Based on the fundamental design of a modern language model that uses a deep neural network, explain why the model assigns a probability value to every single token in its vocabulary for the next-token prediction, including highly unlikely ones like 'keyboard' in this context.

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In the era of deep learning, a typical approach to language modeling is to estimate token probabilities using a deep neural network. Neural networks trained to accomplish this task receive a sequence of context tokens, $$x_0,...,x_{i-1}$$, as input and produce a distribution over the vocabulary $$\mathcal{V}$$, which is denoted by $$\Pr(\cdot|x_0,...,x_{i-1})$$. The probability of the specific token $$x_i$$, denoted as $$\Pr(x_{i}|x_0,...,x_{i-1})$$, is the value of the $$i$$-th entry of this output distribution.

Deep Learning Approach to Language Modeling

When an encoder model parameterized by $$\theta$$ processes an input sequence $$\mathbf{x}$$ and is followed by a Softmax layer parameterized by a weight matrix $$\mathbf{W}$$, it outputs a sequence of probability distributions. This operation is mathematically expressed as: $$ \begin{bmatrix} \mathbf{p}_1^{\mathbf{W},\theta} \\ \vdots \\ \mathbf{p}_m^{\mathbf{W},\theta} \end{bmatrix} = \mathrm{Softmax}_{\mathbf{W}}(\mathrm{Encoder}_{\theta}(\mathbf{x})) $$ In this formula, each $$\mathbf{p}_i^{\mathbf{W},\theta}$$ represents the conditional output distribution $$\Pr(\cdot|\mathbf{x})$$ at sequence position $$i$$. For notation simplicity, the superscripts $$\mathbf{W}$$ and $$\theta$$ affixed to each probability distribution are sometimes dropped.

Probability Distribution Formula for an Encoder-Softmax Language Model

When applying a language model, the generation of text typically follows an autoregressive process. At each individual step, the model takes a token, denoted as $$x_{i-1}$$, as its current input. It then predicts the subsequent token, $$x_i$$, by selecting the one that maximizes the conditional probability $$\Pr(x_i|x_0,...,x_{i-1})$$. This ensures each new prediction is conditioned on the sequence of all previously generated tokens.

Auto-Regressive Generation Process

The inference process in Large Language Models (LLMs) is formally defined as finding the most probable output sequence based on a given user context. Let $$\mathbf{x}$$ denote the input token sequence (conceptually equivalent to a 'prompt'), which comprises $$m+1$$ tokens denoted by $$x_0...x_m$$, where $$x_0$$ is the start symbol $$\langle \mathrm{SOS} \rangle$$. Let $$\mathbf{y}$$ denote the subsequent output token sequence (the response), comprising $$n$$ tokens denoted by $$y_1...y_n$$. The output tokens preceding position $$i$$ are denoted as $$\mathbf{y}_{<i} = y_1...y_{i-1}$$. The primary goal of LLM inference is to maximize the conditional probability $$\Pr(\mathbf{y}|\mathbf{x})$$, evaluating the context $$\mathbf{x}$$ to determine the most likely sequence $$\mathbf{y}$$. Furthermore, the input and output can be concatenated into a single sequence $$[\mathbf{x},\mathbf{y}] = x_0 ... x_m y_1 ... y_n$$ (sometimes represented as $$\mathrm{seq}_{\mathbf{x},\mathbf{y}}$$) to compute joint log-probabilities in decoder-only models.

Formal Definition of LLM Inference

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 language model built with a deep neural network is given the input sequence 'The cat sat on the'. The model's vocabulary consists of the following tokens: {a, cat, hat, mat, on, sat, the}. What does the model produce as its immediate, direct output to predict the very next token?

A language model built with a deep neural network processes the input sequence 'The forecast calls for heavy rain, so the sky is'. The model's output indicates that the token 'overcast' has a higher probability than the token 'blue'. Explain the fundamental nature of this model's direct output that allows it to represent the likelihood of multiple different words as the next token.

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