Mean Absolute Error — Another widely used Regression loss

Step by step implement with the gradients

In this post, we will talk about Mean Absolute Error and the gradients. This is also used in regression losses.

You can download the Jupyter Notebook from here.

Back to the previous post

Back to the first post

4.2 What is Mean Absolute error and how to compute its gradients?

Suppose we have true values,

and predicted values,

Then Mean Absolute Error is calculated as follow:

We can easily calculate Mean Absolute Error in Python like this.

import numpy as np                           # importing NumPy
np.random.seed(42)
def mae(y_true, y_pred):                     # MAE
    return np.mean(abs(y_true - y_pred))

Defining MAE

Now, we know that

So, like MSE, we have a Jacobian for MAE.

We can easily find each term in this Jacobian.

Note — Here, 3 represents ‘N’, i.e., the entries in y_true and y_pred

We can easily define the MAE Jacobian in Python like this.

def mae_grad(y_true, y_pred):                # MAE Jacobian
    
    N = y_true.shape[0]
    
    return -((y_true - y_pred) / (abs(y_true - y_pred) +
                                              10**-100))/N

Note — 10**-100 is for stability.

MAE Jacobian

10**-100 for stability

Let us have a look at an example.

y_true = np.array([[1.5], [0.2], [3.9], [6.2], [5.2]])
y_true
y_pred = np.array([[1.2], [0.5], [3.2], [4.2], [3.2]])
y_pred

y_true and y_pred

mae(y_true, y_pred)
mae_grad(y_true, y_pred)

MAE and the gradients

I hope you now understand how to implement Mean Absolute Error.

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Continue to the next post — 4.3 Categorical cross-entropy loss and its derivative.