Absolute error

As explained in Ensemble Methods for Machine Learning

Absolute error measures prediction error as the absolute difference between a target and its prediction.

Absolute error (AE) measures a regression residual as |y − f(x)|, the absolute difference between the target and prediction. It corresponds to assuming a Laplacian distribution over residuals. section 7.3.1

Use absolute error to understand these practical tradeoffs:

• Penalizes large errors less aggressively than squared error, making it less sensitive to outliers. section 7.3.1
• Lacks second-order differentiability, so it cannot be used directly with Newton boosting. section 7.3.1
• Contrasts with Huber loss, which uses squared error for small residuals and scaled absolute error beyond a threshold. section 7.3.1

Figure 7.10 Loss and log-likelihood functions for three different types of targets: continuous-valued (left), positive continuous-valued (center), and count-valued (right)

Compare the absolute-error curve with squared error and other loss functions across residual sizes. section 7.3.1

How Ensemble Methods for Machine Learning covers it

How deep it goes

Defines the formula, likelihood interpretation, outlier behavior, and differentiability limit for Newton boosting.

Examples

Visualizes absolute error alongside squared error and other loss functions in Figure 7.10.

What you can do after

Evaluate regression loss choices by comparing outlier sensitivity, likelihood assumptions, and differentiability.

How it connects

Connects absolute error to squared error, Huber loss, pseudo-Huber loss, Laplacian distributions, and boosting.

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Also explained in

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  liveBookmanning.com
  Grokking Machine LearningA regression error metric equal to the sum of the absolute differences between observed and predicted target values.

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Related concepts

• squared losscontrasts with · Squared loss is the square of the difference between a predicted or fitted value and its target or observation, often summed across data points.
• objective functionkind of · An objective function assigns values to candidate solutions, predictions, or parameter settings according to a criterion to be minimized or maximized.
• mean absolute errorproduces · Mean absolute error is the arithmetic mean of the absolute differences between corresponding predicted and reference numerical values.
• distancehas part · Distance is a measure of separation between points or entities in a defined space or context.
• forecast errorproduced by · A forecast error is the difference between an observed value and the corresponding forecasted value.
• cross entropy lossalternative to · Cross-entropy loss measures the expected negative log probability that a predicted distribution assigns to outcomes drawn from a target distribution.

This explainer was generated by Manning's knowledge graph from the text of Ensemble Methods for Machine Learning. The section links point to the parts of the book it draws on.