A Step by Step Gradient Boosting Decision Tree Example

Adoption of decision trees is mainly based on its transparent decisions. Also, they overwhelmingly over-perform in applied machine learning studies. Particularly, GBM based trees dominate Kaggle competitions nowadays. Some kaggle winner researchers mentioned that they just used a specific boosting algorithm. However, some practitioners think GBM as a black box just like neural networks. They might just consume LightGBM without understanding its background. This is against decision tree’s nature. We will mention the basic idea of GBDT / GBRT and apply it on a step by step example.

Before Getting Started

Lecture notes of Zico Colter from Carnegie Mellon University and lecture notes of Cheng Li from Northeastern University guide me to understand the concept. Moreover, Tianqi Chen‘s presentation reinforce to make sense. Also, I referenced all sources help me to make clear the subject as a link in this post. I strongly recommend you to visit these links.

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I pushed the core implementation of gradient boosted regression tree algorithm to GitHub. You might want to clone the repository and run it by yourself.

Objective

The following video covers the idea behind GBM. It is very important to know what exactly gradient boosting is before getting started.

Gradient Boosting in Python

This blog post mentions the deeply explanation of gradient boosting algorithm and we will solve a problem step by step. On the other hand, you might just want to run gradient boosting. Its mathematical background might not attract your attention.

Herein, you can find the python implementation of Gradient Boosting algorithm here. This package supports regular decision tree algorithms such as ID3, C4.5, CART, CHAID or Regression Trees, also bagging methods such as random forest and some boosting methods such as adaboost. You can support this work just by starring the GitHub repository.

We actually apply boosting in real world!

This is very similar to baby step giant step method. We initially create a decision tree for the raw data set. That would be the giant step. Then, it is time to tune and boost. We will create new decision tree based on previous tree’s error. We will apply this approach several times. These would be baby steps. Terence Parr described this process wonderfully in golf playing scenario as illustrated below.

Herein, remember random forest algorithm. We separate data set to n different sub data sets and create different decision trees for these sub data sets. In contrast, data set remains same in GBM. We will create a decision tree, we will feed decision tree algorithm same data set but we will update each instance’s label value as its actual value minus its prediction.

You might think sequential decision trees in gradient boosting.

Summing the predictions

For instance, the following illustration shows that first decision tree returns 2 as a result for the boy. Then, we will build another decision tree based on errors for the first decision tree’s results. It returns 0.9 in this time for the boy. Final decision for the boy would be 2.9 which sums the prediction of sequential trees.

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A Step by Step Example

You might remember that we’ve mentioned regression trees in previous posts. Reading that post will contribute to understand GBM clearly.

We’ve worked on the following data set.

Day	Outlook	Temp.	Humidity	Wind	Decision
1	Sunny	Hot	High	Weak	25
2	Sunny	Hot	High	Strong	30
3	Overcast	Hot	High	Weak	46
4	Rain	Mild	High	Weak	45
5	Rain	Cool	Normal	Weak	52
6	Rain	Cool	Normal	Strong	23
7	Overcast	Cool	Normal	Strong	43
8	Sunny	Mild	High	Weak	35
9	Sunny	Cool	Normal	Weak	38
10	Rain	Mild	Normal	Weak	46
11	Sunny	Mild	Normal	Strong	48
12	Overcast	Mild	High	Strong	52
13	Overcast	Hot	Normal	Weak	44
14	Rain	Mild	High	Strong	30

And we’ve built the following decision tree.

This duty is handled by buildDecisionTree function. We will pass data set, number of inline tabs (this is important in python. we will increase this every inner call and restore it after calling) and file name to store decision rules.

root = 1
buildDecisionTree(df,root,"rules0.py") #generate rules0.py

Running this decision tree algorithm for the data set generates the following decision rules.

def findDecision(obj):
   if Outlook == 'Rain':
      if Wind == 'Weak':
         return 47.666666666666664
      if Wind == 'Strong':
        return 26.5
   if Outlook == 'Sunny':
      if Temperature == 'Hot':
         return 27.5
      if Temperature == 'Mild':
         return 41.5
      if Temperature == 'Cool':
         return 38
   if Outlook == 'Overcast':
      return 46.25

Building this decision tree was covered in a previous post. That’s why, I skipped how the tree is built. If it is hard to understand, I strongly recommend you to read that post.

Let’s check the Day 1 and Day 2 instances. They both have sunny outlook and hot temperature. Built decision tree says that decision will be 27.5 for sunny outlook and hot temperature. However, day 1 should be 25 and day 2 should be 30. This means that the error (or residual) is 25 – 27.5 = -2.5 for day 1 and 30 – 27.5 = +2.5 for day 2. The following days have similar errors. We will boost these errors.

Loss function

This is not a must but we will use mean squared error as loss function.

loss = (1/2) x (y – y’)²

where y is the actual value and y’ is the prediction.

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Gradient refers to gradient descent in gradient boosting. We will update each prediction as partial derivative of loss function with respect to the prediction. Let’s find this derivative first.

∂ loss / ∂ y’ = ∂((1/2) x (y – f(x))²)/∂y’ = 2. (1/2) . (y – y’) . ∂(-y’)/∂y’ = 2. (1/2) . (y – y’) . (-1) = y’ – y

Now, we can update predictions by applying the following formula. Here, α is learning rate.

y’ = y’ – α . (∂ loss / ∂ y’)

Please focus on the updating term only. I set α to 1 to make formula simpler.

– α . (∂ loss / ∂ y’) = – α . (y’ – y) = α . (y – y’) = y – y’

This is the label that we are going to build a new decision tree.

Epoch 1

Remember that error was -2.5 for day 1 and +2.5 for day 2. Similarly, we’ll find the errors based on the built decision tree’s results and actual labels for the following days.

import rules0 as myrules
for i, instance in df.iterrows():
   params = [] #features for current line stored in params list
   for j in range(0, columns-1):
      params.append(instance[j])

   prediction = myrules.findDecision(params) #apply rules(i-1) for data(i-1)
   actual = instance[columns-1]

   gradient = actual - prediction
   
   instance[columns-1] = gradient
   df.loc[i] = instance
#end of for loop
df.to_csv("data1.py", index=False)

Then, a new data set will be created and residual for each line set to its decision column.

Day	Outlook	Temp.	Humidity	Wind	Decision
1	Sunny	Hot	High	Weak	-2.5
2	Sunny	Hot	High	Strong	2.5
3	Overcast	Hot	High	Weak	-0.25
4	Rain	Mild	High	Weak	-2.66
5	Rain	Cool	Normal	Weak	4.333
6	Rain	Cool	Normal	Strong	-3.5
7	Overcast	Cool	Normal	Strong	-3.25
8	Sunny	Mild	High	Weak	-6.5
9	Sunny	Cool	Normal	Weak	0
10	Rain	Mild	Normal	Weak	-1.66
11	Sunny	Mild	Normal	Strong	6.5
12	Overcast	Mild	High	Strong	5.75
13	Overcast	Hot	Normal	Weak	-2.25
14	Rain	Mild	High	Strong	3.55

Now, it is time to build a new decision tree based on the data set above. The following code block will generate decision rules for the current data frame.

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root = 1
buildDecisionTree(df,root,"rules1.py")

Running regression tree algorithm creates the following decision rules.

def findDecision(Outlook, Temperature, Humidity, Wind):
   if Wind == 'Weak':
      if Temperature == 'Hot':
         return -1.6666666666666667
      if Temperature == 'Mild':
         return -3.6111111111111094
      if Temperature == 'Cool':
         return 2.166666666666668
   if Wind == 'Strong':
      if Temperature == 'Mild':
         return 5.25
      if Temperature == 'Cool':
         return -3.375
      if Temperature == 'Hot':
         return 2.5

Epoch 2

Let’s look for predictions of day 1 and day 2 again. Now, built decision tree says that day 1 has weak wind and hot temperature and it is -1.666 but its actual value was -2.5 in the 2nd data set. This means that error is -2.5 – (-1.666) = -0.833.

Similarly, the tree says that day 2 has strong wind and hot temperature, that’s why, it is predicted as 2.5 whereas its actual value is 2.5, too. In this case, error is equal to 2.5 – 2.5 = 0. In this way, I calculate each instance’s prediction and subtract from its actual value again.

Day	Outlook	Temp.	Humidity	Wind	Golf Players
1	Sunny	Hot	High	Weak	-0.833
2	Sunny	Hot	High	Strong	0.0
3	Overcast	Hot	High	Weak	1.416
4	Rain	Mild	High	Weak	0.944
5	Rain	Cool	Normal	Weak	2.166
6	Rain	Cool	Normal	Strong	-0.125
7	Overcast	Cool	Normal	Strong	0.125
8	Sunny	Mild	High	Weak	-2.888
9	Sunny	Cool	Normal	Weak	-2.166
10	Rain	Mild	Normal	Weak	1.944
11	Sunny	Mild	Normal	Strong	1.25
12	Overcast	Mild	High	Strong	0.5
13	Overcast	Hot	Normal	Weak	-0.583
14	Rain	Mild	High	Strong	-1.75

This time, the following rules will be created for the data set above.

def findDecision(Outlook, Temperature, Humidity, Wind):
   if Outlook == 'Rain':
      if Wind == 'Weak':
         return 1.685185185185186
      if Wind == 'Strong':
         return -0.9375
   if Outlook == 'Sunny':
      if Wind == 'Weak':
         return -1.962962962962964
      if Wind == 'Strong':
         return 0.625
   if Outlook == 'Overcast':
      return 0.3645833333333334

Epoch 5

I skipped epochs from 3 to 5 because same procedures are applied in each step.

Thereafter, I summarize each epoch’s predictions in the table shown below. I’m going to calculate predictions cumulatively and sum values from epoch 1 to epoch 5 in each line to find the final prediction.

Day	Actual	epoch 1	epoch 2	epoch 3	epoch 4	epoch 5	prediction
1	25	27.5	-1.667	-1.963	0.152	5.55E-17	24.023
2	30	27.5	2.5	0.625	0.152	5.55E-17	30.777
3	46	46.25	-1.667	0.365	0.152	5.55E-17	45.1
4	45	47.667	-3.611	1.685	-0.586	-1.88E-01	44.967
5	52	47.667	2.167	1.685	0.213	1.39E-17	51.731
6	23	26.5	-3.375	-0.938	0.213	1.39E-17	22.4
7	43	46.25	-3.375	0.365	0.213	1.39E-17	43.452
8	35	41.5	-3.611	-1.963	-0.586	-7.86E-02	35.261
9	38	38	2.167	-1.963	0.213	1.39E-17	38.416
10	46	47.667	-3.611	1.685	0.442	-1.88E-01	45.995
11	48	41.5	5.25	0.625	0.442	-7.86E-02	47.739
12	52	46.25	5.25	0.365	-0.586	7.21E-01	52
13	44	46.25	-1.667	0.365	0.152	5.55E-17	45.1
14	30	26.5	5.25	-0.938	-0.586	-1.88E-01	30.038

For instance, predictions will be changed over epoch as illustrated below.

1st Epoch = 27.5

2nd Epoch = 27.5 – 1.667 = 25.833

3rd Epoch = 27.5 – 1.667 – 1.963 = 23.87

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4th Epoch = 27.5 – 1.667 – 1.963 + 0.152 = 24.022

Absolute error was |25-27.5| = 2.5 in 1st round for 1st day but we can reduce it to |25-24.023| = 0.97 in 5th round. As seen, each instance’s prediction closes to its actual value when it is boosted.

BTW, learning rate (α) and number of iterations (epoch) should be tuned for different problems.

Errors over iterations

I pivot mean absolute error value for each epoch.

Day	epoch 1	epoch 2	epoch 3	epoch 4	epoch 5
1	2.5	0.833	1.13	0.977	0.977
2	2.5	0	0.625	0.777	0.777
3	0.25	1.417	1.052	0.9	0.9
4	2.667	0.944	0.741	0.155	0.033
5	4.333	2.167	0.481	0.269	0.269
6	3.5	0.125	0.813	0.6	0.6
7	3.25	0.125	0.24	0.452	0.452
8	6.5	2.889	0.926	0.34	0.261
9	0	2.167	0.204	0.416	0.416
10	1.667	1.944	0.259	0.183	0.005
11	6.5	1.25	0.625	0.183	0.261
12	5.75	0.5	0.135	0.721	0
13	2.25	0.583	0.948	1.1	1.1
14	3.5	1.75	0.813	0.227	0.038
MAE	3.011111	1.112963	0.599383	0.48669	0.406115

The result seems interesting when I plot the total error over epochs.

We can definitely say that boosting works well.

Random Forest vs Gradient Boosting

The both random forest and gradient boosting are an approach instead of a core decision tree algorithm itself. They require to run core decision tree algorithms. They also build many decision trees in the background. So, we will discuss how they are similar and how they are different in the following video.

Adaboost vs Gradient Boosting

The both gradient boosting and adaboost are boosting techniques for decision tree based machine learning models. We will discuss how they are similar and how they are different than each other.

Feature Importance

Decision trees are naturally explainable and interpretable algorithms. However, it is hard to explain GBM. Herein, feature importance offers to understand models better.

Conclusion

So, the intuition behind gradient boosting is covered in this post. XGBoost and LightGBM are common variants of gradient boosting. Even though, decision trees are very powerful machine learning algorithms, a single tree is not strong enough for applied machine learning studies. However, experiments show that its sequential form GBM dominates most of applied ML challenges.

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