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In sparse attention, the attention weights are normalized exclusively over the subset of indices defined by the set $$G$$. This redistribution of probability mass means that the weights which would have been assigned to ignored tokens are now allocated among the tokens in set $$G$$. As a result, the sparse attention weights, $$\alpha'_{i,j}$$, for any token $$j \in G$$ are greater than their counterparts, $$\alpha_{i,j}$$, in a standard dense attention calculation: $$\alpha'_{i,j} > \alpha_{i,j}$$

Comparison of Sparse and Dense Attention Weights

An attention mechanism calculates the following dense attention weights for a specific query token against four preceding key tokens:
- Token 1: 0.10
- Token 2: 0.40
- Token 3: 0.20
- Token 4: 0.30

The mechanism is then modified to be sparse, considering only Token 2 and Token 4. The attention scores are re-calculated and re-normalized over just this smaller set. Explain the fundamental reason why the new, sparse attention weight for Token 2 will be greater than its original dense weight of 0.40.

Effect of Sparsity on Attention Weights

An engineer is analyzing a language model's attention mechanism. For a specific query token, the model initially calculates the following normalized attention weights (dense attention) over four previous tokens:

- Token 1: 0.1
- Token 2: 0.5
- Token 3: 0.3
- Token 4: 0.1

To improve efficiency, the engineer modifies the mechanism to only attend to Token 2 and Token 3, ignoring Tokens 1 and 4. The attention scores for Tokens 2 and 3 are re-calculated and re-normalized over this smaller set. 

Predict the effect of this change on the attention weight for Token 2. Will the new, sparse attention weight for Token 2 be greater than, less than, or equal to its original weight of 0.5? Justify your answer based on the principle of probability mass distribution.

Analyzing Attention Weight Redistribution

An attention mechanism calculates normalized weights for a query token against four previous tokens, resulting in weights α₁, α₂, α₃, and α₄. Now, a new version of the mechanism is implemented which is constrained to only consider the second and fourth tokens. The attention scores are re-calculated and re-normalized over just this smaller set, resulting in new weights α'₂ and α'₄. Assuming all original weights were greater than zero, what is the relationship between the new weight for the second

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