The brain-ish machine
Neurons, weights and biases
A neuron multiplies each input by a weight (how much it cares), adds the results, adds a bias (its default mood), and passes the total on.
Think of it as a judge scoring a dish: taste counts a lot, price counts against it, plus a personal generosity bonus.
Show me the math
The weighted sum is a dot product: multiply matching numbers, then add them up.
Layers and networks
Put many neurons side by side to make a layer, then chain layers, so each one turns the previous layer's numbers into slightly more useful numbers.
Think of it as an assembly line, or a team of detectives: the first desk spots edges, the next spots shapes, the last decides “that's a 7”.
Scale check 3Blue1Brown's digit reader takes 784 pixels in and gives 10 numbers out. With two hidden layers of 16 neurons it has 13,002 weights and biases.
Show me the math
A whole layer is a matrix times a vector: many dot products computed at once.
Activation functions
After adding up its inputs, each neuron bends the total with a simple rule, such as ReLU: “if negative, output 0”. Bends let the network learn curves instead of only straight lines.
Think of it as a dimmer switch with a floor. Below zero the light stays off; above zero it gets brighter in step.
Show me the math
Sigmoid squashes any number into 0 to 1. Tanh squashes it into −1 to 1. Google's course notes that ReLU often works a little better than both, partly because it is less prone to the vanishing gradient problem and is cheap to compute.
How it learns
Loss function
The loss is one number that says how wrong a guess was. Big loss means a bad guess. Zero means perfect.
Think of it as a golf score: lower is better, and the whole game is bringing it down.
Show me the math
For LLMs the loss is cross-entropy: the negative natural log of the probability the model gave to the right token. Small probability on the right answer means a big loss.
Gradient descent and backpropagation
Training asks, over and over: which small change to each weight would lower the loss? Backpropagation works that out. Gradient descent then nudges every weight a little that way. Repeat billions of times.
Think of it as walking downhill blindfolded in fog. You feel the slope under your feet and step downhill. The step size is the learning rate. Backpropagation is the efficient way to feel the slope for millions of dials at once, working backward from the mistake to see who is to blame.
Show me the math
The derivative is a slope. The gradient is all the slopes at once, and it points uphill, so we step the opposite way. The chain rule lets backpropagation reuse each result as it moves backward.
Turning words into numbers
Tokens and tokenization
Before the model sees text, it is chopped into tokens (common words, word pieces, or single bytes), and each token gets an ID number.
Think of it as LEGO bricks. Common words come as one big pre-built brick (“the”). Rare words are snapped together from smaller bricks (“st” + “raw” + “berry”).
Rule of thumb For English, one token is about 4 characters or 0.75 words (OpenAI docs). GPT-2's vocabulary holds 50,257 tokens.
Go deeper
Byte-Pair Encoding (BPE) starts from single characters and repeatedly merges the most frequent neighboring pair. The stepper above runs that loop on a tiny corpus.
Embeddings
Each token ID becomes a long list of numbers, a point in a “meaning space” where similar words sit close together.
Think of it as a map where words are addresses. “Cat” lives next door to “dog” and far from “bus”, and directions on the map carry meaning.
Scale check This map has 2 numbers per word. GPT-3 uses 12,288.
Show me the math
A vector is a list of numbers, read as a point or an arrow. “Close” is measured by distance, or by the dot product.
Next-token prediction
An LLM's whole job: look at the text so far, give every possible next token a probability, pick one, add it to the text, and repeat.
Think of it as phone autocomplete with a far bigger brain, or as finishing a torn movie script one word at a time.
Show me the math
Softmax turns any list of raw scores into probabilities that are positive and add up to 100%. The model itself is deterministic. A prompt gives different answers across runs because the pick is random, weighted by those probabilities.
The transformer
The transformer and attention
A transformer lets every token look at the tokens before it, decide how much each one matters (attention), and blend in their information. Many layers repeat this, and all tokens are processed in parallel.
Think of it as a party. Each word holds up a query (“who describes me?”), wears a key name-tag (“I'm an adjective”), and offers a value (what it can tell you). When a query and a key match well, you listen more to that word's value. Softmax turns the match scores into percentages.
Scale check GPT-3 runs 96 attention heads per layer across 96 layers. Its MLP blocks, not attention, hold about 116 billion of its 175 billion parameters (3Blue1Brown).
Show me the math
Q·Kᵀ is a grid of dot products (query against every key). Dividing by √d keeps the scores from growing too large. Softmax turns each row into percentages. Multiplying by V blends the values using those percentages. Q, K and V come from three learned matrices.
- 3Blue1Brown: Dot products and duality
- 3Blue1Brown: Matrix multiplication as composition
- Wikipedia: Softmax function
- 3Blue1Brown: Attention in transformers, step by step
- 3Blue1Brown: Transformers, the tech behind LLMs
- Jay Alammar: The Illustrated Transformer
- Jay Alammar: The Illustrated GPT-2
- Transformer Explainer: a live GPT-2 in your browser
- LLM Visualization: a 3D walk-through
Making a chatbot
Pretraining on large text
The model starts with random weights and spends months reading a huge amount of text, playing “guess the next token” and being corrected by backpropagation each time.
Think of it as reading every textbook in the library. In Andrej Karpathy's textbook analogy, pretraining is the reading, fine-tuning is the worked examples, and reinforcement learning is the practice problems.
Scale check A person reading non-stop would need over 2,600 years to get through GPT-3's training text (3Blue1Brown). The result is a base model, an internet-text simulator that continues documents but is not yet a helpful assistant.
Go deeper
For the full stack in one sitting, Andrej Karpathy's video “Deep Dive into LLMs like ChatGPT” (about 3.5 hours) covers pretraining data, tokenization, the network, inference, fine-tuning and reinforcement learning.
Fine-tuning (instruction tuning)
The base model is trained further on a smaller, carefully written set of example conversations (question, then ideal answer), so it learns to behave like a helpful assistant.
Think of it as an apprentice who has read the whole library and now studies worked examples from a master: “when asked this, answer like that.”
RLHF: learning from human feedback
People compare pairs of answers and pick the better one. A reward model learns to predict those choices, and the LLM is then tuned to write answers the reward model scores highly.
Think of it as a cooking show. Judges taste two dishes and point to the better one, a sous-chef learns to predict the judges' taste, and the cook practices against the sous-chef thousands of times.
Why it matters In OpenAI's InstructGPT study, people preferred answers from a 1.3-billion-parameter model tuned with human feedback over answers from the 175-billion-parameter GPT-3, despite the 100× size gap.
Talking to it
Inference and sampling
When you chat, the frozen model produces probabilities for the next token. A sampling rule picks one, and the loop repeats until a stop token. Temperature controls how bold the pick is.
Think of it as a spice dial. Low always orders the house favorite. High tries unusual dishes. Top-p says “only consider the dishes that together make up the top slice of popularity”.
Show me the math
Below 1, dividing sharpens the distribution. Above 1, it flattens it. Top-p (nucleus sampling) keeps the smallest set of tokens whose probabilities add up to p, then renormalizes.
The whole machine on one screen
Math notions, in plain words
| Notion | Plain words | Primary link | Backup link |
|---|---|---|---|
| Dot product | Multiply matching numbers and add: how much two arrows point the same way. | 3Blue1Brown | Khan Academy |
| Vector | A list of numbers, read as an arrow or a point. | 3Blue1Brown | Jay Alammar |
| Matrix | A grid of numbers that turns one list of numbers into another. | 3Blue1Brown | 3Blue1Brown |
| Derivative | The slope: how much the output changes when you nudge the input. | Khan Academy | Simple Wikipedia |
| Gradient | All the slopes at once. It points uphill, so we step the other way. | Khan Academy | Wikipedia |
| Chain rule | Passing blame backward through each step of a calculation. | 3Blue1Brown | 3Blue1Brown |
| Activation (ReLU, sigmoid) | A bend that lets networks learn curves. | Google ML Crash Course | Wikipedia |
| Softmax | Turns any scores into percentages that sum to 100%. | Wikipedia | 3Blue1Brown |
| Loss (cost) | One number for “how wrong”. | 3Blue1Brown | 3Blue1Brown |
Go deeper, live
What this page simplifies
- Every number in the widgets is a toy value unless a source is named. Real embeddings, vocabularies and networks are vastly larger.
- Token counts depend on the tokenizer. The 4-characters-per-token rule is for English with OpenAI tokenizers. Code and other languages use more.
- Pretrain, then fine-tune, then RLHF is the classic recipe. Many labs now add reinforcement learning on checkable tasks and preference methods such as DPO.
- GPT-3 figures are illustrative. Current frontier models do not publish such details.