Scaled Dot-Product Attention

Scaled dot-product attention is an attention mechanism where the dot products are scaled down by $\sqrt{d_k}$. Formally we have a query $Q$, a key $K$ and a value $V$ and calculate the attention as:

$$ {\text{Attention}}(Q, K, V) = \text{softmax}\left(\frac{QK^{T}}{\sqrt{d_k}}\right)V $$

If we assume that $q$ and $k$ are $d_k$-dimensional vectors whose components are independent random variables with mean $0$ and variance $1$, then their dot product, $q \cdot k = \sum_{i=1}^{d_k} u_iv_i$, has mean $0$ and variance $d_k$. Since we would prefer these values to have variance $1$, we divide by $\sqrt{d_k}$.