Movie Recommendation Engine¶

The purpose of this notebook is to implement a recommendation engine end-to-end.

I have briefly touched on the math and the logic behind this factinating domain of Machine Learning in the previous post Bridging Recommender Systems and Dimensionality Reduction, therefore I am going to save time and typing here.

As a brief refresh, there are three main ways to approch a recommender system challenge: content based, collaborative filtering and low rank factorization or latent factor models. The third is the one we are about to discover. The exercise consists in predicting missing movie ratings (on a scale of 1 to 5) in a $m \times u$ (1682 $\times$ 943) matrix of m movies and u users (dataset).

The matrices¶

Here some additional details about the matrices we are going to encounter below. The matrix Y (a $m \times u$ matrix) stores the ratings $y^{(i,j)}$ (from 1 to 5). 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. The objective of the model is to predict movie ratings for the movies that users have not yet rated, that is, the entries with $R_{(i, j)} = 0$. This will allow us to recommend the movies with the highest predicted ratings to the user. Throughout the exercise, we will also be working with the matrices, X and $\Theta$. The i-th row of X corresponds to the feature vector $x^{(i)}$ for the i-th movie, and the j-th row of $\Theta$ corresponds to one parameter vector $\theta^{(j)}$, for the j-th user. Both $x^{(i)}$ and $\theta^{(j)}$ are n-dimensional vectors. For the purposes of this exercise, we will use n = 100, and therefore, $x^{(i)} \in \mathcal{R}^{n}$ and $\theta^{(j)} \in \mathcal{R}^{n}$. Correspondingly, X is a $m \times n$ matrix and $\Theta$ is a $u \times n$ matrix.

The learning algorithm¶

The low rank factorization algorithm in the setting of movie recommendations considers a set of n-dimensional parameter vectors $x^{(1)}, \cdots , x^{(m)}$ and $\theta^{(1)}, \cdots , \theta^{(u)}$, where the model predicts the rating for movie i by user j as $y^{(i,j)} = (\theta^{(j)})^T x^{(i)}$. Given a dataset that consists of a set of ratings produced by some users on some movies, we wish to learn the parameter vectors $x^{(1)}, \cdots , x^{(m)}, \theta^{(1)}, \cdots , \theta^{(u)}$ that produce the best fit (minimizes the squared error).

The regularized cost function...¶

The cost function we want to minimize is

$J(x^{(1)}, \cdots , x^{(m)}, \theta^{(1)}, \cdots , \theta^{(u)}) = \frac{1}{2} \sum_{(i,j):r(i,j)=1}\Big((\theta^{(j)})^T x^{(i)} - y^{(i,j)}\Big)^2 + \Big( \frac{\lambda}{2} \sum_{j=1}^u \sum_{k=1}^n (\theta_k^{(j)})^2 \Big) + \Big( \frac{\lambda}{2} \sum_{i=1}^m \sum_{k=1}^n (x_k^{(i)})^2 \Big)$

whose gradient w.r.t. $\Theta$ and $X$ is respectively equal to

$\frac{\partial J}{\partial x_k^{(i)}} = \sum_{j:r(i,j)=1}\Big((\theta^{(j)})^T x^{(i)} - y^{(i,j)}\Big)\theta_k^{(j)} + \lambda x_k^{(i)}$

$\frac{\partial J}{\partial \theta_k^{(j)}} = \sum_{i:r(i,j)=1}\Big((\theta^{(j)})^T x^{(i)} - y^{(i,j)}\Big)x_k^{(i)} + \lambda \theta_k^{(j)}$

I think we are all set. Now we can start playing around with real code.

In [184]:
import scipy.io
import random
import numpy as np
import pandas as pd
import time
import seaborn as sns
import pickle
import matplotlib.pyplot as plt
plt.style.use('ggplot')
%matplotlib inline


Functions¶

In [185]:
def cofiCostFunction(weights, Y, R, num_users, num_movies, num_features, lamb):
"""
X - num_movies  x num_features matrix of movie features
Theta - num_users  x num_features matrix of user features
Y - num_movies x num_users matrix of user ratings of movies
R - num_movies x num_users matrix, where R(i, j) = 1 if the
i-th movie was rated by the j-th user

X_grad - num_movies x num_features matrix, containing the
partial derivatives w.r.t. to each element of X
Theta_grad - num_users x num_features matrix, containing the
partial derivatives w.r.t. to each element of Theta

"""

X = weights[0:num_movies * (num_features)].reshape(num_movies, num_features) #contains n features per movie
Theta = weights[num_movies * (num_features):].reshape(num_users, num_features) #contains n features per user

cost = 0.5 * (((R * X.dot(Theta.T)) - Y)**2).sum() + lamb/2 * (Theta**2).sum() + lamb/2 * (X**2).sum()

for m in range(num_movies):
#get all users who have rated movie m; this is a boolean array with size (1, users_watched_movie_m)
users_watched_movie_m = R[m,:] == 1

#from Theta containing n features per user, get all the features for all users_watched_movie_m
#size (users_watched_movie_m, n_features)
Theta_m = Theta[users_watched_movie_m,:]
#from Y containing (movies, users), get all the ratings for all users_watched_movie_m
#size (1, users_watched_movie_m)
Y_m = Y[m, users_watched_movie_m].reshape((1, users_watched_movie_m.sum()))

#X[m,:] --> (1, n_features)
#X[m,:].dot(Theta_m.T) --> (1, users_watched_movie_m)
#X[m,:].dot(Theta_m.T) - Y_m --> (1, users_watched_movie_m)
#(X[m,:].dot(Theta_m.T) - Y_m).dot(Theta_m) --> (1, n_features)
X_grad[m, :] = (X[m,:].dot(Theta_m.T) - Y_m).dot(Theta_m) + lamb * X[m,:]

for u in range(num_users):
#get all movies rated by user u
movies_rated_by_user_u = R[:,u] == 1

#from X containing n features per movie, get all the features for all movies_rated_by_user_u
#size (movies_rated_by_user_u, n_features)
X_u = X[movies_rated_by_user_u,:]
#from Y containing (movies, users), get all the ratings for all movies_rated_by_user_u
#size (movies_rated_by_user_u, 1)
Y_u = Y[movies_rated_by_user_u, u].reshape((movies_rated_by_user_u.sum(), 1))
Theta_u = Theta[u, :].reshape((1, num_features))

#X_u --> (movies_rated_by_user_u, n_features)
#X_u.dot(Theta_u.T) --> (movies_rated_by_user_u, 1)
#X_u.dot(Theta_u.T) - Y_u --> (movies_rated_by_user_u, 1)
#(X_u.dot(Theta_u.T) - Y_u).T.dot(X_u) --> (1, n_features)
Theta_grad[u, :] = (X_u.dot(Theta_u.T) - Y_u).T.dot(X_u) + lamb * Theta_u

d = {}
with open('C:\\Users\\pochetti\Machine_Learning\\mlclass-ex8-005\\mlclass-ex8\movie_ids.txt') as f:
for line in f:
(key, val) = line.split(' ', 1)
d[int(key)] = val.rstrip()
return d

def normalizeRatings(Y, R):
"""returns normalized Y (and its mean) so that each movie
has a rating of 0 on average"""
movie_av = np.zeros((Y.shape[0], 1))

for m in range(Y.shape[0]):
to_be_averaged = R[m,:] == 1
movie_av[m] = Y[m, to_be_averaged].mean()

return Y - movie_av, movie_av


Running the model on pre-trained weights¶

This is just to check that the cost function works as expected. The printed value is a benchmark calculated on smaller matrices and offered during Andrew Ng's course, and my result perfectly matches his. So, good news, we can go ahead.

In [186]:
data = scipy.io.loadmat('C:\\Users\\pochetti\Machine_Learning\\mlclass-ex8-005\\mlclass-ex8\ex8_movies.mat')
R = data['R']
Y = data['Y']

Theta = params['Theta']
X = params['X']

#  Reduce the data set size just for checking purposes
num_users = 4
num_movies = 5
num_features = 3
X = X[0:num_movies, 0:num_features]
Theta = Theta[0:num_users, 0:num_features]
Y = Y[0:num_movies, 0:num_users]
R = R[0:num_movies, 0:num_users]

weights = np.vstack((X.reshape(X.size, 1), Theta.reshape(Theta.size, 1)))

#  Evaluate cost function
cost, gradient = cofiCostFunction(weights, Y, R, num_users, num_movies, num_features, 1.5)
print('Regularized Cost @lambda 10:', cost)

Regularized Cost @lambda 10: 31.34405624427422


Implementing the Low Rank Factorization Algorithm¶

Loading the whole dataset (Y and R) and "visualizing" the $movies \times users$ (1682 $\times$ 943) Y matrix. imshow nicely allows us to highlight the 1 to 5 ratings and the missing entries. As you can see the matrix is (as expected) pretty sparse.

In [187]:
data = scipy.io.loadmat('C:\\Users\\pochetti\Machine_Learning\\mlclass-ex8-005\\mlclass-ex8\ex8_movies.mat')
R = data['R']
Y = data['Y']

plt.figure(figsize=(10, 8))
plt.imshow(Y, extent=[0, 5, 0, 5], cmap='Blues')
plt.colorbar()
plt.title('Movies x Users matrix')
plt.show()

In [188]:
# loading the movie list with ratings.
# this is in order to input my personal reviews

my_ratings = np.zeros((Y.shape[0], 1))
my_ratings[0] = 3
my_ratings[1] = 4
my_ratings[7] = 1
my_ratings[10]= 5
my_ratings[21]= 5
my_ratings[26]= 4
my_ratings[27]= 5
my_ratings[34]= 1
my_ratings[55]= 5
my_ratings[1234]= 3
my_ratings[1167]= 2
my_ratings[1088]= 3
my_ratings[1001]= 2
my_ratings[819]= 1
my_ratings[688]= 5
my_ratings[627]= 4
my_ratings[526]= 3
my_ratings[456]= 1

print("Let's rate some movies I have watched...\nThis is important to get future recommendations!\n")

for movie in range(my_ratings.shape[0]):
if my_ratings[movie] > 0:
print('Movie: "%s". My rating: %d' % (movieList[movie+1], my_ratings[movie]))

Let's rate some movies I have watched...
This is important to get future recommendations!

Movie: "Toy Story (1995)". My rating: 3
Movie: "GoldenEye (1995)". My rating: 4
Movie: "Babe (1995)". My rating: 1
Movie: "Seven (Se7en) (1995)". My rating: 5
Movie: "Braveheart (1995)". My rating: 5
Movie: "Bad Boys (1995)". My rating: 4
Movie: "Apollo 13 (1995)". My rating: 5
Movie: "Free Willy 2: The Adventure Home (1995)". My rating: 1
Movie: "Pulp Fiction (1994)". My rating: 5
Movie: "Free Willy 3: The Rescue (1997)". My rating: 1
Movie: "Gandhi (1982)". My rating: 3
Movie: "Sleepers (1996)". My rating: 4
Movie: "Jackal, The (1997)". My rating: 5
Movie: "Space Jam (1996)". My rating: 1
Movie: "Pest, The (1997)". My rating: 2
Movie: "Speed 2: Cruise Control (1997)". My rating: 3
Movie: "Little Buddha (1993)". My rating: 2
Movie: "Big Bang Theory, The (1994)". My rating: 3

In [189]:
#  Y is a 1682x943 matrix, containing ratings (1-5) of 1682 movies on
#  943 users
#
#  R is a 1682x943 matrix, where R(i,j) = 1 if and only if user j gave a
#  rating to movie i
R = data['R']
Y = data['Y']

#  Add our own ratings to the data matrix
Y = np.hstack((Y, my_ratings))
R = np.hstack((R, np.where(my_ratings != 0, 1, 0)))

#  Normalize Ratings
Ynorm, Ymean = normalizeRatings(Y, R)

#  Useful Values
num_users = Y.shape[1]
num_movies = Y.shape[0]
num_features = 100

####################################################
# Guessing Initial Parameters (Theta, X)
####################################################
# This is done by performing an SVD on Y,
# then reconstructing Theta by multiplying s and V
# and then truncating X and Theta's columns to account for the right number of features.
# It is generally a good idea to initialize the parameters with SVD
# as this decomposition guarantees by definition the lowest possible
# reconstruction erro of Y, so it helps us going in the right direction.
# At the end of the day we have to produce X and Theta so that the RMSE
# between Y and Theta x X over the non null values of Y is as small as possible.
# The problem with SVD is that it assumes all the missing values are 0, hence the
# decomposition will yield ineherently wrong results. This is why we need
# an additional optimization on top of it.
X, s, V = np.linalg.svd(Ynorm, full_matrices=False)

features = s.shape[0]
S = np.zeros((features, features))
S[:features, :features] = np.diag(s)

Theta = np.dot(S, V)[:,:num_features]
X = X[:,:num_features]

initial_weights = np.vstack((X.reshape(X.size, 1), Theta.reshape(Theta.size, 1)))

print('Y dimensions:', Y.shape)
print('R dimensions:', R.shape)
print('X dimensions:', X.shape)
print('Theta dimensions:', Theta.shape)

Y dimensions: (1682, 944)
R dimensions: (1682, 944)
X dimensions: (1682, 100)
Theta dimensions: (944, 100)

In [190]:
# Set Regularization
lamb = 10
start = time.time()

opt_results = scipy.optimize.minimize(fun=cofiCostFunction, x0=initial_weights.squeeze(),
args=(Ynorm, R, num_users, num_movies, num_features, lamb),
jac=True, method='TNC', options={'disp':True})
end = time.time()
opt_results

Out[190]:
     fun: 7392279.00696755
jac: array([ -1.74412433e-06,  -8.55478169e-07,  -2.55634779e-06, ...,
9.30204157e-08,  -1.35419401e-08,   4.25925383e-07])
message: 'Converged (|f_n-f_(n-1)| ~= 0)'
nfev: 1164
nit: 55
status: 1
success: True
x: array([-0.3450651 , -0.33919942, -0.22434563, ...,  0.05467321,
0.08744782, -0.10401336])
In [191]:
total = end-start
print("Training executed in", round(total/60, 2), "minutes")

Training executed in 4.86 minutes


After training the model, we can now make recommendations by computing the predictions matrix and then display the top movies I really cannot miss!

In [192]:
# Unfold the returned weights into X and Theta
X = opt_results.x[0:num_movies * (num_features)].reshape(num_movies, num_features)
Theta = opt_results.x[num_movies * (num_features):].reshape(num_users, num_features)

predicted = X.dot(Theta.T)
mine = predicted[:,num_users - 1] + Ymean.flatten()
top = np.argsort(mine)[::-1][:20]


Top 20 movies I never watched and I cannot miss!¶

In [193]:
print('Top Recommended Movies\n')
for rank, index in enumerate(top):
if index not in [movie for (movie, rating) in enumerate(my_ratings) if rating > 0]:
print('%d: "%s". Predicted rating: %2.2f' % (rank+1, movieList[index+1], mine[index]))

Top Recommended Movies

1: "Star Kid (1997)". Predicted rating: 5.00
2: "Aiqing wansui (1994)". Predicted rating: 5.00
3: "Someone Else's America (1995)". Predicted rating: 5.00
4: "They Made Me a Criminal (1939)". Predicted rating: 5.00
5: "Santa with Muscles (1996)". Predicted rating: 5.00
6: "Prefontaine (1997)". Predicted rating: 5.00
7: "Entertaining Angels: The Dorothy Day Story (1996)". Predicted rating: 5.00
8: "Marlene Dietrich: Shadow and Light (1996)". Predicted rating: 5.00
9: "Saint of Fort Washington, The (1993)". Predicted rating: 5.00
10: "Great Day in Harlem, A (1994)". Predicted rating: 5.00
11: "Pather Panchali (1955)". Predicted rating: 4.64
12: "Shawshank Redemption, The (1994)". Predicted rating: 4.60
13: "Titanic (1997)". Predicted rating: 4.54
14: "Maya Lin: A Strong Clear Vision (1994)". Predicted rating: 4.52
15: "Usual Suspects, The (1995)". Predicted rating: 4.50
16: "Everest (1998)". Predicted rating: 4.50
18: "Anna (1996)". Predicted rating: 4.49
19: "Some Mother's Son (1996)". Predicted rating: 4.49
20: "Schindler's List (1993)". Predicted rating: 4.48