Recommender Systems (Collaborative Filtering)

9 min readApr 7, 2024

The goal of a collaborative filtering recommender system is to generate two vectors: For each user, a ‘parameter vector’ that embodies the movie tastes of a user. For each movie, a feature vector of the same size which embodies some description of the movie. The dot product of the two vectors plus the bias term should produce an estimate of the rating the user might give to that movie.

Existing ratings are provided in matrix form as shown. Y contains ratings; 0.5 to 5 inclusive in 0.5 steps. 0 if the movie has not been rated. R has a 1 where movies have been rated. Movies are in rows, users in columns. Each user has a parameter vector w^user and bias. Each movie has a feature vector x^movie. These vectors are simultaneously learned by using the existing user/movie ratings as training data. One training example is shown above: w^(1).x^(1)+b^(1)=4. It is worth noting that the feature vector x^movie must satisfy all the users while the user vector w^user must satisfy all the movies. This is the source of the name of this approach — all the users collaborate to generate the rating set.

Once the feature vectors and parameters are learned, they can be used to predict how a user might rate an unrated movie. This is shown in the diagram above. The equation is an example of predicting a rating for user one on movie zero.

In this tutorial, we will implement the function cofiCostFunc that computes the collaborative filtering objective function. After implementing the objective function, we will use a TensorFlow custom training loop to learn the parameters for collaborative filtering. The first step is to detail the data set and data structures that will be used in the lab.

Movie ratings dataset

The data set is derived from the MovieLens “ml-latest-small” dataset.
[F. Maxwell Harper and Joseph A. Konstan. 2015. The MovieLens Datasets: History and Context. ACM Transactions on Interactive Intelligent Systems (TiiS) 5, 4: 19:1–19:19. https://doi.org/10.1145/2827872]

The original dataset has 9000 movies rated by 600 users. The dataset has been reduced in size to focus on movies from the years since 2000. This dataset consists of ratings on a scale of 0.5 to 5 in 0.5 step increments. The reduced dataset has n_u=443 users, and n_m=4778 movies.

Below, you will load the movie dataset into the variables Y and R.

The matrix Y (a n_m×n_u matrix) stores the ratings y^(I,j). The matrix R is an binary-valued indicator matrix, where R^(I,j)=1 if user j gave a rating to movie i, and R(I,j)=0 otherwise.

Throughout this part of the exercise, you will also be working with the matrices, X, W and b:

We will start by loading the movie ratings dataset to understand the structure of the data. We will load Y and R with the movie dataset.
We’ll also load X, W, and b with pre-computed values.

Collaborative filtering learning algorithm

cost function

for all i,j where r(i,j) equals 1 could be written as:

Implimentation

Import the libraries:


import numpy as np
import tensorflow as tf
from tensorflow import keras

Load Data:


X, W, b, num_movies, num_features, num_users = load_precalc_params_small()
Y, R = load_ratings_small()

print("Y", Y.shape, "R", R.shape)
print("X", X.shape)
print("W", W.shape)
print("b", b.shape)
print("num_features", num_features)
print("num_movies",   num_movies)
print("num_users",    num_users)

Y (4778, 443) R (4778, 443)

X (4778, 10)

W (443, 10)

b (1, 443)

num_features 10

num_movies 4778

num_users 443


#  From the matrix, we can compute statistics like average rating.
tsmean =  np.mean(Y[0, R[0, :].astype(bool)])
print(f"Average rating for movie 1 : {tsmean:0.3f} / 5" )

Average rating for movie 1 : 3.400 / 5

Cost Function

def cofi_cost_func(X, W, b, Y, R, lambda_):
    """
    Returns the cost for the content-based filtering
    Args:
      X (ndarray (num_movies,num_features)): matrix of item features
      W (ndarray (num_users,num_features)) : matrix of user parameters
      b (ndarray (1, num_users)            : vector of user parameters
      Y (ndarray (num_movies,num_users)    : matrix of user ratings of movies
      R (ndarray (num_movies,num_users)    : matrix, where R(i, j) = 1 if the i-th movies was rated by the j-th user
      lambda_ (float): regularization parameter
    Returns:
      J (float) : Cost
    """
    nm, nu = Y.shape
    J = 0 
    for j in range(nu):
        w = W[j,:]
        b_j = b[0,j]
        
        for i in range(nm):
            x = X[i,:]
            y = Y[i,j]
            r = R[i,j]
            J += np.square(r * (np.dot(w,x) + b_j - y ))
        
    J += lambda_* (np.sum(np.square(W)) + np.sum(np.square(X)))
    
    J = J/2
    return J

Vectorized Implementation

It is important to create a vectorized implementation to compute J, since it will later be called many times during optimization. The linear algebra utilized is not the focus of this series, so the implementation is provided. If you are an expert in linear algebra, feel free to create your version without referencing the code below.

def cofi_cost_func_v(X, W, b, Y, R, lambda_):
    """
    Returns the cost for the content-based filtering
    Vectorized for speed. Uses tensorflow operations to be compatible with custom training loop.
    Args:
      X (ndarray (num_movies,num_features)): matrix of item features
      W (ndarray (num_users,num_features)) : matrix of user parameters
      b (ndarray (1, num_users)            : vector of user parameters
      Y (ndarray (num_movies,num_users)    : matrix of user ratings of movies
      R (ndarray (num_movies,num_users)    : matrix, where R(i, j) = 1 if the i-th movies was rated by the j-th user
      lambda_ (float): regularization parameter
    Returns:
      J (float) : Cost
    """
    j = (tf.linalg.matmul(X, tf.transpose(W)) + b - Y)*R
    J = 0.5 * tf.reduce_sum(j**2) + (lambda_/2) * (tf.reduce_sum(X**2) + tf.reduce_sum(W**2))
    return J

Evaluate Cost Funciton


# Evaluate cost function
J = cofi_cost_func_v(X_r, W_r, b_r, Y_r, R_r, 0);
print(f"Cost: {J:0.2f}")

# Evaluate cost function with regularization 
J = cofi_cost_func_v(X_r, W_r, b_r, Y_r, R_r, 1.5);
print(f"Cost (with regularization): {J:0.2f}")

Cost: 13.67
Cost (with regularization): 28.09

Learning movie recommendations

After we have finished implementing the collaborative filtering cost function, we can start training your algorithm to make movie recommendations for yourself.

movieList, movieList_df = load_Movie_List_pd()

my_ratings = np.zeros(num_movies)          #  Initialize my ratings

# Check the file small_movie_list.csv for id of each movie in our dataset
# For example, Toy Story 3 (2010) has ID 2700, so to rate it "5", you can set
my_ratings[2700] = 5 

#Or suppose you did not enjoy Persuasion (2007), you can set
my_ratings[2609] = 2;

# We have selected a few movies we liked / did not like and the ratings we
# gave are as follows:
my_ratings[929]  = 5   # Lord of the Rings: The Return of the King, The
my_ratings[246]  = 5   # Shrek (2001)
my_ratings[2716] = 3   # Inception
my_ratings[1150] = 5   # Incredibles, The (2004)
my_ratings[382]  = 2   # Amelie (Fabuleux destin d'Amélie Poulain, Le)
my_ratings[366]  = 5   # Harry Potter and the Sorcerer's Stone (a.k.a. Harry Potter and the Philosopher's Stone) (2001)
my_ratings[622]  = 5   # Harry Potter and the Chamber of Secrets (2002)
my_ratings[988]  = 3   # Eternal Sunshine of the Spotless Mind (2004)
my_ratings[2925] = 1   # Louis Theroux: Law & Disorder (2008)
my_ratings[2937] = 1   # Nothing to Declare (Rien à déclarer)
my_ratings[793]  = 5   # Pirates of the Caribbean: The Curse of the Black Pearl (2003)
my_rated = [i for i in range(len(my_ratings)) if my_ratings[i] > 0]

print('\nNew user ratings:\n')
for i in range(len(my_ratings)):
    if my_ratings[i] > 0 :
        print(f'Rated {my_ratings[i]} for  {movieList_df.loc[i,"title"]}');

New user ratings:

Rated 5.0 for Shrek (2001)
Rated 5.0 for Harry Potter and the Sorcerer’s Stone (a.k.a. Harry Potter and the Philosopher’s Stone) (2001)
Rated 2.0 for Amelie (Fabuleux destin d’Amélie Poulain, Le) (2001)
Rated 5.0 for Harry Potter and the Chamber of Secrets (2002)
Rated 5.0 for Pirates of the Caribbean: The Curse of the Black Pearl (2003)
Rated 5.0 for Lord of the Rings: The Return of the King, The (2003)
Rated 3.0 for Eternal Sunshine of the Spotless Mind (2004)
Rated 5.0 for Incredibles, The (2004)
Rated 2.0 for Persuasion (2007)
Rated 5.0 for Toy Story 3 (2010)
Rated 3.0 for Inception (2010)
Rated 1.0 for Louis Theroux: Law & Disorder (2008)
Rated 1.0 for Nothing to Declare (Rien à déclarer) (2010)

Now, let’s add these reviews to Y and R and normalize the ratings.


# Reload ratings and add new ratings
Y, R = load_ratings_small()
Y    = np.c_[my_ratings, Y]
R    = np.c_[(my_ratings != 0).astype(int), R]

# Normalize the Dataset
Ynorm, Ymean = normalizeRatings(Y, R)

Let’s prepare to train the model. Initialize the parameters and select the Adam optimizer.


#  Useful Values
num_movies, num_users = Y.shape
num_features = 100

# Set Initial Parameters (W, X), use tf.Variable to track these variables
tf.random.set_seed(1234) # for consistent results
W = tf.Variable(tf.random.normal((num_users,  num_features),dtype=tf.float64),  name='W')
X = tf.Variable(tf.random.normal((num_movies, num_features),dtype=tf.float64),  name='X')
b = tf.Variable(tf.random.normal((1,          num_users),   dtype=tf.float64),  name='b')

# Instantiate an optimizer.
optimizer = keras.optimizers.Adam(learning_rate=1e-1)

Let’s now train the collaborative filtering model. This will learn the parameters X, W, and b.

The operations involved in learning w, b, and x simultaneously do not fall into the typical ‘layers’ offered in the TensorFlow neural network package.

Recalling the steps of gradient descent.

repeat until convergence:
compute forward pass
compute the derivatives of the loss relative to parameters
update the parameters using the learning rate and the computed derivatives

TensorFlow has the marvellous capability of calculating the derivatives for you. This is shown below. Within the tf.GradientTape() section, operations on Tensorflow Variables are tracked. When tape.gradient() is later called, it will return the gradient of the loss relative to the tracked variables. The gradients can then be applied to the parameters using an optimizer. This is a very brief introduction to a useful feature of TensorFlow and other machine learning frameworks. Further information can be found by investigating "custom training loops" within the framework of interest.


iterations = 200
lambda_ = 1
for iter in range(iterations):
    # Use TensorFlow’s GradientTape
    # to record the operations used to compute the cost 
    with tf.GradientTape() as tape:

        # Compute the cost (forward pass included in cost)
        cost_value = cofi_cost_func_v(X, W, b, Ynorm, R, lambda_)

    # Use the gradient tape to automatically retrieve
    # the gradients of the trainable variables with respect to the loss
    grads = tape.gradient( cost_value, [X,W,b] )

    # Run one step of gradient descent by updating
    # the value of the variables to minimize the loss.
    optimizer.apply_gradients( zip(grads, [X,W,b]) )

    # Log periodically.
    if iter % 20 == 0:
        print(f"Training loss at iteration {iter}: {cost_value:0.1f}")

Training loss at iteration 0: 2321191.3 Training loss at iteration 20: 136168.7 Training loss at iteration 40: 51863.3 Training loss at iteration 60: 24598.8 Training loss at iteration 80: 13630.4 Training loss at iteration 100: 8487.6 Training loss at iteration 120: 5807.7 Training loss at iteration 140: 4311.6 Training loss at iteration 160: 3435.2 Training loss at iteration 180: 2902.1

Recommendations

Below, we compute the ratings for all the movies and users and display the movies that are recommended. These are based on the movies and ratings entered as my_ratings[] above. To predict the rating of movie i for user j, we compute w^(j)⋅x^(i)+b^(j). This can be computed for all ratings using matrix multiplication.

# Make a prediction using trained weights and biases
p = np.matmul(X.numpy(), np.transpose(W.numpy())) + b.numpy()

#restore the mean
pm = p + Ymean

my_predictions = pm[:,0]

# sort predictions
ix = tf.argsort(my_predictions, direction='DESCENDING')

for i in range(17):
    j = ix[i]
    if j not in my_rated:
        print(f'Predicting rating {my_predictions[j]:0.2f} for movie {movieList[j]}')

print('\n\nOriginal vs Predicted ratings:\n')
for i in range(len(my_ratings)):
    if my_ratings[i] > 0:
        print(f'Original {my_ratings[i]}, Predicted {my_predictions[i]:0.2f} for {movieList[i]}')filter=(movieList_df["number of ratings"] > 20)
movieList_df["pred"] = my_predictions
movieList_df = movieList_df.reindex(columns=["pred", "mean rating", "number of ratings", "title"])
movieList_df.loc[ix[:300]].loc[filter].sort_values("mean rating", ascending=False)

filter=(movieList_df["number of ratings"] > 20)
movieList_df["pred"] = my_predictions
movieList_df = movieList_df.reindex(columns=["pred", "mean rating", "number of ratings", "title"])
movieList_df.loc[ix[:300]].loc[filter].sort_values("mean rating", ascending=False)