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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.

Analyzing Language Model Outputs

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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