LLMs Explained From First Principles: Vectors, Attention, Backpropagation, and Scaling Limits

A widely shared technical explainer cuts through the hype to reveal the mathematical engine powering modern large language models (LLMs) like GPT-4 and Claude: Google's Transformer architecture. At its core, the process begins by converting text into numerical vectors—lists of real numbers positioned in a high-dimensional space. These vectors are transformed via learned matrices into queries, keys, and values, the fundamental components for the model's attention mechanism. The heart of the Transformer, attention calculates similarity scores between tokens using dot products, applies a softmax function to create a probability distribution, and then produces a new context-aware vector for each token as a weighted sum of values. This allows every token to become a blend of relevant information from across the input sequence.

To capture diverse linguistic patterns, the architecture employs multi-head attention, running multiple of these operations in parallel. Since Transformers lack inherent sequence awareness, sinusoidal positional encodings are added to give tokens a sense of order, using sine and cosine functions at different frequencies—a concept rooted in Fourier analysis. Following attention, each token is independently processed by a feed-forward neural network, applying non-linear transformations. The entire system is optimized through backpropagation and calculus, with its remarkable capabilities like reasoning emerging purely from scaling these interconnected layers of linear algebra and probability. The article underscores that the 'magic' of AI is not mystical but a specific, scalable arrangement of these foundational mathematical principles.