What is the Difference Between Entropy and GDU?

Entropy and GDU (Gradient-Derived Uncertainty) are both concepts related to uncertainty, but they are used in different contexts:

Entropy:

• Definition: Entropy is a measure of uncertainty or disorder in a system. In information theory, it quantifies the unpredictability of a probability distribution.
• Formula (Shannon Entropy):

H(X)=−∑ p(x) log⁡p(x) 

where p(x) is the probability of outcome x.

• Usage:
  • In machine learning, entropy is often used in decision trees to determine the best split.
  • In statistics, it measures randomness in a dataset.
  • In physics, it describes the disorder of a system.

Gradient-Derived Uncertainty (GDU):

• Definition: GDU is a method of measuring uncertainty in deep learning models, especially in the context of neural networks and Bayesian deep learning.
• How It Works:
  • GDU quantifies uncertainty by analyzing gradients of the loss function with respect to model parameters.
  • It provides insights into model confidence, helping to identify when a model is unsure about its predictions.
• Usage:
  • Used in uncertainty quantification for deep learning models.
  • Helps in active learning, where a model selects uncertain samples for labeling.
  • Useful in Bayesian deep learning to estimate epistemic uncertainty.

Key Differences:

1. Entropy in Decision Trees

Example: Classifying Emails as Spam or Not Spam

Imagine we have a dataset where we classify emails as Spam or Not Spam based on certain words. Suppose our dataset is split as follows:

Step 1: Calculate Entropy

Entropy measures how “mixed” the classes are. If a set contains only spam or only non-spam emails, entropy is 0 (perfectly pure). If the set is equally mixed, entropy is 1 (maximum disorder).

Entropy formula:

H(X)=−∑ p(x) log⁡2p(x)

For the word “Discount”:

H=−((30/40)*log⁡2(30/40)+(10/40)*log⁡2(10/40))

H≈−(0.75×−0.415+0.25×−2)

H≈0.811

For the word “Meeting”:

H≈−(0.083×−3.585+0.917×−0.127)

$$H\approx 0.503$$

Step 2: Choosing the Best Split

Since “Discount” has higher entropy, it is a worse predictor of spam than “Meeting,” which has lower entropy. Decision trees use entropy (or information gain, which is entropy reduction) to decide which word gives the best split.

• Higher entropy → The data is more mixed (uncertain), making it a less useful predictor.
• Lower entropy → The data is more pure (less uncertain), making it a better predictor.

• “Discount” (H = 0.811): This means emails containing “Discount” are more mixed (30 spam, 10 not spam). A model using “Discount” won’t be as confident in classifying emails.
• “Meeting” (H = 0.503): Emails with “Meeting” are more consistently “Not Spam” (5 spam, 55 not spam). A model using “Meeting” would make clearer distinctions.

2. Gradient-Derived Uncertainty (GDU) in Deep Learning

Example: Image Classification with a Neural Network

Imagine a neural network classifying images as Cat, Dog, or Horse. When given an unclear image, the model outputs:

Step 1: Compute Entropy (Softmax Output)

Using entropy, we calculate:

H(X)=−∑ p(x) log⁡2p(x)

H=−(0.4*log⁡2(0.4)+0.35*log⁡2(0.35)+0.25*log⁡2(0.25))

H≈1.57

This high entropy suggests that the model is uncertain, but it doesn’t tell why the model is uncertain.

Step 2: Compute Gradient-Derived Uncertainty (GDU)

• GDU looks at how sensitive the loss function is to changes in weights.
• If small changes in weights cause big changes in loss, the model is highly uncertain.
• If the gradients are small, the model is more confident.

Mathematically, GDU is often computed as:

U(x)=∣∣∇θL(x)∣∣2​

where:

• ∇θL(x) is the gradient of the loss with respect to model parameters θ.
• ∣∣⋅∣∣ is the L2 norm (magnitude of the gradient vector).

If the model has high GDU, it means it struggles with this image and should either:

1. Be trained on similar images to improve confidence.
2. Be flagged as uncertain, allowing for human review.

Key Takeaways