Added Binary Focal Cross Entropy (#10674)

* Added Binary Focal Cross Entropy

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* Fixed Issue

* Fixed Issue

* Added BFCE loss to loss_functions.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update machine_learning/loss_functions.py

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

machine_learning/loss_functions.py

@@ -39,6 +39,57 @@ def binary_cross_entropy(
     return np.mean(bce_loss)
 
 
+def binary_focal_cross_entropy(
+    y_true: np.ndarray,
+    y_pred: np.ndarray,
+    gamma: float = 2.0,
+    alpha: float = 0.25,
+    epsilon: float = 1e-15,
+) -> float:
+    """
+    Calculate the mean binary focal cross-entropy (BFCE) loss between true labels
+    and predicted probabilities.
+
+    BFCE loss quantifies dissimilarity between true labels (0 or 1) and predicted
+    probabilities. It's a variation of binary cross-entropy that addresses class
+    imbalance by focusing on hard examples.
+
+    BCFE = -Σ(alpha * (1 - y_pred)**gamma * y_true * log(y_pred)
+                + (1 - alpha) * y_pred**gamma * (1 - y_true) * log(1 - y_pred))
+
+    Reference: [Lin et al., 2018](https://arxiv.org/pdf/1708.02002.pdf)
+
+    Parameters:
+    - y_true: True binary labels (0 or 1).
+    - y_pred: Predicted probabilities for class 1.
+    - gamma: Focusing parameter for modulating the loss (default: 2.0).
+    - alpha: Weighting factor for class 1 (default: 0.25).
+    - epsilon: Small constant to avoid numerical instability.
+
+    >>> true_labels = np.array([0, 1, 1, 0, 1])
+    >>> predicted_probs = np.array([0.2, 0.7, 0.9, 0.3, 0.8])
+    >>> binary_focal_cross_entropy(true_labels, predicted_probs)
+    0.008257977659239775
+    >>> true_labels = np.array([0, 1, 1, 0, 1])
+    >>> predicted_probs = np.array([0.3, 0.8, 0.9, 0.2])
+    >>> binary_focal_cross_entropy(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.")
+    # Clip predicted probabilities to avoid log(0)
+    y_pred = np.clip(y_pred, epsilon, 1 - epsilon)
+
+    bcfe_loss = -(
+        alpha * (1 - y_pred) ** gamma * y_true * np.log(y_pred)
+        + (1 - alpha) * y_pred**gamma * (1 - y_true) * np.log(1 - y_pred)
+    )
+
+    return np.mean(bcfe_loss)
+
+
 def categorical_cross_entropy(
     y_true: np.ndarray, y_pred: np.ndarray, epsilon: float = 1e-15
 ) -> float: