regression tends to have low variance, if the ratio of n to p is moderately large.

Bootstrap aggregation, or just bagging, is a general-purpose procedure for reducing the variance of a statistical learning method.

More bagging

Bagging typically results in improved accuracy over prediction using a single tree. Unfortunately, however, it can be difficult to interpret the resulting model. Recall that one of the advantages of decision trees is the attractive and easily interpreted diagram that results.

However, when we bag a large number of trees, it is no longer possible to represent the resulting statistical learning procedure using a single tree, and it is no longer clear which variables are most important to the procedure. Thus, bagging improves prediction accuracy at the expense of interpretability. Although the collection of bagged trees is much more difficult to interpret than a single tree, one can obtain an overall summary of the importance of each predictor using the MSE (for bagging regression trees) or the Gini index (for bagging classification trees). In the case of bagging regression trees, we can record the total amount that the MSE is decreased due to splits over a given predictor, averaged over all B possible trees. A large value indicates an important predictor. Similarly, in the context of bagging classification trees, we can add up the total amount that the Gini index is decreased by splits over a given predictor, averaged over all B trees.

Making your own Bootstrap: Changing the Level of the Decision Tree

Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points n).

In [1]:

%matplotlib inline

# Common imports
import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.pipeline import make_pipeline
from sklearn.utils import resample
from sklearn.tree import DecisionTreeRegressor
import pandas as pd
from sklearn.tree import DecisionTreeClassifier
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler, OneHotEncoder
from sklearn.compose import ColumnTransformer
from IPython.display import Image 
import os

# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"

if not os.path.exists(PROJECT_ROOT_DIR):
    os.mkdir(PROJECT_ROOT_DIR)

if not os.path.exists(FIGURE_ID):
    os.makedirs(FIGURE_ID)

if not os.path.exists(DATA_ID):
    os.makedirs(DATA_ID)

def image_path(fig_id):
    return os.path.join(FIGURE_ID, fig_id)

def data_path(dat_id):
    return os.path.join(DATA_ID, dat_id)

def save_fig(fig_id):
    plt.savefig(image_path(fig_id) + ".png", format='png')


n = 100
n_boostraps = 100
maxdepth = 8

# Make data set.
x = np.linspace(-3, 3, n).reshape(-1, 1)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
error = np.zeros(maxdepth)
bias = np.zeros(maxdepth)
variance = np.zeros(maxdepth)
polydegree = np.zeros(maxdepth)
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)

from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)

# we produce a simple tree first as benchmark
simpletree = DecisionTreeRegressor(max_depth=3) 
simpletree.fit(X_train_scaled, y_train)
simpleprediction = simpletree.predict(X_test_scaled)
for degree in range(1,maxdepth):
    model = DecisionTreeRegressor(max_depth=degree) 
    y_pred = np.empty((y_test.shape[0], n_boostraps))
    for i in range(n_boostraps):
        x_, y_ = resample(X_train_scaled, y_train)
        model.fit(x_, y_)
        y_pred[:, i] = model.predict(X_test_scaled)#.ravel()

    polydegree[degree] = degree
    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
    print('Polynomial degree:', degree)
    print('Error:', error[degree])
    print('Bias^2:', bias[degree])
    print('Var:', variance[degree])
    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
 
mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2))
print("Simple tree:",mse_simpletree)
plt.xlim(1,maxdepth)
plt.plot(polydegree, error, label='MSE')
plt.plot(polydegree, bias, label='bias')
plt.plot(polydegree, variance, label='Variance')
plt.legend()
save_fig("baggingboot")
plt.show()

Polynomial degree: 1
Error: 0.05485044745873867
Bias^2: 0.05363014989229746
Var: 0.0012202975664411882
0.05485044745873867 >= 0.05363014989229746 + 0.0012202975664411882 = 0.05485044745873865
Polynomial degree: 2
Error: 0.04754825003279861
Bias^2: 0.0362312015777108
Var: 0.01131704845508782
0.04754825003279861 >= 0.0362312015777108 + 0.01131704845508782 = 0.04754825003279862
Polynomial degree: 3
Error: 0.028256917047283964
Bias^2: 0.019709199043491926
Var: 0.008547718003792035
0.028256917047283964 >= 0.019709199043491926 + 0.008547718003792035 = 0.02825691704728396
Polynomial degree: 4
Error: 0.02417252675174287
Bias^2: 0.016541517177965183
Var: 0.007631009573777696
0.02417252675174287 >= 0.016541517177965183 + 0.007631009573777696 = 0.02417252675174288
Polynomial degree: 5
Error: 0.020350773309798075
Bias^2: 0.013742894355267554
Var: 0.006607878954530523
0.020350773309798075 >= 0.013742894355267554 + 0.006607878954530523 = 0.02035077330979808
Polynomial degree: 6
Error: 0.019509108923639135
Bias^2: 0.01312013610582818
Var: 0.006388972817810939
0.019509108923639135 >= 0.01312013610582818 + 0.006388972817810939 = 0.01950910892363912
Polynomial degree: 7
Error: 0.020056743323946562
Bias^2: 0.012815095479733507
Var: 0.007241647844213045
0.020056743323946562 >= 0.012815095479733507 + 0.007241647844213045 = 0.020056743323946552
Simple tree: 0.5601973572808581

Random forests

Random forests provide an improvement over bagged trees by way of a small tweak that decorrelates the trees.

As in bagging, we build a number of decision trees on bootstrapped training samples. But when building these decision trees, each time a split in a tree is considered, a random sample of m predictors is chosen as split candidates from the full set of p predictors. The split is allowed to use only one of those m predictors.

A fresh sample of m predictors is taken at each split, and typically we choose

m\approx \sqrt{p}.

In building a random forest, at each split in the tree, the algorithm is not even allowed to consider a majority of the available predictors.

The reason for this is rather clever. Suppose that there is one very strong predictor in the data set, along with a number of other moderately strong predictors. Then in the collection of bagged variable importance random forest trees, most or all of the trees will use this strong predictor in the top split. Consequently, all of the bagged trees will look quite similar to each other. Hence the predictions from the bagged trees will be highly correlated. Unfortunately, averaging many highly correlated quantities does not lead to as large of a reduction in variance as averaging many uncorrelated quantities. In particular, this means that bagging will not lead to a substantial reduction in variance over a single tree in this setting.

Random Forest Algorithm

The algorithm described here can be applied to both classification and regression problems.

We will grow of forest of say B trees.

1. For b=1:B

• Draw a bootstrap sample from the training data organized in our \boldsymbol{X} matrix.

• We grow then a random forest tree T_b based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached

1. we select m \le p variables at random from the p predictors/features

2. pick the best split point among the m features using for example the CART algorithm and create a new node

3. split the node into daughter nodes

4. Output then the ensemble of trees \{T_b\}_1^{B} and make predictions for either a regression type of problem or a classification type of problem.

Random Forests Compared with other Methods on the Cancer Data

In [2]:

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
from sklearn.svm import SVC
from sklearn.linear_model import LogisticRegression
from sklearn.tree import DecisionTreeClassifier
from sklearn.ensemble import BaggingClassifier

# Load the data
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
#define methods
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
# Support vector machine
svm = SVC(gamma='auto', C=100)
# Decision Trees
deep_tree_clf = DecisionTreeClassifier(max_depth=None)
#Scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Logistic Regression
logreg.fit(X_train_scaled, y_train)
print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Support Vector Machine
svm.fit(X_train_scaled, y_train)
print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Decision Trees
deep_tree_clf.fit(X_train_scaled, y_train)
print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))


from sklearn.ensemble import RandomForestClassifier
from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import cross_validate
# Data set not specificied
#Instantiate the model with 500 trees and entropy as splitting criteria
Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy")
Random_Forest_model.fit(X_train_scaled, y_train)
#Cross validation
accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
print(accuracy)
print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))


import scikitplot as skplt
y_pred = Random_Forest_model.predict(X_test_scaled)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = Random_Forest_model.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
plt.show()

(426, 30)
(143, 30)
Test set accuracy Logistic Regression with scaled data: 0.96
Test set accuracy SVM with scaled data: 0.96
Test set accuracy with Decision Trees and scaled data: 0.87

[0.93333333 0.73333333 0.93333333 1.         1.         0.92857143
 1.         0.92857143 0.92857143 1.        ]
Test set accuracy with Random Forests and scaled data: 0.98

Recall that the cumulative gains curve shows the percentage of the overall number of cases in a given category gained by targeting a percentage of the total number of cases.

Similarly, the receiver operating characteristic curve, or ROC curve, displays the diagnostic ability of a binary classifier system as its discrimination threshold is varied. It plots the true positive rate against the false positive rate.

Compare Bagging on Trees with Random Forests

In [3]:

bag_clf = BaggingClassifier(
    DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
    n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)

In [4]:

bag_clf.fit(X_train, y_train)
y_pred = bag_clf.predict(X_test)
from sklearn.ensemble import RandomForestClassifier
rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
rnd_clf.fit(X_train, y_train)
y_pred_rf = rnd_clf.predict(X_test)
np.sum(y_pred == y_pred_rf) / len(y_pred)

Out [4]:

0.9790209790209791

Boosting, a Bird's Eye View

The basic idea is to combine weak classifiers in order to create a good classifier. With a weak classifier we often intend a classifier which produces results which are only slightly better than we would get by random guesses.

This is done by applying in an iterative way a weak (or a standard classifier like decision trees) to modify the data. In each iteration we emphasize those observations which are misclassified by weighting them with a factor.

What is boosting? Additive Modelling/Iterative Fitting

Boosting is a way of fitting an additive expansion in a set of elementary basis functions like for example some simple polynomials. Assume for example that we have a function

f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),

where \beta_m are the expansion parameters to be determined in a minimization process and b(x;\gamma_m) are some simple functions of the multivariable parameter x which is characterized by the parameters \gamma_m.

As an example, consider the Sigmoid function we used in logistic regression. In that case, we can translate the function b(x;\gamma_m) into the Sigmoid function

\sigma(t) = \frac{1}{1+\exp{(-t)}},

where t=\gamma_0+\gamma_1 x and the parameters \gamma_0 and \gamma_1 were determined by the Logistic Regression fitting algorithm.

As another example, consider the cost function we defined for linear regression

C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.

In this case the function f(x) was replaced by the design matrix \boldsymbol{X} and the unknown linear regression parameters \boldsymbol{\beta}, that is \boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta}. In linear regression we can simply invert a matrix and obtain the parameters \beta by

\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.

In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \beta_m and \gamma_m.

Iterative Fitting, Regression and Squared-error Cost Function

The way we proceed is as follows (here we specialize to the squared-error cost function)

1. Establish a cost function, here {\cal C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2 with f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m).

2. Initialize with a guess f_0(x). It could be one or even zero or some random numbers.

3. For m=1:M

a. minimize \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2 wrt \gamma and \beta

b. This gives the optimal values \beta_m and \gamma_m

c. Determine then the new values f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)

We could use any of the algorithms we have discussed till now. If we use trees, \gamma parameterizes the split variables and split points at the internal nodes, and the predictions at the terminal nodes.

Squared-Error Example and Iterative Fitting

To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.

For simplicity we assume also that our functions b(x;\gamma)=1+\gamma x.

This means that for every iteration m, we need to optimize

(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.

We start our iteration by simply setting f_0(x)=0. Taking the derivatives with respect to \beta and \gamma we obtain

\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,

and

\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.

We can then rewrite these equations as (defining \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) with \boldsymbol{e} being the unit vector)

\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0,

which gives us \beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w}). Similarly we have

\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,

which leads to \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}). Inserting for \beta gives us an equation for \gamma. This is a non-linear equation in the unknown \gamma and has to be solved numerically.

The solution to these two equations gives us in turn \beta_1 and \gamma_1 leading to the new expression for f_1(x) as f_1(x) = \beta_1(1+\gamma_1x). Doing this M times results in our final estimate for the function f.

Iterative Fitting, Classification and AdaBoost

Let us consider a binary classification problem with two outcomes y_i \in \{-1,1\} and i=0,1,2,\dots,n-1 as our set of observations. We define a classification function G(x) which produces a prediction taking one or the other of the two values \{-1,1\}.

The error rate of the training sample is then

\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).

The iterative procedure starts with defining a weak classifier whose error rate is barely better than random guessing. The iterative procedure in boosting is to sequentially apply a weak classification algorithm to repeatedly modified versions of the data producing a sequence of weak classifiers G_m(x).

Here we will express our function f(x) in terms of G(x). That is

f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),

will be a function of

G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).

Adaptive Boosting, AdaBoost

In our iterative procedure we define thus

f_m(x) = f_{m-1}(x)+\beta_mG_m(x).

The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the exponential cost/loss function defined as

C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.

We optimize \beta and G for each value of m=1:M as we did in the regression case. This is normally done in two steps. Let us however first rewrite the cost function as

C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},

where we have defined w_i^m= \exp{(-y_if_{m-1}(x_i))}.

Building up AdaBoost

First, for any \beta > 0, we optimize G by setting

G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),

which is the classifier that minimizes the weighted error rate in predicting y.

We can do this by rewriting

\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,

which can be rewritten as

(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,

which leads to

\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},

where we have redefined the error as

\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},

which leads to an update of

f_m(x) = f_{m-1}(x) +\beta_m G_m(x).

This leads to the new weights

w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}

Adaptive boosting: AdaBoost, Basic Algorithm

The algorithm here is rather straightforward. Assume that our weak classifier is a decision tree and we consider a binary set of outputs with y_i \in \{-1,1\} and i=0,1,2,\dots,n-1 as our set of observations. Our design matrix is given in terms of the feature/predictor vectors \boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}]. Finally, we define also a classifier determined by our data via a function G(x). This function tells us how well we are able to classify our outputs/targets \boldsymbol{y}.

We have already defined the misclassification error \mathrm{err} as

\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),

where the function I() is one if we misclassify and zero if we classify correctly.

Basic Steps of AdaBoost

With the above definitions we are now ready to set up the algorithm for AdaBoost. The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.

1. We start by initializing all weights to w_i = 1/n, with i=0,1,2,\dots n-1. It is easy to see that we must have \sum_{i=0}^{n-1}w_i = 1.

2. We rewrite the misclassification error as

\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},

1. Then we start looping over all attempts at classifying, namely we start an iterative process for m=1:M, where M is the final number of classifications. Our given classifier could for example be a plain decision tree.

a. Fit then a given classifier to the training set using the weights w_i.

b. Compute then \mathrm{err} and figure out which events are classified properly and which are classified wrongly.

c. Define a quantity \alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}

d. Set the new weights to w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}.

5. Compute the new classifier G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i).

For the iterations with m \le 2 the weights are modified individually at each steps. The observations which were misclassified at iteration m-1 have a weight which is larger than those which were classified properly. As this proceeds, the observations which were difficult to classifiy correctly are given a larger influence. Each new classification step m is then forced to concentrate on those observations that are missed in the previous iterations.

AdaBoost Examples

Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here.

In [5]:

from sklearn.ensemble import AdaBoostClassifier

ada_clf = AdaBoostClassifier(
    DecisionTreeClassifier(max_depth=2), n_estimators=200,
    algorithm="SAMME.R", learning_rate=0.01, random_state=42)
ada_clf.fit(X_train, y_train)
y_pred = ada_clf.predict(X_test)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = ada_clf.predict_proba(X_test)
skplt.metrics.plot_roc(y_test, y_probas)
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
plt.show()

Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent

Gradient boosting is again a similar technique to Adaptive boosting, it combines so-called weak classifiers or regressors into a strong method via a series of iterations.

In order to understand the method, let us illustrate its basics by bringing back the essential steps in linear regression, where our cost function was the least squares function.

The Squared-Error again! Steepest Descent

We start again with our cost function {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) where we want to minimize This means that for every iteration, we need to optimize

(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.

We define a real function h_m(x) that defines our final function f_M(x) as

f_M(x) = \sum_{m=0}^M h_m(x).

In the steepest decent approach we approximate h_m(x) = -\rho_m g_m(x), where \rho_m is a scalar and g_m(x) the gradient defined as

g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.

With the new gradient we can update f_m(x) = f_{m-1}(x) -\rho_m g_m(x). Using the above squared-error function we see that the gradient is g_m(x_i) = -2(y_i-f(x_i)).

Choosing f_0(x)=0 we obtain g_m(x) = -2y_i and inserting this into the minimization problem for the cost function we have

(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.

Steepest Descent Example

Optimizing with respect to \rho we obtain (taking the derivative) that \rho_1 = -1/2. We have then that

f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i.

We can then proceed and compute

g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,

and find a new value for \rho_2=-1/2 and continue till we have reached m=M. We can modify the steepest descent method, or steepest boosting, by introducing what is called gradient boosting.

Gradient Boosting, algorithm

Steepest descent is however not much used, since it only optimizes f at a fixed set of n points, so we do not learn a function that can generalize. However, we can modify the algorithm by fitting a weak learner to approximate the negative gradient signal.

Suppose we have a cost function C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) where y_i is our target and f(x_i) the function which is meant to model y_i. The above cost function could be our standard squared-error function

C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.

The way we proceed in an iterative fashion is to

1. Initialize our estimate f_0(x).

2. For m=1:M, we

a. compute the negative gradient vector \boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x) at f(x) = f_{m-1}(x);

b. fit the so-called base-learner to the negative gradient h_m(u_m,x);

c. update the estimate f_m(x) = f_{m-1}(x)+h_m(u_m,x);

4. The final estimate is then f_M(x) = \sum_{m=1}^M h_m(u_m,x).

Gradient Boosting, Examples of Regression

In [6]:

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.ensemble import GradientBoostingRegressor
import scikitplot as skplt
from sklearn.metrics import mean_squared_error

n = 100
maxdegree = 6

# Make data set.
x = np.linspace(-3, 3, n).reshape(-1, 1)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)

error = np.zeros(maxdegree)
bias = np.zeros(maxdegree)
variance = np.zeros(maxdegree)
polydegree = np.zeros(maxdegree)
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)

for degree in range(1,maxdegree):
    model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0)  
    model.fit(X_train,y_train)
    y_pred = model.predict(X_test)
    polydegree[degree] = degree
    error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
    bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
    variance[degree] = np.mean( np.var(y_pred) )
    print('Max depth:', degree)
    print('Error:', error[degree])
    print('Bias^2:', bias[degree])
    print('Var:', variance[degree])
    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))

plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
plt.plot(polydegree, bias, label='bias')
plt.plot(polydegree, variance, label='Variance')
plt.legend()
save_fig("gdregression")
plt.show()

Max depth: 1
Error: 0.5069860830630872
Bias^2: 0.28951491799791806
Var: 0.21747116506516934
0.5069860830630872 >= 0.28951491799791806 + 0.21747116506516934 = 0.5069860830630875
Max depth: 2
Error: 0.5222413718621189
Bias^2: 0.28962869031035515
Var: 0.23261268155176382
0.5222413718621189 >= 0.28962869031035515 + 0.23261268155176382 = 0.5222413718621189
Max depth: 3
Error: 0.522240032475565
Bias^2: 0.2896287710119233
Var: 0.2326112614636416
0.522240032475565 >= 0.2896287710119233 + 0.2326112614636416 = 0.5222400324755649
Max depth: 4
Error: 0.5222400329453616
Bias^2: 0.28962877060331055
Var: 0.2326112623420511
0.5222400329453616 >= 0.28962877060331055 + 0.2326112623420511 = 0.5222400329453616
Max depth: 5
Error: 0.5222400329453616
Bias^2: 0.28962877060331055
Var: 0.2326112623420511
0.5222400329453616 >= 0.28962877060331055 + 0.2326112623420511 = 0.5222400329453616

/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  y = column_or_1d(y, warn=True)
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  y = column_or_1d(y, warn=True)
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  y = column_or_1d(y, warn=True)
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  y = column_or_1d(y, warn=True)
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:494: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  y = column_or_1d(y, warn=True)

Gradient Boosting, Classification Example

In [7]:

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
import scikitplot as skplt
from sklearn.ensemble import GradientBoostingClassifier
from sklearn.model_selection import cross_validate

# Load the data
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)

gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0)  
gd_clf.fit(X_train_scaled, y_train)
#Cross validation
accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
print(accuracy)
print("Test set accuracy with Gradient boosting and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))

import scikitplot as skplt
y_pred = gd_clf.predict(X_test_scaled)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("gdclassiffierconfusion")
plt.show()
y_probas = gd_clf.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
save_fig("gdclassiffierroc")
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
save_fig("gdclassiffiercgain")
plt.show()

(426, 30)
(143, 30)

[0.93333333 0.93333333 0.8        0.85714286 1.         0.92857143
 1.         0.92857143 0.92857143 0.92857143]
Test set accuracy with Gradient boosting and scaled data: 0.97

XGBoost: Extreme Gradient Boosting

XGBoost or Extreme Gradient Boosting, is an optimized distributed gradient boosting library designed to be highly efficient, flexible and portable. It implements machine learning algorithms under the Gradient Boosting framework. XGBoost provides a parallel tree boosting that solve many data science problems in a fast and accurate way. See the article by Chen and Guestrin.

The authors design and build a highly scalable end-to-end tree boosting system. It has a theoretically justified weighted quantile sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.

It is now the algorithm which wins essentially all ML competitions!!!

Regression Case

In [8]:

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
import xgboost as xgb
import scikitplot as skplt
from sklearn.metrics import mean_squared_error

n = 100
maxdegree = 6

# Make data set.
x = np.linspace(-3, 3, n).reshape(-1, 1)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)

error = np.zeros(maxdegree)
bias = np.zeros(maxdegree)
variance = np.zeros(maxdegree)
polydegree = np.zeros(maxdegree)
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)

for degree in range(maxdegree):
    model =  xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200)

    model.fit(X_train,y_train)
    y_pred = model.predict(X_test)
    polydegree[degree] = degree
    error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
    bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
    variance[degree] = np.mean( np.var(y_pred) )
    print('Max depth:', degree)
    print('Error:', error[degree])
    print('Bias^2:', bias[degree])
    print('Var:', variance[degree])
    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))

plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
plt.plot(polydegree, bias, label='bias')
plt.plot(polydegree, variance, label='Variance')
plt.legend()
plt.show()

/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/xgboost/compat.py:36: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead.
  from pandas import MultiIndex, Int64Index

[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576: 
Parameters: { "colsaobjective" } might not be used.

  This could be a false alarm, with some parameters getting used by language bindings but
  then being mistakenly passed down to XGBoost core, or some parameter actually being used
  but getting flagged wrongly here. Please open an issue if you find any such cases.


Max depth: 0
Error: 0.35587778675776993
Bias^2: 0.35587778675776993
Var: 0.0
0.35587778675776993 >= 0.35587778675776993 + 0.0 = 0.35587778675776993
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576: 
Parameters: { "colsaobjective" } might not be used.

  This could be a false alarm, with some parameters getting used by language bindings but
  then being mistakenly passed down to XGBoost core, or some parameter actually being used
  but getting flagged wrongly here. Please open an issue if you find any such cases.


Max depth: 1
Error: 0.3001669239476101
Bias^2: 0.2731899981798276
Var: 0.0269769337028265
0.3001669239476101 >= 0.2731899981798276 + 0.0269769337028265 = 0.3001669318826541
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576: 
Parameters: { "colsaobjective" } might not be used.

  This could be a false alarm, with some parameters getting used by language bindings but
  then being mistakenly passed down to XGBoost core, or some parameter actually being used
  but getting flagged wrongly here. Please open an issue if you find any such cases.


Max depth: 2
Error: 0.30000279381576256
Bias^2: 0.2711099029541577
Var: 0.02889288030564785
0.30000279381576256 >= 0.2711099029541577 + 0.02889288030564785 = 0.30000278325980556
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576: 
Parameters: { "colsaobjective" } might not be used.

  This could be a false alarm, with some parameters getting used by language bindings but
  then being mistakenly passed down to XGBoost core, or some parameter actually being used
  but getting flagged wrongly here. Please open an issue if you find any such cases.


Max depth: 3
Error: 0.2999692169766251
Bias^2: 0.2710765113417533
Var: 0.028892725706100464
0.2999692169766251 >= 0.2710765113417533 + 0.028892725706100464 = 0.2999692370478538
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576: 
Parameters: { "colsaobjective" } might not be used.

  This could be a false alarm, with some parameters getting used by language bindings but
  then being mistakenly passed down to XGBoost core, or some parameter actually being used
  but getting flagged wrongly here. Please open an issue if you find any such cases.


Max depth: 4
Error: 0.2999728858924457
Bias^2: 0.2710867359084896
Var: 0.02888614870607853
0.2999728858924457 >= 0.2710867359084896 + 0.02888614870607853 = 0.2999728846145681
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:576: 
Parameters: { "colsaobjective" } might not be used.

  This could be a false alarm, with some parameters getting used by language bindings but
  then being mistakenly passed down to XGBoost core, or some parameter actually being used
  but getting flagged wrongly here. Please open an issue if you find any such cases.


Max depth: 5
Error: 0.29998782047173667
Bias^2: 0.2711004934923823
Var: 0.02888733707368374
0.29998782047173667 >= 0.2711004934923823 + 0.02888733707368374 = 0.29998783056606604

Xgboost on the Cancer Data

As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.

In [9]:

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import cross_validate
import scikitplot as skplt
import xgboost as xgb
# Load the data
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)

xg_clf = xgb.XGBClassifier()
xg_clf.fit(X_train_scaled,y_train)

y_test = xg_clf.predict(X_test_scaled)

print("Test set accuracy with Gradient Boosting and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))

import scikitplot as skplt
y_pred = xg_clf.predict(X_test_scaled)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("xdclassiffierconfusion")
plt.show()
y_probas = xg_clf.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
save_fig("xdclassiffierroc")
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
save_fig("gdclassiffiercgain")
plt.show()


xgb.plot_tree(xg_clf,num_trees=0)
plt.rcParams['figure.figsize'] = [50, 10]
save_fig("xgtree")
plt.show()

xgb.plot_importance(xg_clf)
plt.rcParams['figure.figsize'] = [5, 5]
save_fig("xgparams")
plt.show()

(426, 30)
(143, 30)
[15:38:29] WARNING: /var/folders/nz/j6p8yfhx1mv_0grj5xl4650h0000gp/T/abs_eek2t0c4ro/croots/recipe/xgboost-split_1659548960591/work/src/learner.cc:1115: Starting in XGBoost 1.3.0, the default evaluation metric used with the objective 'binary:logistic' was changed from 'error' to 'logloss'. Explicitly set eval_metric if you'd like to restore the old behavior.
Test set accuracy with Gradient Boosting and scaled data: 1.00

/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/xgboost/sklearn.py:1224: UserWarning: The use of label encoder in XGBClassifier is deprecated and will be removed in a future release. To remove this warning, do the following: 1) Pass option use_label_encoder=False when constructing XGBClassifier object; and 2) Encode your labels (y) as integers starting with 0, i.e. 0, 1, 2, ..., [num_class - 1].
  warnings.warn(label_encoder_deprecation_msg, UserWarning)

Warning:

Output truncated. This notebook contains too many cells to display efficiently.