Learn it here.
Prove it to yourself.

A function maps an input to an output. In y = mx + b, x is the input and the predicted y is the output.

The examples stay fixed while fitting. The parameters m and b change because they control the function's behavior.

Learning means choosing parameter values that make predictions agree with the observed targets. The same distinction scales from a two-parameter line to a transformer with billions of parameters.

Exercise: For (x, y) = (0,1), (1,3), (2,5), (3,7), (4,9), choose m and b for y = mx + b. Show all five predictions and label inputs, targets, parameters, and predictions.

Mastery gate: Explain a parameter without LLM jargon and predict the effect of changing m versus b on a new input.

A loss function compresses many prediction errors into one number that can guide training.

A gradient is a local slope. Gradient descent moves parameters against that slope, scaled by the learning rate.

Exercise: Compute MSE for two line fits, then take one gradient-descent step by hand.

Mastery gate: Predict the symptoms of a learning rate that is too high or too low.

Vectors hold features, matrices transform them, and tensors generalize these arrays to more axes.

Shape annotations expose which dimensions represent batch, sequence, embedding, heads, and vocabulary.

Exercise: Trace the shapes through one linear layer and diagnose one deliberately mismatched axis.

Mastery gate: Read a shape error and identify which axis is wrong.

Stacking only linear transformations still produces one linear transformation. Non-linear activations let networks fit curved decision boundaries.

Backpropagation applies the chain rule from the loss back to every parameter that influenced it.

Exercise: Inspect or train a two-layer MLP on a non-linear toy dataset and explain the gradient path.

Mastery gate: Explain why an activation enables behavior that stacked linear layers cannot.

Self-supervision builds broad predictive capability; supervised and preference methods shape behavior for particular tasks.

More scale can add capacity and knowledge, but cannot repair invalid data or an unreliable evaluation ruler.

Exercise: Classify retained posttrainllm attempts by learning paradigm and name the signal each used.

Mastery gate: Explain why scale cannot rescue a broken evaluation or contaminated dataset.

A tokenizer chooses the discrete units a model sees. Embeddings map each token ID into a learned vector.

A causal language model learns to predict the next token from the tokens before it.

Exercise: Tokenize three prompts and inspect how punctuation and identifiers split.

Mastery gate: Explain how tokenization can affect SQL and tool-call reliability.

Queries express what a position seeks, keys describe what positions offer, and values carry the information to combine.

A transformer block combines attention, an MLP, normalization, residual paths, and positional information.

Exercise: Work one tiny query/key/value attention calculation with explicit shapes.

Mastery gate: Describe what attention can copy or route that a position-wise MLP cannot.

A loss curve only becomes useful when paired with data checks, held-out behavior, gradient health, and reproducible settings.

Tiny-overfit is the first gate: if a model cannot memorize a tiny repeated set, scaling the run hides the bug rather than fixing it.

Exercise: Inspect a tiny-overfit run and diagnose data, learning-rate, capacity, and precision failures separately.

Mastery gate: Distinguish a data bug, optimization bug, and capacity limit from their symptoms.

SFT imitates desired outputs. LoRA learns a low-rank update while keeping the base weights frozen.

Preference methods compare better and worse responses; weak pair construction or missing reference control can collapse useful behavior.

Exercise: Compare the successful SQL SFT run with the failed hygiene preference run and isolate the changed variables.

Mastery gate: Design a bounded recipe with data, target, regression slices, budget, and stop rule.

An evaluation must first pass a frontier-ceiling check: if a capable model cannot satisfy the ruler, the ruler is measuring noise.

Held-out data, regression slices, provenance, and cost measurements turn a score into a decision.

Exercise: Draft a target, frozen baseline, held-out gate, regressions, resource budget, and ship/retry/reject rule.

Mastery gate: Reject an attractive score when its ruler, holdout, provenance, or regression gate is invalid.