Implement a Recommender that trains a latent-factor recommender on rating data with SGD, without sklearn. Where the other Tier 4 problems implement a single metric or tree, this one builds an entire recommender pipeline end to end: baseline → latent factors → early stopping → serving.

Shared spec

Ratings arrive as a list of (user, item, rating) triples, where rating is a float. The same (user, item) pair may appear several times — do not deduplicate. Each appearance is an independent training step.

model = Recommender(epochs=20, lr=0.05, reg=0.02, k=0, seed=0)
model.fit(ratings)
model.predict(user, item)   # float

Prediction formula

$\hat{r}_{ui} = \mu + b_u + b_i + p_u \cdot q_i$

• $\mu$ — the global mean rating. Computed once at the start of fit and never changed. 0.0 when the training data is empty.
• $b_u, b_i$ — user and item biases. A user or item never seen is 0.0.
• $p_u \cdot q_i$ — the latent-factor dot product. When k == 0 this term is always 0 (L1). The latent vectors of a user or item never seen are also treated as zero (so the dot product is 0).

Training — stochastic gradient descent

Repeat epochs times. Each epoch walks ratings once in its original order (no shuffling). For each (u, i, r):

pred = mu + b_u + b_i + p_u·q_i     # predict with the pre-update values
err  = r - pred

b_u  ← b_u + lr·(err - reg·b_u)
b_i  ← b_i + lr·(err - reg·b_i)
p_u  ← p_u + lr·(err·q_i - reg·p_u)   # use the OLD q_i
q_i  ← q_i + lr·(err·p_u - reg·q_i)   # use the OLD p_u — simultaneous update

⚠️ $p_u$ and $q_i$ update simultaneously. Do not feed the freshly updated $p_u$ into the $q_i$ update — both updates must read the values as they stood at the start of this step.

Latent-factor initialization (only when k > 0, and reproducible from seed)

rng = np.random.default_rng(seed)

Create rng once. Walk ratings and, for each user as it first appears, call rng.normal(0, 0.1, size=k) in order to build $p_u$; then walk ratings again and, for each item as it first appears, build $q_i$ with the same call (all users first, then all items — swap the order and the same seed yields different vectors). This initialization happens once, before the training loop starts.

Level structure

Each level must preserve the behavior of the previous ones. Calling fit without val_ratings must behave exactly as in L1 and L2 even after L3.

At this level, ignore k and predict with only $\mu + b_u + b_i$.

Recommender(epochs: int, lr: float, reg: float, k: int, seed: int)
fit(ratings: list[tuple]) -> self
predict(user, item) -> float

• Follow the shared spec's training rules (SGD, original order, no deduplication) exactly. The $p_u \cdot q_i$ term is simply always 0 at this level.
• Calling predict without calling fit returns 0.0 for everything.
• An empty ratings list works without raising, and $\mu$ is 0.0.
• fit returns self.
• Re-fitting the same model discards the previous state completely and starts over (no biases survive).