Considering the scenario below, explain the fundamental reason for the two models' drastically different specialized abilities. Your explanation must center on the concept of the models' tunable parameters, collectively represented as θ.

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A machine learning model, such as a Large Language Model, is a function whose behavior is governed by a set of tunable parameters. These parameters are collectively represented by the symbol θ. The process of training involves finding the optimal values for θ that enable the model to perform its designated task.

Model Parameterization by θ

A simple predictive model is defined by the function `output = (weight_1 * input_1) + (weight_2 * input_2) + bias`. During the training process, the model adjusts its internal values to better predict the output based on the inputs. Which components of this function represent the model's tunable parameters (collectively denoted as θ)?

A newly initialized language model produces nonsensical, random-sounding text. After a period of training on a large text corpus, the same model begins to generate grammatically correct and contextually relevant sentences. In the context of model parameterization (represented by θ), explain what has fundamentally changed within the model to cause this improvement.

Effect of Training on Model Parameters

In the context of knowledge distillation, $P_\theta^s$ denotes the probability distribution of the student model's output. This distribution is conditional on a given context $\mathbf{c}'$ and a latent variable $\mathbf{z}$, and is parameterized by the student model's weights $\theta$. The relationship is formally expressed by the equation: $P_\theta^s = \text{Pr}_\theta^s(\cdot|\mathbf{c}', \mathbf{z})$.

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