LLM as a Mathematical Function

In this tutorial, we will understand a Large Language Model (LLM) as a mathematical function, let’s break it down step by step. A mathematical function is a process that takes an input and applies a set of rules or operations to give an output. You can think of it as a “machine” that transforms one thing into another.

In the case of an LLM, the function is a complex process that works with language instead of numbers. You can relate the idea of a mathematical function to an LLM:

When we simplify the operation of a Large Language Model (LLM) to the form

y = f(x)

LLM:   y = f(x)

So, in a simple mathematical way:

• x = Input (a sequence of words or tokens)
• f(x) = The function performed by the LLM (its learned transformation from input to output)
• y = Output (the model’s prediction or answer)

Input (x)

 The input to the LLM is a sequence of words, sentences, or text. This could be a question you ask or a sentence you want the model to complete. Mathematically, you can think of this as a vector or a set of numbers representing those words.

For example, if you input:

• “What is the capital of France?” This input is converted into a form that the model can understand, often called a tokenized vector. So, x could be something like a sequence of numbers that represent these words.

The Function (f)

The function here represents the model itself. It processes the input x using layers of neural networks (in LLMs, often transformers). It’s a very complex function, built on neural networks, which have learned patterns from large amounts of text data. This function takes the input, processes it through many layers of transformations (involving complex mathematical operations like matrix multiplications, activation functions, etc.), and ultimately tries to understand the context, meaning, and intent behind the input.

It involves operations like:

• Embedding: Converting words into numerical vectors.
• Attention: Learning which parts of the input sequence are most important to focus on when predicting the next word.
• Transformation: Passing these vectors through multiple layers of attention and transformation to generate a context-aware representation.

Output (y)

In mathematical terms, this output y is a sequence of tokens, just like the input x, but it represents the model’s response. After processing the input through its layers, the LLM produces an output. This is the model’s prediction, like answering the question or completing the sentence. The output can also be transformed into a sequence of words or tokens.

For example, if the input is “What is the capital of France?”, the output might be:

• “The capital of France is Paris.“