Large Language Models (LLMs) are built on transformer architectures, but the specifics of how these models actually learn remain opaque to many. We know that they output surprisingly fluent text and can answer complex queries—but behind the scenes, they are simply learning to predict the next token. Here’s a closer look at how transformers learn using vectors, attention heads, and backpropagation—and where stochastic methods like Monte Carlo fit into training dynamics.

The Transformer: A Vector Machine

Transformers operate on vectorized input—token embeddings representing words or subwords in high-dimensional space. Each input token is transformed into three vectors:

• Query ($\mathbf{Q}$): “What am I looking for?”
• Key ($\mathbf{K}$): “What do I offer?”
• Value ($\mathbf{V}$): “What’s my content?”

These are computed via learned weight matrices:

where $\mathbf{x}$ is the input embedding and $\mathbf{W}_q$, $\mathbf{W}_k$, $\mathbf{W}_v$ are trainable matrices per attention head.

Each attention head computes the similarity between queries and keys via scaled dot-product attention, then uses those scores to weigh the value vectors:

Backpropagation: Refining Q, K, and V

Once the model predicts a token (e.g., guesses “ball” instead of the correct “cat”), a loss function like cross-entropy is used to measure error. This scalar error is then backpropagated through the network, computing gradients of the loss with respect to all trainable weights—including those that generated the Q, K, and V vectors.

Backpropagation uses the chain rule to calculate:

These gradients tell us how much each matrix contributed to the final error. The weights are then updated using gradient descent:

This update process happens across all layers: attention, feedforward, and even the token embedding layers. Over many examples, the model gets better at shaping attention to make correct predictions.

Escaping Local Minima: Monte Carlo and Noise

Gradient descent is a local method—it finds a direction downhill in the loss landscape and steps that way. But it can get stuck in local minima or saddle points. That’s where Monte Carlo and stochastic techniques come in.

• Stochastic Gradient Descent (SGD): Already introduces randomness via mini-batch sampling. This noise helps nudge the model out of shallow minima.
• Simulated Annealing: Adds controlled randomness to weight updates. Early in training, it allows higher-energy (worse) steps to escape bad basins, gradually reducing this behavior.
• Bayesian Methods: Treat weights as probability distributions rather than fixed values. Inference is performed by sampling from these distributions—effectively a Monte Carlo process over the network's behavior.
• Reinforcement Learning (e.g., RLHF): Uses rollouts to sample sequences, evaluate rewards, and improve the model. These rollouts are inherently Monte Carlo estimates.

Even dropout, when interpreted as approximate Bayesian inference, can be seen as introducing Monte Carlo sampling into forward passes.

Final Thoughts

Transformer models learn through an intricate but structured dance of vector transformations, attention weighting, and error minimization. The learning process is fundamentally driven by gradient descent, but made robust through stochasticity—both implicit (via data shuffling) and explicit (via sampling methods). As models scale, it’s not just the architectures that grow, but the techniques for navigating complex loss landscapes that must evolve as well.

Understanding this flow—from Q/K/V projections to backpropagated gradients and Monte Carlo nudges—offers a more grounded appreciation of how LLMs like GPT, PaLM, and LLaMA learn to speak our language.