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Common Loss Functions

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Mean Squared Error (MSE)

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MSE=1N∑i=1N(yi−y^i)2MSE = \frac{1}{N} \sum_{i=1}^N (y_i - \hat{y}_i)^2

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Cross-Entropy Loss

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H(y,p)=−∑iyilog⁡(pi)H(y,p) = -\sum_{i} y_i \log(p_i)

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Hinge Loss

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L(y)=max⁡(0,1−yi⋅f(xi))L(y) = \max(0, 1 - y_i \cdot f(x_i))

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Huber Loss

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Lδ(y,y^)={12(y−y^)2for ∣y−y^∣≤δ,δ∣y−y^∣−12δ2otherwiseL_{\delta} (y, \hat{y}) = \begin{cases} \frac{1}{2}(y - \hat{y})^2 & \text{for } |y - \hat{y}| \le \delta, \\ \delta |y - \hat{y}| - \frac{1}{2}\delta^2 & \text{otherwise} \end{cases}

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Binary Cross-Entropy Loss

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BCE=−1N∑i=1Nyilog⁡(pi)+(1−yi)log⁡(1−pi)BCE = -\frac{1}{N} \sum_{i=1}^N y_i \log(p_i) + (1 - y_i) \log(1 - p_i)

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Categorical Cross-Entropy Loss

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CCE=−∑i=1N∑c=1Myiclog⁡(pic)CCE = -\sum_{i=1}^N \sum_{c=1}^M y_{ic} \log(p_{ic})

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Mean Absolute Error (MAE)

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MAE=1N∑i=1N∣yi−y^i∣MAE = \frac{1}{N} \sum_{i=1}^N |y_i - \hat{y}_i|

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Logistic Loss

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logistic=log⁡(1+e−yif(xi))logistic = \log(1 + e^{-y_i f(x_i)})

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Negative Log-Likelihood Loss

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NLL=−log⁡(p^yi)NLL = -\log(\hat{p}_{y_i})

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Kullback-Leibler Divergence Loss

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DKL(P∥Q)=∑p(x)log⁡(p(x)q(x))D_{KL}(P \parallel Q) = \sum p(x)\log(\frac{p(x)}{q(x)})

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