Huber Loss: Combining MSE and MAE
The Huber loss offers a unique blend of the Mean Squared Error (MSE) and Mean Absolute Error (MAE), providing both the smoothness of MSE and the robustness of MAE. The formula for the Huber loss is defined piecewise, allowing it to switch between a quadratic and a linear penalty depending on the size of the prediction error. The Huber loss for a single prediction is given by:
Lδ(y,y^)={21(y−y^)2δ⋅(∣y−y^∣−21δ)if ∣y−y^∣≤δotherwiseHere, y is the true value, y^ is the predicted value, and δ is a positive threshold parameter.
Huber loss behaves like MSE for small errors, promoting smooth optimization and sensitivity to small deviations. For large errors, it switches to MAE behavior, reducing the influence of outliers and providing robustness. This balance makes Huber loss especially useful in datasets with occasional large errors or outliers.
The transition parameter δ is central to how the Huber loss functions. When the absolute error is less than or equal to δ, the loss is quadratic, just like MSE. This means small errors are penalized more strongly, encouraging precise predictions. When the error exceeds δ, the loss becomes linear, similar to MAE, which prevents large errors from having an outsized impact on the optimization process. By tuning δ, you can control the trade-off between sensitivity to small errors and robustness to outliers. A smaller δ makes the loss function more like MAE, while a larger δ makes it more like MSE.
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The Huber loss offers a unique blend of the Mean Squared Error (MSE) and Mean Absolute Error (MAE), providing both the smoothness of MSE and the robustness of MAE. The formula for the Huber loss is defined piecewise, allowing it to switch between a quadratic and a linear penalty depending on the size of the prediction error. The Huber loss for a single prediction is given by:
Lδ(y,y^)={21(y−y^)2δ⋅(∣y−y^∣−21δ)if ∣y−y^∣≤δotherwiseHere, y is the true value, y^ is the predicted value, and δ is a positive threshold parameter.
Huber loss behaves like MSE for small errors, promoting smooth optimization and sensitivity to small deviations. For large errors, it switches to MAE behavior, reducing the influence of outliers and providing robustness. This balance makes Huber loss especially useful in datasets with occasional large errors or outliers.
The transition parameter δ is central to how the Huber loss functions. When the absolute error is less than or equal to δ, the loss is quadratic, just like MSE. This means small errors are penalized more strongly, encouraging precise predictions. When the error exceeds δ, the loss becomes linear, similar to MAE, which prevents large errors from having an outsized impact on the optimization process. By tuning δ, you can control the trade-off between sensitivity to small errors and robustness to outliers. A smaller δ makes the loss function more like MAE, while a larger δ makes it more like MSE.
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