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The moving average approach for creating memory summary vectors can be extended to a cumulative average of the keys and values. Instead of averaging over a fixed-size window of recent key-value pairs, this method extends the moving average to include all positions from the beginning of the sequence up to the current position $$i$$. This results in a summary that incorporates the entire history of the sequence at each step.

Cumulative Average of Keys and Values for Memory Component

A cumulative average can be calculated using a recursive formula. This means the average at the current step can be computed based on the value of the average from the preceding step and the current data point, which is more efficient than recalculating the sum over the entire history at each step.

Recursive Formula for Cumulative Average

A language model's memory component can be summarized using two different averaging techniques. Technique 1 calculates the summary by averaging all key-value pairs from the start of the sequence up to the current position. Technique 2 calculates the summary by averaging only the key-value pairs within a fixed-size window of the most recent positions. Compare these two techniques, explaining the primary advantage and disadvantage of Technique 1 (the cumulative approach) regarding its representation of the sequence's history.

Analysis of Memory Summary Techniques

Consider two different methods for creating a memory summary vector at each step in a sequence:

*   **Method 1:** The summary is the average of *all* key-value pairs from the beginning of the sequence up to the current position.
*   **Method 2:** The summary is the average of only the *last 50* key-value pairs.

For each scenario described below, determine which method (1 or 2) is more suitable and briefly justify your choice by explaining the trade-offs of each method in that context.

Selecting a Memory Summarization Strategy

The cumulative average of the keys and values for a memory component ($$\mathrm{Mem}$$) is formulated as a tuple containing the cumulative average of key vectors ($$\mathbf{k}_j$$) and value vectors ($$\mathbf{v}_j$$) up to a given index $$i$$. This is achieved by summing all vectors from the start of the sequence (index $$j=0$$) to the current index $$i$$, and then normalizing by the total number of elements, $$i+1$$. The mathematical representation is: $$\mathrm{Mem} = \Big( \frac{\sum_{j=0}^{i} \mathbf{k}_j}{i+1}, \frac{\sum_{j=0}^{i} \mathbf{v}_j}{i+1} \Big)$$

Formula for Memory as a Cumulative Average of Keys and Values

A language model's memory component is designed to create a summary vector at each step by calculating the average of all key-value pairs from the start of the sequence up to that current step. When this model is processing a very long sequence, what is the effect on the summary vector's representation of information from the very beginning of the sequence as the model approaches the end?

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