CART Gini cost function

For CART, as you know, the general goal is to find the split point for a particular feature that minimizes the impurity of the resulting subsets. To minimize it, you should consider all possible split points for a given feature and calculate the impurity of the resulting left and right subsets (G_left and G_right) after splitting the data at the split point. You could then use the formula you provided to calculate the impurity cost for each feature and each split point. This calculates a weighted sum of impurities, where the weights are determined by the number of instances in the left and right subsets (m_left and m_right), normalized by the total number of instances (m). Then, you can iterate through all features and all possible split points for each feature, calculating J(k, t_k) for each combination. Now, you select the optimal split (the feature k and split point t_k) that result in the lowest J(k, t_k) value for that node in the decision tree.

You continue this process for other nodes in the tree until a stopping criterion (e.g. maximum depth, minimum samples per leaf, etc.) is met and you call it good.

This is essentially a brute-force approach to finding the best split for each node in the decision tree by considering all possible split points for all features and selects the one that minimizes the impurity cost function. In practice, people use optimization techniques and heuristics to make this process more efficient.