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During autoregressive inference, the Value matrix (V) in the KV cache is expanded at each step. The new value vector, $\mathbf{v}_{i'}$, corresponding to the current token, is appended to the existing matrix of values. This update operation is expressed by the formula: $$\mathbf{V} = \text{Append}(\mathbf{V}, \mathbf{v}_{i'})$$

Formula for Updating the Value Matrix in the KV Cache

An engineer is implementing the attention mechanism for an autoregressive model. Instead of appending the new value vector to the Value matrix in the cache at each step, they decide to replace the oldest value vector with the new one once the cache reaches a fixed size. Analyze the primary consequence of this 'fixed-size, replacement' strategy on the model's ability to generate long, coherent sequences of text.

Analysis of KV Cache Update Methods

During autoregressive inference, the Value matrix in the cache is updated by replacing the oldest value vector with the newly computed value vector for the current token.

An autoregressive language model is generating a sequence of tokens. The attention mechanism has already processed the first three tokens, resulting in the following Value matrix `V` stored in its cache, where each row corresponds to a token:

`V = [[0.1, 0.8], [0.5, 0.2], [0.9, 0.3]]`

For the fourth token, the model computes a new value vector:

`v_new = [0.4, 0.6]`

According to the standard update rule for the cache, what will the new Value matrix `V` be after this fourth token is processed?

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