Best Split Finder

mediumFREEDecision TreesMachine Learning 1

Find the best threshold for splitting a feature to maximize information gain.

Given a list of feature values and corresponding labels, try every midpoint between consecutive sorted unique feature values as a potential split threshold. For each threshold, split the data into left ( $\leq$ threshold) and right ( $>$ threshold) subsets.

Information gain is the parent entropy minus the size-weighted entropies of the two subsets, with entropy measured in bits (log base 2).

Return the threshold that gives the highest information gain, along with the gain value, as a tuple (threshold, gain). Both rounded to 4 decimal places. If several thresholds tie for the highest gain, return the smallest one.

If there's no possible split (all values identical), or no threshold gives a gain greater than 0, return (None, 0.0).

Example

Input

feature_values = [1, 2, 3, 4]
labels = [0, 0, 1, 1]

Output

(2.5, 1.0)

Reasoning

feature_values holds one number per sample ( $1, 2, 3, 4$ ) and labels holds its class ( $0, 0, 1, 1$ ). The candidate thresholds are the points halfway between neighbouring distinct values: $1.5$ , $2.5$ and $3.5$ .
The parent set has two of each class, so its entropy is $H = -\tfrac{1}{2}\log_2\tfrac{1}{2} - \tfrac{1}{2}\log_2\tfrac{1}{2} = 1$ bit.
At $1.5$ : the left side is $\{0\}$ ( $H = 0$ ) and the right side is $\{0, 1, 1\}$ with $H = -\tfrac{1}{3}\log_2\tfrac{1}{3} - \tfrac{2}{3}\log_2\tfrac{2}{3} \approx 0.9183$ . The gain is $1 - \tfrac{3}{4}(0.9183) \approx 0.3113$ . By symmetry $3.5$ also gives $0.3113$ .
At $2.5$ : the left side is $\{0, 0\}$ and the right side is $\{1, 1\}$ . Both are pure, so the gain is $1 - \tfrac{2}{4}\cdot 0 - \tfrac{2}{4}\cdot 0 = 1.0$ , the largest possible here.
Notice that a gain of $1.0$ equals the parent entropy: the split removes all uncertainty, giving (2.5, 1.0).

Constraints

feature_values: list of numeric values
labels: list of class labels (same length)
Split: left = values <= threshold, right = values > threshold
Return (best_threshold, best_gain) rounded to 4 decimal places

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Concept primer

Background Knowledge

The problem of finding the best threshold for splitting a feature is a fundamental concept in Decision Trees, a type of supervised learning algorithm. In Decision Trees, the goal is to recursively partition the data into smaller subsets based on the features, with the aim of maximizing the information gain at each split. The information gain is a measure of how much a split reduces the impurity or uncertainty in the data. In this context, impurity refers to the amount of mixing of different classes in a node.

The information gain is typically calculated using entropy, a measure of the amount of uncertainty or randomness in a probability distribution. The entropy of a discrete random variable is calculated as $-\sum_{i=1}^{n} p_i \log_2 p_i$ , where $p_i$ is the probability of each class. The information gain is then calculated as the difference in entropy between the parent node and the weighted average of the entropy of the child nodes. The goal is to find the split that results in the highest information gain, which corresponds to the largest reduction in impurity.

To solve this problem, we need to understand how to calculate the entropy of a node, how to split the data into left and right subsets based on a given threshold, and how to calculate the information gain for each possible split. We also need to be able to handle the case where all feature values are identical, in which case there is no possible split.

Algorithm/Approach

The general approach to solving this problem involves iterating over all possible midpoints between consecutive unique feature values, splitting the data into left and right subsets based on each midpoint, and calculating the information gain for each split. The algorithm pattern is a simple iterative approach, where we try each possible split and keep track of the split that results in the highest information gain.

Step-by-Step Strategy

To implement the solution, we can follow these steps:

Sort the unique feature values and calculate the midpoints between consecutive values.
For each midpoint, split the data into left and right subsets based on the midpoint.
Calculate the entropy of the parent node and the entropy of the left and right child nodes.
Calculate the information gain for each split by subtracting the weighted average of the entropy of the child nodes from the entropy of the parent node.
Keep track of the split that results in the highest information gain and return the corresponding threshold and gain value.

Common Pitfalls

Some common pitfalls to watch out for when implementing this solution include:

Failing to handle the case where all feature values are identical.
Incorrectly calculating the entropy of the parent or child nodes.
Failing to weight the entropy of the child nodes by the proportion of samples in each node.
Not rounding the threshold and gain value to 4 decimal places as required.

Time & Space Complexity

The time complexity of this solution is O(n log n) due to the sorting of the feature values, where n is the number of unique feature values. The space complexity is O(n) for storing the sorted feature values and the midpoints. Note that the time complexity may be higher if the calculation of the entropy and information gain involves additional iterations over the data.

solution.py

Python 3.13

Test results