Binary Classification Loss

In machine learning, evaluating the performance of binary classification models requires a robust loss function that measures the discrepancy between predicted probabilities and actual class labels. The Binary Classification Loss (also known as Log Loss or Logarithmic Loss) is the standard metric used for this purpose, providing a smooth, differentiable objective that heavily penalizes confident but incorrect predictions.

Given a list of true binary labels y_true (where each element is either 0 or 1) and a corresponding list of predicted probabilities y_pred (where each element is a value between 0 and 1 representing the model's confidence that the sample belongs to class 1), compute the mean binary classification loss across all samples.

Mathematical Formulation:

For a single sample with true label y and predicted probability p, the binary classification loss is computed as:

$$L(y, p) = -[y \cdot \log(p) + (1 - y) \cdot \log(1 - p)]$$

For a dataset with N samples, the mean loss is:

$$\text{Mean Loss} = \frac{1}{N} \sum_{i=1}^{N} L(y_i, p_i)$$

Numerical Stability:

Taking the logarithm of values exactly equal to 0 or 1 results in undefined or infinite values. To ensure numerical stability, clip the predicted probabilities to the range [epsilon, 1 - epsilon] before computing the logarithm. The default value for epsilon is 1e-15.

Your Task:

Write a Python function that:

• Accepts a list of true binary labels, a list of predicted probabilities, and an optional epsilon parameter
• Computes the mean binary classification loss across all samples
• Returns the result as a float rounded to 4 decimal places

Behavior:

• Both input lists will always have the same length
• True labels will contain only 0s and 1s
• Predicted probabilities will be values between 0 and 1 (inclusive)