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In autoregressive language models, next tokens are predicted based solely on their preceding context (the 'left-context'). Accordingly, the output of the attention mechanism for a single token at position $$i$$ is calculated using only information from tokens $${}0$$ to $$i$$. This output is formulated as the product of the attention weight row vector for token $$i$$ and the matrix of corresponding value vectors up to that position:

$$ \mathrm{Att}_{\mathrm{qkv}}(\mathbf{q}_i,\mathbf{K}_{\le i},\mathbf{V}_{\le i}) = \begin{bmatrix} \alpha_{i,0} & \dots & \alpha_{i,i} \end{bmatrix} \begin{bmatrix} \mathbf{v}_0 \\ \vdots \\ \mathbf{v}_{i} \end{bmatrix} $$

This matrix multiplication is equivalent to the weighted sum of the value vectors:

$$ \mathrm{Att}_{\mathrm{qkv}}(\mathbf{q}_i,\mathbf{K}_{\le i},\mathbf{V}_{\le i}) = \sum_{j = 0}^{i} \alpha_{i,j} \mathbf{v}_{j} $$

In these equations, the keys and values up to position $$i$$ are respectively defined as the matrices $$ \mathbf{K}_{\le i} = \begin{bmatrix} \mathbf{k}_0 \\ \vdots \\ \mathbf{k}_{i} \end{bmatrix} $$ and $$ \mathbf{V}_{\le i} = \begin{bmatrix} \mathbf{v}_0 \\ \vdots \\ \mathbf{v}_{i} \end{bmatrix} $$.

Causal Attention Output for a Single Token

Based on the provided value vectors and attention weights, which token's meaning will have the most influence on the final attention output vector for the token at position 2? Justify your answer by explaining how the output is calculated.

Interpreting Causal Attention Output

A developer is implementing an autoregressive model and finds a bug. For a sequence of 4 tokens (indexed 0, 1, 2, 3), the attention output for the token at position 2 is being calculated using the following weighted sum: 

`Output_at_pos_2 = (0.1 * v_0) + (0.2 * v_1) + (0.6 * v_2) + (0.1 * v_3)`

where `v_j` is the value vector for the token at position `j`. Identify the specific term in this expression that violates the core principle of this type of attention mechanism and explain why its inclusion is incorrect for an autoregressive task.

Debugging a Causal Attention Calculation

In the original version of self-attention, the attention weights are assumed to be dense. This means that for a given query at position $$i$$, most of the values in the attention weight vector $$\begin{bmatrix} \alpha_{i,0} & ... & \alpha_{i,i} \end{bmatrix}$$ are non-zero. Consequently, the query must compute its output by attending to nearly all key-value pairs up to position $$i$$.

Dense Attention Assumption

In an autoregressive model, the attention output for a token is a weighted sum of the value vectors of itself and all preceding tokens. Consider a sequence of three tokens (at positions 0, 1, and 2). The value vectors are given as v_0 = [1, 2], v_1 = [3, 0], and v_2 = [4, 5]. The attention weights for the token at position 2, which determine the contribution of each token in the context, are α_2,0 = 0.1, α_2,1 = 0.6, and α_2,2 = 0.3. Based on this information, what is the attention output vector

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