🛒 Recommendation Engine — Collaborative Filtering Live

The rating matrix is deliberately sparse — each user has only rated a handful of the catalogue, so the vast majority of entries are simply unknown, not zero. Training the model to predict a 0 or an average value for every unobserved entry would actively teach it the wrong thing (that unrated products are disliked). Instead, the squared-error loss sums only over the (user, item) pairs that actually have a star rating: L = Σ(observed) (r_ui − r̂_ui)². Gradients only flow through cells with real feedback, and the latent vectors for unrated pairs are pulled into sensible positions purely by sharing rows and columns with observed cells.

The training objective adds λ(‖u‖² + ‖p‖²) to the squared-error loss, penalising latent vectors with large magnitude. Without it, a user or product with very few observed ratings can drift to an extreme vector that fits those one or two data points almost perfectly but predicts wildly for everything else — classic overfitting on sparse data. Regularization shrinks vectors toward zero unless the data provides strong repeated evidence to push them further out, trading a little training accuracy for much better generalisation to the unrated products that recommendations actually depend on.

When you rate a product for the selected user, that single (user, item, rating) triple is added to the observed set and the optimiser runs a further batch of SGD passes focused on the updated data. Because the user and product vectors are shared parameters, a handful of extra gradient steps is enough to nudge the user's latent position toward products with similar factors — you can watch the point move in the scatter plot and the top-N predicted list reorder within a second, without retraining from scratch.

If every entry of the rating matrix were observed, singular value decomposition (SVD) could compute the optimal low-rank factors directly in closed form. But real rating matrices are extremely sparse — a user has rated a tiny fraction of the catalogue — so classical SVD is undefined for the missing entries. Stochastic gradient descent sidesteps this cleanly: it only ever touches observed cells, scales to millions of ratings because each update is O(K) per rating, and naturally supports incremental updates when new ratings arrive, which is exactly what the live retraining in this simulation relies on.

The synthetic ratings in this simulation are generated from ground-truth latent factors plus random noise, so even a perfect model cannot predict the exact noisy stars every time — there is an irreducible error floor set by the noise variance. Regularization also deliberately trades a little training-set accuracy for generalisation, further preventing RMSE from reaching zero. In practice this is healthy: a model that drives observed-set RMSE to zero with only two latent dimensions and a handful of ratings per user is almost certainly memorising noise rather than learning taste.