The SoftMax Derivative, Step-by-Step!!!

Summary

In this StatQuest episode, Josh Starmer walks viewers through the process of computing the derivative of the softmax function, step by step. He explains the concept of softmax as a way to predict probabilities for different categories (like species of iris flowers) and demonstrates how to calculate its derivatives with respect to different outputs, using the quotient rule for differentiation. The video includes a detailed breakdown of how the predicted probabilities for Setosa, Versicolor, and Virginica change based on raw output values, with numerical examples to solidify the understanding. Josh also promotes his StatQuest study guides and Patreon support at the end.

Takeaways

• 😀 The video explains how to calculate the derivative of the softmax function step by step.
• 😀 The softmax function is used to transform raw output values from a neural network into predicted probabilities.
• 😀 The quotient rule is used to find the derivative of the softmax function, specifically for setosa in the example.
• 😀 The derivative of the softmax function with respect to the raw output value of setosa involves the predicted probability of setosa and other probabilities.
• 😀 The derivative of the predicted probability for setosa with respect to its own raw output value is the predicted probability for setosa times (1 - predicted probability for setosa).
• 😀 The video provides an example where the predicted probability for setosa is 0.69, and the derivative of the softmax function gives a value of 0.21.
• 😀 Derivatives with respect to other classes like versicolor and virginica are also explored, showing how they influence the softmax output for setosa.
• 😀 The derivative of the predicted probability for setosa with respect to versicolor is calculated as the negative predicted probability for setosa times the predicted probability for versicolor.
• 😀 A similar approach is used to calculate the derivative with respect to virginica, leading to the value -0.15.
• 😀 The video promotes further study resources, including StatQuest study guides and contributions via Patreon, channel membership, or purchasing merchandise.

Q & A

What is the primary focus of this StatQuest video?

The primary focus of the video is explaining the derivative of the softmax function, particularly the derivative with respect to the raw output values for different classes in a neural network.

What is softmax, and why is it used in this context?

Softmax is a mathematical function used in machine learning and neural networks to convert raw output values (logits) into predicted probabilities. It ensures that the predicted probabilities for each class sum to 1, which is necessary for classification tasks.

Why is the quotient rule used in calculating the derivative of the softmax function?

The quotient rule is used because the softmax function involves a ratio of exponentials (numerator/denominator), and the derivative of such a function requires applying the quotient rule to find the rate of change of the predicted probabilities with respect to the raw output values.

What does the derivative of the softmax with respect to the raw output value for setosa tell us?

The derivative of the softmax with respect to the raw output value for setosa gives us the rate of change of the predicted probability for setosa as the raw output value for setosa changes. In this case, it is the predicted probability for setosa multiplied by (1 - the predicted probability for setosa).

How are the predicted probabilities for setosa, versicolor, and virginica related in the softmax function?

The predicted probabilities for setosa, versicolor, and virginica are interdependent, meaning the raw output values for each of the classes influence the softmax output for all classes. The derivatives with respect to each other show how changing one output affects the others.

What happens when calculating the derivative of the softmax with respect to versicolor?

When calculating the derivative of the softmax with respect to versicolor, the first term in the quotient rule becomes zero because versicolor does not appear in the numerator for setosa's probability. The result is a product of the negative predicted probability for setosa and the predicted probability for versicolor.

What is the value of the derivative of the predicted probability for setosa with respect to versicolor, and how is it calculated?

The derivative of the predicted probability for setosa with respect to versicolor is -0.07, calculated by multiplying the predicted probability for setosa (0.69) by the predicted probability for versicolor (0.1).

Why is the term 'virginica' left as homework in this video?

The term 'virginica' is left as homework because the process for calculating its derivative with respect to setosa follows the same steps as the calculation for versicolor, and the video only walks through versicolor to demonstrate the method.

What are some ways viewers can support StatQuest as mentioned in the video?

Viewers can support StatQuest by subscribing to the channel, contributing to the Patreon campaign, becoming a channel member, buying original songs or merchandise (such as t-shirts or hoodies), or making a donation.

What are the main mathematical concepts highlighted in this video?

The main mathematical concepts discussed in the video are the softmax function, its derivatives with respect to the raw output values for different classes, the quotient rule for derivatives, and algebraic manipulation to simplify the expressions.