(I actually asked the following question on Stack Overflow and Cross Validated Exchange for more than a month:
but, still, there has been no response so far.)

This question refers to the following step in the classical procedure of Adaboost classification.
For each boosting round $b$, we define $$ c_b = \text{argmin}_{c \in \mathcal{S}} \Bigg[ \frac{\sum_{i=1}^n w^{(b)}_i \mathbf{1}_{\big\{ c(x_i) \neq y_i \big\}}}{\sum_{i=1}^n w^{(b)}_i }\Bigg],$$ where $W:= \big[\{ w^{(b)}_i \}_{i=1}^n \big]$ is the array of weights corresponding to each boosting round $b$ and $\mathcal{S}$ is the set of stumps (decision trees of depth 1 corresponding to one feature). Suppose that we assign an array $W$ and generate training points $\mathbf{x}$ and $\mathbf{y}$ (with $\mathbf{y}$ only taking values -1, 1) as follows:

W = [0.05, 0.032732683535398856, 0.05, 0.05, 0.032732683535398856, 
0.05, 0.05, 0.05, 0.032732683535398856, 0.05, 
0.05, 0.05, 0.05, 0.05, 0.05, 
0.05, 0.05, 0.032732683535398856, 0.032732683535398856, 0.032732683535398856]

from sklearn.datasets import make_blobs
x,y = make_blobs(n_samples = 20, n_features = 5, centers = 2, cluster_std = 20.0, random_state = 100) 
y[y==0] = -1

Then my textbook uses the following code A to generate $c_b$.

from sklearn.tree import DecisionTreeClassifier
clf = DecisionTreeClassifier(max_depth=1)
clf.fit(x, y, sample_weight = W)  # Here clf is the weak classifier c_b. 
training_pred = clf.predict(x)

However, the following code B based on the definition of $c_b$ gives a different result:

 import numpy as np
 from sklearn.tree import DecisionTreeClassifier

 error_rate = 100000

 for k in range(5):

        clf = DecisionTreeClassifier(max_depth=1)
        clf.fit(x[:,[k]], y)

        local_training_pred = clf.predict(x[:,[k]])

        local_error_rate = 0

        for i in range(len(x)):
            if (local_training_pred[i] != y[i]):
                local_error_rate += (W[i])/np.sum(W)

        if local_error_rate < error_rate:
            error_rate = local_error_rate
            training_pred = local_training_pred


Here the code compares the error rate of each stump; selects the one with the lowest error rate and then computes the prediction of the training set $\mathbf{x}$ under that stump.

Nonetheless, Codes A and B do not return the same result for our choice of $W$. Does anyone know the reason behind this? Have I actually mistaken the definition of stumps?



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