Making Recommendations

Recommender is what you experience when sites make recommendations whether a movie, a product….

Movie Recommendations

Let’s say we have a small dataset of movie reviews:

  • nu = no of users
  • nm = no of movies
  • r(i,j) =1 if user j has rated movie i
  • y(i,j) = rating given by user j to movie i which is defined only if r(i,j)=1
  • Note not every user has rated all the movies

Let’s predict how users will rate the movies that have not been rated

Before we proceed let’s raise another possibility:

  • What if we have more features regarding these movies such as (x1=romance & x2=action) would that help predict what the users will rate the movies
  • What if we don’t have any additional features?

We’ll cover the first option first:

Added Features

  • The values of whether it is that features vary from 0 -1 with 0 being least or not and 1 being definitely
  • Here is an example, if a movie has x1=0.9 it means it is highly romantic and if x2=0.1 means it is not an action movie
  • Let n=number of features

Let’s predict the rating of Cute puppies of love for Alice

  • we will use the linear regression algorithm W * X(i) + b
  • Let’s pretend the value of w= [5, 0] and b =0 and from the table X(3)=[0.99, 0]
  • Note since we have two features X that’s why we needed two values for W
  • So the algorithm will give us 4.95
  • Which seems reasonable since she rated the two other romantic movies high as well
  • The superscript of 1 stands for the fact that Alice if the first user in the dataset, and so each user will have their own 1-4 since we have 4 users
  • So the generalized algorithm to predict the rating for any user will have the (j) superscript as shown in the image

Cost Function

  • As you see below it will be the value predicted - the actual value rated
  • But since there are some cases when users have not rated all the movies so how can we sum over all the data rows, well we won’t
  • This is it differs from linear regression because we only sum over the RATED movies

  • To learn the parameters we need to minimize the cost function

Blank Features

What if we haven’t learned the values of the added features? Let’s say a movie just came out and for some reason nobody has rated the added features.

  • Can we predict what each user will rate the added features
  • Let’s pretend we know the values of w and b for each user as shown in the image
  • As you see all the values of b are 0 so let’s ignore b from the formula
  • So if you have all the ratings from all users for a certain movie you can predict x1
  • Then you can predict x2
  • And now that you have both x values you can try to predict the ratings that are missing for all the users to all movies as shown above
  • If you don’t have all ratings to a movie from all users you cannot predict x values

Cost Function

Collaborative Filtering

Let’s put both scenarios detailed above together.

Gradient Descent

Binary Labels

Sometimes we just ask the user if they like something or not, this is a simple binary model. So we can revert back to what we’ve already learned to learn if we should recommend an item or not.

  • Some users have not commented on the item
  • Here is the breakdown

  • Now that we are using binary values we just need to predict the probability which is a binary classification model

Cost Function

Now the cost function for binary classification will become:

Mean Normalization

It will make the algorithm make better prediction and speed it up if we normalize the reviews by the users

  • So let’s take all the data/reviews and put them in a 2D matrix
  • We take all the rows and average them out, if we have a movie that hasn’t been shown to anyone we can average out the columns instead
  • Of course we skip the ? marks
  • Now we get a vector of average ratings
  • Then we take the original ratings and subtract the mean/average values from it and now we have a new matrix
  • Then to predict the new rating we use the same formula as we did above but in order to not have a negative rating (since the rating system is from 0-5) we must add back the mean/average value for each row/movie which is \(\mu\) i
  • So user 5 will have a rating of 2.5 for the first movie i=1

TF Implementation

  • First we specify the optimizer: Adam in this case
  • Choose number of iterations
  • Loop through the number of iterations we call the GradientTape which will record the values
  • In the loop you will see the cost function J which takes inputs: X, W, B, Ynorm are the normalized mean. Which values have a rating, number of users, number of movies, as well as the regularization parameter \(\lambda\)
  • This cost_value will figure out the derivatives
  • Then we feed the derivatives into the gradient to give us the gradient
  • We run it, and repeat it by updating the values of the variables to minimize the loss using the optimizer we specified