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Understanding Transformers · Part 16 of 16

End-to-End Walkthrough and Cheat Sheet

One prompt, followed from text to next token — and a one-page reference for everything in the series.

The full path: “The cat sat on the”

  1. Tokenize. The tokenizer splits the text into vocabulary tokens and returns integer IDs (Part 6).
  2. Embed. Each ID looks up a learned embedding vector; positional information is added or applied (Part 6).
  3. Enter block 1. Layer normalization prepares the vectors; causal self-attention lets each position use itself and earlier positions (Parts 8, 13).
  4. Mix information. Queries compare with Keys, softmax makes attention weights, and weighted Values update each token (Part 8).
  5. Transform locally. The feedforward network does nonlinear, position-wise computation (Part 11).
  6. Repeat. Residual connections carry the representation up through dozens of blocks, each building on the last (Parts 5, 12, 13).
  7. Produce logits. A final layer normalization is applied after the last block, and the last position's vector is projected to one score per vocabulary token (Part 10).
  8. Convert to probabilities. Softmax might give “mat” high probability, “floor” moderate, and unrelated tokens almost none (Part 10).
  9. Select and append. A decoding strategy picks a token; the model appends it and runs again, reusing the KV cache (Parts 14, 15).

The famous equation, unpacked

Attention(Q,K,V) = softmax(Q KT / √dk) V

Q: what each token is looking for. K: what each token can be matched on. QKT: every Query–Key compatibility score. √dk: a scale correction. softmax: normalized attention weights. V: the information being blended. It is matrix multiplication, dot-product similarity, scaling, normalization, and a weighted average — nothing more mystical than that.

Compact math reference

ConceptFormula
Linear layery = Wx + b
Gradient updatew ← w − η · ∇L
Softmaxexp(zi) / Σj exp(zj)
Attentionsoftmax(QKT / √dk) V
Residualx ← x + F(x)
Next-token loss− log P(correct token)

What you should now be able to explain

  1. Why next-token prediction can teach broad linguistic and factual competence.
  2. How vectors and matrices represent and transform information.
  3. How derivatives, backpropagation, and gradient descent train the weights.
  4. Why nonlinear activations and depth are necessary.
  5. How tokens become embeddings and pick up contextual meaning.
  6. How Queries, Keys, Values, softmax, and multi-head attention work.
  7. How residuals, normalization, and the FFN form a transformer block.
  8. How a prompt turns into logits and generates one token at a time.
Where to go next. With this foundation, the original “Attention Is All You Need” paper, an annotated transformer implementation, or a small PyTorch project that codes single-head attention by hand are all within reach. Later topics — RoPE, FlashAttention, mixture-of-experts, LoRA, quantization — are extensions of what's here.
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