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Binary Cross-Entropy Loss

EasyClassification

Compute the binary cross-entropy loss (log loss) for a set of predictions.

Given true labels yi∈{0,1}y_i \in \{0, 1\} and predicted probabilities y^i∈(0,1)\hat{y}_i \in (0, 1):

BCE=−1n∑i=1n[yilog⁡(y^i)+(1−yi)log⁡(1−y^i)]BCE = -\frac{1}{n}\sum_{i=1}^{n}[y_i \log(\hat{y}_i) + (1 - y_i)\log(1 - \hat{y}_i)]

To avoid log⁡(0)\log(0), clip predictions to the range [ϵ,1−ϵ][\epsilon, 1-\epsilon] where ϵ=10−7\epsilon = 10^{-7}.

Return the loss rounded to 4 decimal places.

Example:

Input:
y_true = [1, 0, 1]
y_pred = [0.9, 0.1, 0.8]
Output:
0.1446
Reasoning:
  • First, we clip the predicted probabilities to the range [ϵ,1−ϵ][\epsilon, 1-\epsilon] where ϵ=10−7\epsilon = 10^{-7}: ypred=[0.9,0.1,0.8]y_{pred} = [0.9, 0.1, 0.8] remains the same since all values are within the range.
  • Then, we calculate the binary cross-entropy loss for each sample:
    • For yi=1y_i = 1 and y^i=0.9\hat{y}_i = 0.9, the loss is −log⁡(0.9)-\log(0.9)
    • For yi=0y_i = 0 and y^i=0.1\hat{y}_i = 0.1, the loss is −log⁡(1−0.1)=−log⁡(0.9)-\log(1-0.1) = -\log(0.9)
    • For yi=1y_i = 1 and y^i=0.8\hat{y}_i = 0.8, the loss is −log⁡(0.8)-\log(0.8)
  • Next, we calculate the total loss by summing the individual losses and dividing by the number of samples n=3n=3: BCE=−13[log⁡(0.9)+log⁡(0.9)+log⁡(0.8)]BCE = -\frac{1}{3}[\log(0.9) + \log(0.9) + \log(0.8)]
  • The final output is the loss rounded to 4 decimal places: BCE≈0.1446BCE \approx 0.1446

Constraints:

  • y_true: list of 0s and 1s
  • y_pred: list of predicted probabilities (0 to 1)
  • Clip predictions to [1e-7, 1-1e-7] before computing log
  • Return a single float rounded to 4 decimal places
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