Based on the case study, first calculate the boundary value. Then, from the following list of relative offsets [8, 15, 31, 32, 45], identify which offsets would be grouped into the category for values less than or equal to this boundary.

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In the calculation of T5 relative position bias buckets, a key mathematical component is the expression: $$\frac{n_b+1}{2} - 1$$. This term establishes the upper boundary for the initial set of buckets that utilize a one-to-one mapping with query-key offsets, where $$n_b + 1$$ represents the total number of available buckets.

Formula Component for T5 Bias Bucketing

In a specific implementation of a relative position encoding scheme, the following expression is used as part of the logic to determine a boundary for grouping offsets: `(n_b + 1) / 2 - 1`. If the hyperparameter `n_b` is set to 31, what is the value of this expression?

In a relative position encoding mechanism, a hyperparameter `n_b` is used to define the number of parameter buckets. A specific calculation, `(n_b + 1) / 2 - 1`, is used to determine a boundary for categorizing relative distances. Describe the relationship between `n_b` and the result of this calculation. Specifically, if `n_b` is increased, what happens to the calculated boundary value, and what does this imply about the range of distances being grouped together?

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