A language model is generating a sequence. At a specific step i, it computes the following probabilities for the next token over its vocabulary V = {'run', 'walk', 'jump', 'sit', 'sleep'}. Given a setting of K=3, which of the following sets correctly represents the selection pool V_i according to the formal definition: $V_i = \underset{y_i \in V}{\text{argTopK}} \, \text{Pr}(y_i|\mathbf{x}, \mathbf{y}_{<i})$

Probabilities:

• Pr('run' | ...) = 0.15
• Pr('walk' | ...) = 0.40
• Pr('jump' | ...) = 0.05
• Pr('sit' | ...) = 0.35
• Pr('sleep' | ...) = 0.05

A developer is implementing the selection mechanism for a text generation model based on the formal definition: $V_i = \underset{y_i \in V}{\text{argTopK}} \, \text{Pr}(y_i|\mathbf{x}, \mathbf{y}_{<i})$ For a vocabulary V = {'cat', 'dog', 'ran', 'sat'} and K=2, the model computes the next-token probabilities as: Pr('cat'|...) = 0.1, Pr('dog'|...) = 0.5, Pr('ran'|...) = 0.3, Pr('sat'|...) = 0.1. The developer's code returns the set {0.5, 0.3} as the selection pool $V_i$. What is the fundamental error in this output when compared to the formal definition?

Interpreting the Formal Definition of Top-k Selection