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One of the simplest forms of self-attention incorporating relative positional embedding modifies the attention weight calculation while maintaining the standard weighted sum for the output. The attention output vector is computed as: $$ \mathrm{Att}_{\mathrm{qkv}}(\mathbf{q}_i, \mathbf{K}_{\le i}, \mathbf{V}_{\le i}) = \sum_{j=0}^{i} \alpha(i,j) \mathbf{v}_j $$ The attention weight $$ \alpha(i, j) $$ is calculated by adding a relative positional encoding bias term $$ \mathrm{PE}(i, j) $$ to the query-key product: $$ \alpha(i, j) = \mathrm{Softmax}\left(\frac{\mathbf{q}_i \mathbf{k}_j^\top + \mathrm{PE}(i, j)}{\sqrt{d}} + \mathrm{Mask}(i, j)\right) $$ The only difference between this approach and the original self-attention model is the addition of the $$ \mathrm{PE}(i,j) $$ bias term.

Formula for Attention Weight with Relative Positional Encoding

The output of a causal attention mechanism for a specific query vector $\mathbf{q}_i$ is calculated as a weighted sum of value vectors $\mathbf{v}_j$ from all positions $j$ up to and including the current position $i$. The formula is expressed as: $$ \text{Att}_{\text{qkv}}(\mathbf{q}_i, \mathbf{K}_{\leq i}, \mathbf{V}_{\leq i}) = \sum_{j=0}^{i} \alpha(i, j) \mathbf{v}_j $$ Here, $\alpha(i, j)$ represents the attention weight assigned to the value vector at position $j$ when computing the output for position $i$.

Formula for Causal Attention

In a sequence processing model, the unnormalized attention score between a query at position `i` and a key at position `j` is calculated using the formula: `Score(i, j) = (q_i ⋅ k_j + PE(i, j)) / √d`. What is the primary function of the `PE(i, j)` term in this calculation?

Consider the following formula for calculating an unnormalized attention score between a query at position `i` and a key at position `j` in a sequence: `Score(i, j) = (q_i ⋅ k_j^T + PE(i, j)) / √d + Mask(i, j)`. Explain the distinct contribution of the `PE(i, j)` term and the `Mask(i, j)` term to the final attention weight. How would the model's behavior likely change if each of these terms were individually removed?

Analyzing Components of an Attention Score Formula

A language model is being developed for a machine translation task. During testing, it is observed that while the model generates grammatically plausible sentences, it frequently scrambles the word order, leading to nonsensical translations, especially for longer input sentences. The engineering team has confirmed that the attention weights are being calculated using the formula shown in the case details. Based on an analysis of this formula, identify the most probable cause for the model's failure to preserve correct word order and explain your reasoning.

Diagnosing a Language Model's Performance Issue

In self-attention with relative positional embeddings, the bias term $$ \mathrm{PE}(i, j) $$ added to the query-key product can intuitively be interpreted as a distance penalty between positions $$ i $$ and $$ j $$. To reflect that tokens further apart should generally have less influence on each other, the value of $$ \mathrm{PE}(i, j) $$ decreases as the token at position $$ i $$ moves further away from the token at position $$ j $$.

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