Machine learning, derived

Each lesson starts from one concrete question, builds the answer on a small example you can change, asks you to predict before it tells you, and ends with code you write yourself.

• 01 Cross-entropy and KL, from surprise

  Why −log p, why the best model copies the frequencies, entropy as the floor under training loss, KL as the gap, the softmax gradient q − p, forward vs reverse KL. Game: bet on the next word.

• 02 From a reward to a gradient: REINFORCE, PPO, GRPO

  The log-derivative trick, baselines and advantages, PPO's ratio and clipping, GRPO's group-relative advantages, per-token credit, length bias and the KL leash. Game: predict the push.

• 03 Backpropagation, by assigning blame

  Real numbers forward into a loss, then blame traced back through every weight and ReLU gate, an update, and a gradient check against nudging. Why sigmoid pairs with cross-entropy. Game: the blame game.

• 04 Attention: score, normalize, mix

  Dot products you can drag in 2-D, softmax as a temperature, outputs as convex mixes, why √d, causal masks, and the KV cache with a memory calculator. Game: be the query.

• 05 Bayes from counts

  1,000 messages as dots: build the counts, let the evidence select the group, then name it. Likelihood vs posterior, recall vs precision, odds and log-odds, recalibrating downsampled models. Game: call the odds.

• 06 k-means, against Lloyd's algorithm

  The objective, why nearest-center and mean updates, Lloyd as coordinate descent, local minima, k-means++ measured over 300 runs, feature scaling, empty clusters, and k-means inside ANN indexes. Game: place the centers.

• 07 A/B tests and the cost of peeking

  p-values built from A/A tests, the standard error of a rate difference, power and sample size, sample-ratio mismatch, clustered units, multiple metrics, novelty, CUPED and sequential tests. Game: ship or not.

• 08 Optimizers: learning rate, momentum, Adam

  The 2/λ stability limit, condition number, momentum's √κ speed-up, noise floors and learning-rate decay, Adam's bias correction and diagonal scaling, warmup, AdamW. Game: descend the valley.

• 09 Retrieval vs ranking

  The recall ceiling, why two-tower models can't use cross features, in-batch negatives and the logQ correction, position bias, cold start, NDCG vs MRR vs recall, pointwise/pairwise/listwise losses. Game: budget the funnel.

• 10 PCA and SVD: what a projection keeps

  Projection and the Pythagorean trade between kept variance and error, eigenvectors of the covariance, SVD and Eckart–Young, LoRA's rank bet, scree plots, scaling and outliers. Game: find the axis.

• 11 Leakage and the leaderboard that lies

  What validation promises, prediction time and point-in-time features, target, temporal, group, preprocessing and selection leakage, bias vs variance, leakage vs drift. Game: validate, then deploy.