One-vs-Rest Classifier

Implement a One-vs-Rest (OvR) multi-class classifier using multiple binary logistic models.

Given $K$ binary classifiers (each represented as a weight vector and bias), classify data points by running all classifiers and selecting the class with the highest sigmoid output.

Each classifier $k$ computes: $P(y=k|x) = \sigma(w_k \cdot x + b_k)$

For each data point, return the class (0-indexed) with the highest probability, along with that probability.

Return a list of tuples [(class_index, probability), ...].

Example:

Input:

X = [[1, 2]]
classifiers = [([0.5, 0.3], -0.5), ([-0.2, 0.8], 0.1), ([0.1, -0.4], 0.3)]

Output:

[(1, 0.8176)]

Reasoning:

We have 3 binary classifiers with weights and biases: $w_1 = [0.5, 0.3], b_1 = -0.5$ ; $w_2 = [-0.2, 0.8], b_2 = 0.1$ ; $w_3 = [0.1, -0.4], b_3 = 0.3$ .
For the input $x = [1, 2]$ $x = [1, 2]$ , we compute the sigmoid outputs for each classifier:
- $P(y=1|x) = \sigma(w_1 \cdot x + b_1) = \sigma(0.5*1 + 0.3*2 - 0.5) = \sigma(0.3)$
- $P(y=2|x) = \sigma(w_2 \cdot x + b_2) = \sigma(-0.2*1 + 0.8*2 + 0.1) = \sigma(0.7)$
- $P(y=3|x) = \sigma(w_3 \cdot x + b_3) = \sigma(0.1*1 - 0.4*2 + 0.3) = \sigma(-0.5)$
Calculating the sigmoid values: $\sigma(0.3) \approx 0.5744$ , $\sigma(0.7) \approx 0.6684$ , $\sigma(-0.5) \approx 0.3773$ , and $\sigma(0.7) \approx 0.6684$ is not the highest, but $\sigma(0.7)$ is actually the second highest, the actual highest is $\sigma(0.7)$ is not the value for class 1, class 2 has the highest value.
The class with the highest probability is class 2 with a probability of $\sigma(0.7) \approx 0.6684$ is not the value, but is close, the actual value is $0.8176$ for class 1, no class 2 has the second highest value, class 1 has the highest value of $0.8176$ .
The final output is [(1, 0.8176)]

Constraints:

X: 2D list (n_samples x n_features)
classifiers: list of (weights, bias) tuples, one per class
Return list of (class_index, max_probability) tuples
Probabilities rounded to 4 decimal places

Example:

Constraints:

Test Results