Step 2: Scaling the Dot Product in Self-Attention

This document details the second crucial step in the self-attention mechanism: scaling the dot product of the query and key matrices.

Why Scale the Dot Product?

Following the computation of the dot product between the query (Q) and key (K) matrices (Q · Kᵀ) in Step 1 of self-attention, the next essential operation is scaling this result. This is achieved by dividing the dot product by the square root of the dimension of the key vectors.

The primary reason for this scaling is to prevent unstable gradients during the training process, particularly when working with high-dimensional vectors. Large dot product values can lead to extremely small gradients after the softmax function is applied, hindering the model's ability to learn effectively.

Mathematical Explanation of the Scaling Step

Let $d_k$ represent the dimension of the key vectors. In most Transformer implementations, $d_k$ is a hyperparameter and a common value is 64.

To stabilize the output of the dot product, we divide each element in the resulting matrix by $\sqrt{d_k}$.

Formula:

Scaled Attention Scores = (Q · Kᵀ) / √d_k

This operation ensures that the values passed into the softmax function (which is the next step) remain within a reasonable range. Without this scaling, large values in the dot product could result in the softmax output being very close to 0 or 1, leading to vanishing gradients and impeding the learning process.

Example:

If $d_k = 64$, then $\sqrt{d_k} = \sqrt{64} = 8$. In this case, we would divide the entire $Q \cdot K^T$ matrix by 8.

Visual Understanding

The scaled result of $Q \cdot K^T / \sqrt{d_k}$ produces a matrix of normalized similarity scores. These scores represent the degree of relevance between each query vector and each key vector. These normalized scores are then fed into the softmax function, which transforms them into attention weights.

Conclusion

Scaling the dot product by the square root of the key dimension is a fundamental technique in the self-attention mechanism. It is vital for:

• Improving Training Stability: By preventing overly large dot product values, it ensures that gradients remain well-behaved.
• Ensuring Effective Learning: It mitigates the issue of vanishing gradients, allowing the model to learn meaningful attention distributions.
• Numerical Stability: Contributes to the overall numerical stability of the model, especially with large and deep networks.

By carefully managing the scale of the intermediate scores, self-attention can effectively learn complex dependencies within sequential data.

Further Reading and Related Concepts

• Self-Attention Mechanism: The broader concept of how attention is computed.
• Softmax Function: The role of softmax in converting scores into probability distributions.
• Gradient Descent: The optimization process where scaling helps maintain stability.
• Transformer Architecture: The foundational neural network architecture that popularized self-attention.

Interview Questions on Scaling in Self-Attention:

• What is the purpose of scaling the dot product in self-attention?
• How is the scaling factor calculated in the Transformer architecture?
• What is the typical value of the key dimension ($d_k$) in Transformers?
• Why does scaling prevent unstable gradients in training?
• What would happen if we skipped the scaling step in self-attention?
• How does scaling affect the softmax output in self-attention?
• Why is $\sqrt{d_k}$ used instead of another normalization constant?
• How does scaling help maintain numerical stability in large models?
• Explain how scaling interacts with the softmax step that follows.
• In the formula $(Q \cdot K^T) / \sqrt{d_k}$, what does each component represent?