Add KL divergence loss algorithm (TheAlgorithms#11238)

* Add KL divergence loss algorithm

* Apply suggestions from code review

---------

Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

Author: ZJUGuoShuai

machine_learning/loss_functions.py

@@ -629,6 +629,40 @@ def smooth_l1_loss(y_true: np.ndarray, y_pred: np.ndarray, beta: float = 1.0) ->
     return np.mean(loss)
 
 
+def kullback_leibler_divergence(y_true: np.ndarray, y_pred: np.ndarray) -> float:
+    """
+    Calculate the Kullback-Leibler divergence (KL divergence) loss between true labels
+    and predicted probabilities.
+
+    KL divergence loss quantifies dissimilarity between true labels and predicted
+    probabilities. It's often used in training generative models.
+
+    KL = Σ(y_true * ln(y_true / y_pred))
+
+    Reference: https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence
+
+    Parameters:
+    - y_true: True class probabilities
+    - y_pred: Predicted class probabilities
+
+    >>> true_labels = np.array([0.2, 0.3, 0.5])
+    >>> predicted_probs = np.array([0.3, 0.3, 0.4])
+    >>> kullback_leibler_divergence(true_labels, predicted_probs)
+    0.030478754035472025
+    >>> true_labels = np.array([0.2, 0.3, 0.5])
+    >>> predicted_probs = np.array([0.3, 0.3, 0.4, 0.5])
+    >>> kullback_leibler_divergence(true_labels, predicted_probs)
+    Traceback (most recent call last):
+        ...
+    ValueError: Input arrays must have the same length.
+    """
+    if len(y_true) != len(y_pred):
+        raise ValueError("Input arrays must have the same length.")
+
+    kl_loss = y_true * np.log(y_true / y_pred)
+    return np.sum(kl_loss)
+
+
 if __name__ == "__main__":
     import doctest