A variant of the gradient descent algorithm for collaborative filtering (Figure 17.2) can be used to predict P(rating > threshold) for various values of threshold in {1, 2, 3, 4}. Modify the code so that it learns such a probability.

[Hint: Make the prediction the sigmoid of the linear function as in logistic regression.] Does this modification work better for the task of recommending the top-n movies, for, say n = 10, where the aim is to have the maximum number of movies rated 5 in the top-n list? Which threshold works best? What if the top-n is judged by the number of movies rated 4 or 5?