Recommender Systems

Introduction

Introduction

Today, we look at a topic that has become enormously important: recommender systems.

In Part I, we will:

  • Define recommender systems
  • Review the challenges they pose
  • Discuss two classic methods:
    • Collaborative Filtering
    • Matrix Factorization

In Part II, we will:

  • Delve into more recent deep learning methods.

This section draws heavily on

  • These slides by Alex Smola
  • Matrix Factorization Techniques for Recommender Systems, (Koren et al. 2009)
  • Collaborative Filtering with Temporal Dynamics, (Koren 2009)

What are Recommender Systems?

The concept of recommender systems emerged in the late 1990s / early 2000s as social life moved online:

  • online purchasing and commerce
  • online discussions and ratings
  • social information sharing

In these systems the amount of content was exploding and users were having a hard time finding things they were interested in.

Users wanted recommendations.

Over time, the problem has only gotten worse:

An enormous need has emerged for systems to help sort through products, services, and content items.

This often goes by the term personalization.

Some examples (as of Fall 2024):

  • Movie recommendation (Netflix ~6.5K movies and shows, YouTube ~14B videos)
  • Related product recommendation (Amazon ~600M products)
  • Web page ranking (Google >>100B pages)
  • Social content filtering (Facebook, Twitter)
  • Services (Airbnb, Uber, TripAdvisor)
  • News content recommendation (Apple News)
  • Priority inbox & spam filtering (Gmail)
  • Online dating (Match.com)

A more formal view:

  • User - requests content
  • Objects - instances of content
  • Context - ratings, purchases, views, device, location, time, history
  • Interface - browser, mobile

Inferring Preferences

Unfortunately, users generally have a hard time explaining what types of content they prefer.

Some early systems worked by interviewing users to ask what they liked. Those systems did not work very well.

A very interesting article about the earliest personalization systems is User Modeling via Stereotypes by Elaine Rich, dating from 1979.

Instead, modern systems work by capturing user’s opinions about specific items.

This can be done actively:

  • When a user is asked to rate a movie, product, or experience,

or it can be done passively:

  • By noting which items a user chooses to purchase (for example).

Challenges

  • The biggest issue is scalability: typical data for this problem is huge.
    • Millions of objects
    • 100s of millions of users
  • Changing user base
  • Changing inventory (movies, stories, goods)
  • Available features
  • Imbalanced dataset
    • User activity / item reviews are power law distributed

This data is a subset of the data presented in: “From amateurs to connoisseurs: modeling the evolution of user expertise through online reviews,” by J. McAuley and J. Leskovec. WWW, 2013

Example

Let’s look at a dataset for testing recommender systems consisting of Amazon movie reviews:

We’ll download a compressed pickle file containing the data if it is not already present.

Code 
# This is a 647 MB file, delete it after use
import gdown
url = "https://drive.google.com/uc?id=14GakA7oOjbQp7nxcGApI86WlP3GrYTZI"
pickle_output = "train.pkl.gz"

import os.path
if not os.path.exists(pickle_output):
    gdown.download(url, pickle_output)
Downloading...
From (original): https://drive.google.com/uc?id=14GakA7oOjbQp7nxcGApI86WlP3GrYTZI
From (redirected): https://drive.google.com/uc?id=14GakA7oOjbQp7nxcGApI86WlP3GrYTZI&confirm=t&uuid=d7b88cec-0b66-4122-a380-81616a23d215
To: /Users/tgardos/Source/courses/ds701/DS701-Course-Notes/train.pkl.gz

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We’ll load the data into a pandas DataFrame.

Code 
import gzip
import time

start_time = time.time()
with gzip.open(pickle_output, 'rb') as f:
    df = pd.read_pickle(f)
elapsed_time = time.time() - start_time
print(f"Elapsed read time: {elapsed_time:.2f} seconds")
Elapsed read time: 2.73 seconds

Run df.info() to see the column names and data types.

Code 
df.info()
<class 'pandas.DataFrame'>
RangeIndex: 1697533 entries, 0 to 1697532
Data columns (total 9 columns):
 #   Column                  Dtype  
---  ------                  -----  
 0   Id                      int64  
 1   ProductId               object 
 2   UserId                  object 
 3   HelpfulnessNumerator    int64  
 4   HelpfulnessDenominator  int64  
 5   Score                   float64
 6   Time                    int64  
 7   Summary                 object 
 8   Text                    object 
dtypes: float64(1), int64(4), object(4)
memory usage: 116.6+ MB

where

  • HelpfulnessNumerator: The number of users who found the review helpful (the “yes” votes)
  • HelpfulnessDenominator: The total number of users who voted on whether the review was helpful (both “yes” and “no” votes)

Now we can count the number of users and movies:

Code 
from IPython.display import display, Markdown

n_users = df["UserId"].unique().shape[0]
n_movies = df["ProductId"].unique().shape[0]
n_reviews = len(df)
display(Markdown(f'There are:\n'))
display(Markdown(f'* {n_reviews:,} reviews\n* {n_movies:,} movies\n* {n_users:,} users'))

display(Markdown(f'There are {n_users * n_movies:,} potential reviews, meaning sparsity of {(n_reviews/(n_users * n_movies)):0.4%}'))

There are:

  • 1,697,533 reviews
  • 50,052 movies
  • 123,960 users

There are 6,204,445,920 potential reviews, meaning sparsity of 0.0274%

where

\[ \text{sparsity} = \frac{\text{\# of reviews}}{\text{\# of users} \times \text{\# of movies}} = \frac{\text{\# of reviews}}{\text{\# of potential reviews}} \]

Reviews are Sparse

Only 0.02% of the reviews are available – 99.98% of the reviews are missing.

Code 
display(Markdown(f'There are on average {n_reviews/n_movies:0.1f} reviews per movie' +
     f' and {n_reviews/n_users:0.1f} reviews per user'))

There are on average 33.9 reviews per movie and 13.7 reviews per user

Sparseness is Skewed

Although on average a movie receives 34 reviews, almost all movies have even fewer reviews.

Code 
plt.figure(figsize=(10, 4))  # Set the figure size
reviews_per_movie = df.groupby('ProductId').count()['Id'].values
frac_below_mean = np.sum(reviews_per_movie < (n_reviews/n_movies))/len(reviews_per_movie)
plt.plot(sorted(reviews_per_movie, reverse=True), '.-')
xmin, xmax, ymin, ymax = plt.axis()
plt.hlines(n_reviews/n_movies, xmin, xmax, 'r', lw = 3)
plt.ylabel('Number of Ratings', fontsize = 14)
plt.xlabel('Movie', fontsize = 14)
plt.legend(['Number of Ratings', 'Average Number of Ratings'], fontsize = 14)
plt.title(f'Amazon Movie Reviews\nNumber of Ratings Per Movie\n' +
          f'{frac_below_mean:0.0%} of Movies Below Average', fontsize = 16);

Likewise, although the average user writes 14 reviews, almost all users write even fewer reviews.

Code 
plt.figure(figsize=(10, 4))  # Set the figure size
reviews_per_user = df.groupby('UserId').count()['Id'].values
frac_below_mean = np.sum(reviews_per_user < (n_reviews/n_users))/len(reviews_per_user)
plt.plot(sorted(reviews_per_user, reverse=True), '.-')
xmin, xmax, ymin, ymax = plt.axis()
plt.hlines(n_reviews/n_users, xmin, xmax, 'r', lw = 3)
plt.ylabel('Number of Ratings', fontsize = 14)
plt.xlabel('User', fontsize = 14)
plt.legend(['Number of Ratings', 'Average Number of Ratings'], fontsize = 14)
plt.title(f'Amazon Movie Reviews\nNumber of Ratings Per User\n' +
          f'{frac_below_mean:0.0%} of Users Below Average', fontsize = 16);

Objective Function

Ultimately, our goal is to predict the rating that a user would give to an item.

For that, we need to define a loss or objective function.

A typical objective function is root mean square error (RMSE)

\[ \text{RMSE} = \sqrt{\frac{1}{|S|} \sum_{(i,u)\in S} (\hat{r}_{ui} - r_{ui})^2}, \]

where

  • \(r_{ui}\) is the rating that user \(u\) gives to item \(i\), and
  • \(S\) is the subset of items that have ratings.

OK, now we know the problem and the data available. How can we address the problem?

The earliest method developed is called collaborative filtering.

Colaborative Filtering

Collaborative Filtering

The central idea of collaborative filtering is that the set of known recommendations can be considered to be a bipartite graph.

The nodes of the bipartite graph are users (\(U\)) and items (\(V\)).

Each edge corresponds to a known rating \(r_{ui}.\)

Then recommendations are formed by traversing or processing the bipartite graph.

There are at least two ways this graph can be used.

Two ways to form a rating for item \((u, i)\):

  1. Using user-user similarity:
    • look at users that have similar item preferences to user \(u\)
    • look at how those users rated item \(i\)

⟹ Good for many users, fewer items.
(e.g., NetFix had ~280M subscribers, ~6.5K movies/shows)

  1. Using item-item similarity:
    • look at other items that have been liked by similar users as item \(i\)
    • look at how user \(u\) rated those items

⟹ Good for many items, fewer users
(e.g. Amazon had ~300M accounts, ~600M products)

Item-Item CF

For item-item similarity, we’ll look at item-item Collaborative Filtering (CF).

The questions are:

  • How do we judge “similarity” of items?
  • How do we form a predicted rating?

Here is another view of the ratings graph, this time as a matrix that includes missing entries:

Let’s say we want to predict the value of this unknown rating:

We’ll consider two other items, namely items 3 and 6 (for example).

Note that we are only interested in items that this user has rated.

We will discuss strategies for assessing similarity shortly.

How did we choose these two items?

We used \(k\)-nearest neighbors. Here \(k\) = 2.

For now, let’s just say we determine the similarities as:

\[ s_{13} = 0.2 \]

\[ s_{16} = 0.3 \]

These similarity scores tell us how much weight to put on the rating of the other items.

So we can form a prediction of \(\hat{r}_{15}\) as:

\[ \hat{r}_{15} = \frac{s_{13} \cdot r_{35} + s_{16} \cdot r_{65}}{s_{13} + s_{16}} = \frac{0.2 \cdot 2 + 0.3 \cdot 3}{0.2 + 0.3} = 2.6 \]

Similarity

How should we assess similarity of items?

A reasonable approach is to consider items similar if their ratings are correlated:, i.e.,

\[\rho(X,Y) = \frac{E\left[(X-\mu_X)(Y-\mu_Y)\right]}{\sigma_X \sigma_Y}.\]

However, note that two items will not have ratings in the same positions.

So we want to compute correlation only over the users who rated both the items.

Example

Let’s put the ratings in python lists:

Code 
import numpy as np
from IPython.display import display, Markdown

ratings_item_i = [1, np.nan, np.nan, 5, 5, 3, np.nan, np.nan, np.nan, 4, 2, np.nan, np.nan, np.nan, np.nan, 4, np.nan, 5, 4, 1, np.nan]
ratings_item_j = [np.nan, np.nan, 4, 2, 5, np.nan, np.nan, 1, 2, 5, np.nan, np.nan, 2, np.nan, np.nan, 3, np.nan, np.nan, np.nan, 5, 4]

display(Markdown(f'Ratings for item $i$:\n\n{ratings_item_i}'))
display(Markdown(f'Ratings for item $j$:\n\n{ratings_item_j}'))

Ratings for item \(i\):

[1, nan, nan, 5, 5, 3, nan, nan, nan, 4, 2, nan, nan, nan, nan, 4, nan, 5, 4, 1, nan]

Ratings for item \(j\):

[nan, nan, 4, 2, 5, nan, nan, 1, 2, 5, nan, nan, 2, nan, nan, 3, nan, nan, nan, 5, 4]

Example, continued

Let’s drop the non-common ratings:

Code 
# Create new lists where only numbers are kept that are not np.nan in both lists
filtered_ratings_item_i = [rating_i for rating_i, rating_j in zip(ratings_item_i, ratings_item_j) if not np.isnan(rating_i) and not np.isnan(rating_j)]
filtered_ratings_item_j = [rating_j for rating_i, rating_j in zip(ratings_item_i, ratings_item_j) if not np.isnan(rating_i) and not np.isnan(rating_j)]

display(Markdown(f'Common ratings for item $i$: {filtered_ratings_item_i}'))
display(Markdown(f'Common ratings for item $j$: {filtered_ratings_item_j}'))

Common ratings for item \(i\): [5, 5, 4, 4, 1]

Common ratings for item \(j\): [2, 5, 5, 3, 5]

Example, continued

Code 
display(Markdown(f'Common ratings for item $i$: {filtered_ratings_item_i}'))
display(Markdown(f'Common ratings for item $j$: {filtered_ratings_item_j}'))

Common ratings for item \(i\): [5, 5, 4, 4, 1]

Common ratings for item \(j\): [2, 5, 5, 3, 5]

Now we can compute the Pearson correlation coefficient:

Code 
rho = np.corrcoef(filtered_ratings_item_i, filtered_ratings_item_j)[0,1]
display(Markdown(f'Pearson correlation coefficient: {rho:0.2f}'))

Pearson correlation coefficient: -0.43

Which is a moderate negative correlation, meaning that as item \(i\) gets rated higher, item \(j\) gets rated lower.

Pearson Correlation Coefficient

The Pearson correlation coefficient, often denoted as \(r\), is a measure of the linear correlation between two variables \(X\) and \(Y\). It quantifies the degree to which a linear relationship exists between the variables. The value of \(r\) ranges from -1 to 1, where:

  • \(r = 1\) indicates a perfect positive linear relationship,
  • \(r = -1\) indicates a perfect negative linear relationship,
  • \(r = 0\) indicates no linear relationship.

The formula for the Pearson correlation coefficient is:

\[ r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}} \]

Where: - \(X_i\) and \(Y_i\) are the individual sample points, - \(\bar{X}\) and \(\bar{Y}\) are the means of the \(X\) and \(Y\) samples, respectively.

The Pearson correlation coefficient is sensitive to outliers and assumes that the relationship between the variables is linear and that the data is normally distributed.

Similarity for Binary Data

In some cases we will need to work with binary \(r_{ui}\).

For example, purchase histories on an e-commerce site, or clicks on an ad.

In this case, an appropriate replacement for Pearson \(r\) is the Jaccard similarity coefficient or Intersection over Union.

\[ J_{Sim}(\mathbf{x}, \mathbf{y}) = \frac{|\mathbf{x} \cap \mathbf{y}|}{|\mathbf{x} \cup \mathbf{y}|}. \]

See the lecture on similarity measures.

Improving CF in the presence of bias

One problem with the story so far arises due to bias.

  • Some items are significantly higher or lower rated
  • Some users rate substantially higher or lower in general

These properties interfere with similarity assessment.

Bias correction is crucial for CF recommender systems.

We need to include

  • Per-user offset
  • Per-item offset
  • Global offset

Representing biases

Hence we need to form a per-item bias of:

\[ b_{ui} = \mu + \alpha_u + \beta_i \]

where

  • \(b_{ui}\) is the bias of user \(u\) for item \(i\).
  • \(\mu\) is the global average rating across all items and users.
  • \(\alpha_u\) is the offset of user \(u\) and
  • \(\beta_i\) is the offset of item \(i\).

If we gather all these elements together we can form:

  • \(\boldsymbol{\alpha}\) an \(n\times 1\) vector of per-user offsets, and
  • \(\boldsymbol{\beta}\) an \(m\times 1\) vector of per-item offsets.

Estimating biases

How can we estimate the parameters \(\boldsymbol{\alpha}\), \(\boldsymbol{\beta}\), and \(\mu\)?

Let’s assume for a minute that we had a fully-dense matrix of ratings \(R\).

Recall that each of the \(m\) rows of \(R\) represents an item and each of the \(n\) columns represents a user.

Estimating biases, continued

One way to do this is to minimize the squared error between the ratings and the biases:

\[ \min_{\boldsymbol{\alpha},\boldsymbol{\beta},\mu} \Vert R - \mathbf{1}\boldsymbol{\alpha}^T + \boldsymbol{\beta}\mathbf{1}^T + \mu1\Vert^2 + \lambda(\Vert\boldsymbol{\alpha}\Vert^2 + \Vert\boldsymbol{\beta}\Vert^2). \]

and include a regularization term to minimize the magnitude of the biases.

  • Here, bold-faced \(\mathbf{1}\) represents appropriately sized vectors of ones, and non-boldfaced \(1\) is an \(m\times n\) matrix of ones.

  • So \(\mathbf{1}\boldsymbol{\alpha}^T\) is an \(m\times n\) matrix where each row is the bias for a user.

  • Similarly, \(\boldsymbol{\beta}\mathbf{1}^T\) is an \(m\times n\) matrix where each column is the bias for an item.

  • And \(\mu1\) is an \(m\times n\) matrix where each element is \(\mu\).

While this is not a simple ordinary least squares problem, there is a strategy for solving it.

Assume we hold \(\beta\mathbf{1}^T\) and \(\mu1\) constant.

Then the remaining problem is

\[ \min_{\alpha} \Vert R - \mathbf{1}\alpha^T \Vert^2 + \lambda \Vert\alpha\Vert^2, \]

which (for each column of \(R\)) is a standard regularized least squares problem solved via Ridge regression.

Aside: Ridge Regression

Ridge Regression is a regularized least squares method that adds a penalty term to prevent overfitting.

Standard Least Squares: \[ \min_{\boldsymbol{\beta}} \Vert X\boldsymbol{\beta} - \mathbf{y}\Vert_2^2 \]

Ridge Regression: \[ \min_{\boldsymbol{\beta}} \Vert X\boldsymbol{\beta} - \mathbf{y}\Vert_2^2 + c\Vert\boldsymbol{\beta}\Vert_2^2 \]

The penalty term \(c\Vert\boldsymbol{\beta}\Vert_2^2\) shrinks the coefficients \(\boldsymbol{\beta}\) towards zero.

Why Ridge Regression?

  • Addresses multicollinearity (when features are highly correlated)
  • Prevents coefficient magnitudes from becoming too large
  • Always has a unique solution: \(\hat{\boldsymbol{\beta}} = (X^TX + cI)^{-1}X^T\mathbf{y}\)

The hyperparameter \(c\):

  • When \(c = 0\): ordinary least squares
  • When \(c \rightarrow \infty\): all coefficients shrink to zero
  • Choose \(c\) via cross-validation to balance fitting vs. regularization

Back to Solving the problem

This sort of problem is called jointly convex in that it is convex in each of the variables \(\alpha\), \(\beta\), and \(\mu\).

The strategy for solving is:

  1. Hold \(\alpha\) and \(\beta\) constant, solve for \(\mu\).
  2. Hold \(\alpha\) and \(\mu\) constant, solve for \(\beta\).
  3. Hold \(\beta\) and \(\mu\) constant, solve for \(\alpha\).

Each of the three steps will reduce the overall error. As a result, we iterate over them until convergence.

The last issue is that the matrix \(R\) is not dense - in reality we only have a small subset of its entries.

We simply need to adapt the least-squares solution to only consider the entries in \(R\) that we know.

As a result, the actual calculation is as follows…

Step 1:

\[ \mu = \frac{\sum_{(u, i) \in R} (r_{ui} - \alpha_u - \beta_i)}{|R|} \]

Step 2:

\[ \alpha_u = \frac{\sum_{i \in R(u)}(r_{ui} - \mu - \beta_i)}{\lambda + |R(u)|} \]

Step 3:

\[ \beta_i = \frac{\sum_{u \in R(i)}(r_{ui} - \mu - \alpha_u)}{\lambda + |R(i)|} \]

Step 4: If not converged, go to Step 1.

Here \(i \in R(u)\) means the set of items rated by user \(u\) and \(u \in R(i)\) means the set of users who have rated item \(i\) and \(|R(u)|\) is the number of ratings.

Now that we have learned the biases, we can do a better job of estimating correlation:

\[ \hat{\rho}_{ij} = \frac{\sum_{u\in U(i,j)}(r_{ui} - b_{ui})(r_{uj}-b_{uj})} {\sqrt{\sum_{u\in U(i,j)}(r_{ui} - b_{ui})^2\sum_{u\in U(i,j)}(r_{uj}-b_{uj})^2}}, \]

where

  • \(b_{ui} = \mu + \alpha_u + \beta_i\), and
  • \(U(i,j)\) are the users who have rated both \(i\) and \(j\).

And using biases we can also do a better job of estimating ratings:

\[ \hat{r}_{ui} = b_{ui} + \frac{\sum_{j \in n_k(i, u)} s_{ij}(r_{uj} - b_{uj})}{\sum_{j \in n_k(i, u)} s_{ij}}, \]

where

  • \(b_{ui} = \mu + \alpha_u + \beta_i\),
  • \(n_k(i, u)\) are the \(k\) nearest neighbors to \(i\) that were rated by user \(u\) and
  • \(s_{ij}\) is the similarity between items \(i\) and \(j\), estimated as above.

Negative Similarity Scores?

When using correlation coefficient as the similarity score, negative values have important implications for the weighted calculation.

Looking at the weighted rating formula:

\[ \hat{r}_{ui} = b_{ui} + \frac{\sum_{j \in n_k(i, u)} s_{ij}(r_{uj} - b_{uj})}{\sum_{j \in n_k(i, u)} s_{ij}}, \]

How Negative Similarity Scores Work

When \(s_{ij}\) is negative (negative correlation), here’s what happens:

  1. Numerator: The term \(s_{ij}(r_{uj} - b_{uj})\) becomes negative when \(s_{ij} < 0\). This means:

    • If user \(u\) rated item \(j\) above its bias (\(r_{uj} - b_{uj} > 0\)), the negative similarity will push the prediction for item \(i\) down
    • If user \(u\) rated item \(j\) below its bias (\(r_{uj} - b_{uj} < 0\)), the negative similarity will push the prediction for item \(i\) up

    This makes intuitive sense: if items \(i\) and \(j\) are negatively correlated, then liking one means you’re likely to dislike the other.

  2. Denominator: When \(s_{ij}\) is negative, it contributes a negative value to the sum in the denominator. This can potentially lead to issues:

    • If all similarities are negative, the denominator becomes negative
    • If there’s a mix of positive and negative similarities, they might partially cancel out, potentially making the denominator close to zero or even zero

Common Solutions

To address these issues, practitioners typically use one of these approaches:

  1. Use only positive similarities: Filter out items with negative correlation (only use \(k\)-NN where \(s_{ij} > 0\))

  2. Use absolute values in denominator: \(\displaystyle\hat{r}_{ui} = b_{ui} + \frac{\sum_{j \in n_k(i, u)} s_{ij}(r_{uj} - b_{uj})}{\sum_{j \in n_k(i, u)} |s_{ij}|}\)

  3. Shift and scale similarities: Transform correlation values from \([-1, 1]\) to \([0, 1]\) using \(s'_{ij} = \frac{s_{ij} + 1}{2}\)

The first approach (filtering to positive similarities) is most common because negatively correlated items are typically less informative for prediction than positively correlated items in most domains.

Assessing CF

This completes the high level view of CF.

Working with user-user similarities is analogous.

Strengths:

  • Essentially no training.
    • The reliance on \(k\)-nearest neighbors helps in this respect.
  • Easy to update with new users, items, and ratings.
  • Explainable:
    • “We recommend Minority Report because you liked Blade Runner and Total Recall.”

Weaknesses:

  • Accuracy can be a problem – resulting in poor recommendations
  • Scalability can be a problem – compute grows (think \(k\)-NN)

Matrix Factorization Approaches

Matrix Factorization

Note that standard CF forces us to consider similarity among items, or among users, but does not take into account both.

Can we use both kinds of similarity simultaneously?

We can’t use both the rows and columns of the ratings matrix \(R\) at the same time – the user and item vectors live in different vector spaces.

Figure 1

(Koren et al. 2009)

What we could try to do is find a single vector space in which we represent both users and items, along with a similarity function, such that:

  • users who have similar item ratings are similar in the vector space
  • items who have similar user ratings are similar in the vector space
  • when a given user highly rates a given item, that user and item are similar in the vector space.

We saw this idea previously, in an SVD lecture.

This new vector space is called a latent space, and the user and item representations are called latent vectors.

This notion of a shared latent space is also central to deep learning recommender approaches (Naumov et al. 2019) we will look at later.

Now, however, we are working with a matrix which is only partially observed. That is, we only know some of the entries in the ratings matrix.

Nonetheless, we can imagine a situation like this:

where we decompose the ratings matrix \(R\) into two matrices.

We want the product of the two matrices to be as close as possible to the known values of the ratings matrix.

What this setup implies is that our similarity function is the inner product.

Which means that to predict an unknown rating, we take the inner product of latent vectors:

Taking, for example, the 2nd row of “items” and the 5th row of “users”…

We have

\[ (-0.5 \cdot -2)+(0.6 \cdot 0.3)+(0.5 \cdot 2.4) = 2.43, \]

so:

Solving Matrix Factorization

Notice that in this case we’ve decided that the factorization should be rank 3, i.e., low-rank.

So we want something like an SVD.

(Recall that SVD gives us the most-accurate-possible low-rank factorization of a matrix).

However, we can’t use the SVD algorithm directly, because we don’t know all the entries in \(R\).

Indeed, the unseen entries in \(R\) are exactly what we want to predict.

Here is what we want to solve:

\[ \min_{U,V} \Vert (R - UV^T)_S\Vert^2 + \lambda(\Vert U\Vert^2 + \Vert V\Vert^2) \]

where:

  • \(R\) is \(m\times n\),
  • \(U\) is the \(m\times k\) items matrix,
  • \(V\) is the \(n\times k\) users matrix and
  • \(k\) is the rank of the factorization and dimensionality of the latent space.

The \((\cdot)_S\) notation means that we are only considering the subset of matrix entries that correspond to known reviews (the set \(S\)).

Note that as usual, we add \(\ell_2\) penalization to avoid overfitting (Ridge regression).

Once again, this problem is jointly convex in that it is convex in each of the variables \(U\) and \(V\).

In particular, if we hold either \(U\) or \(V\) constant, then the result is a simple ridge regression.

So one commonly used algorithm for this problem is called alternating least squares (ALS):

  1. Hold \(U\) constant, and solve for \(V\)
  2. Hold \(V\) constant, and solve for \(U\)
  3. If not converged, go to Step 1.

The only thing we’ve left out at this point is how to deal with the missing entries of \(R\).

It’s not hard, but the details aren’t that interesting, so we’ll give you code instead!

ALS in Practice

The entire Amazon reviews dataset is too large to work with easily, and it is too sparse.

Hence, we will take the densest rows and columns of the matrix.

Code 
# print(df.shape)

# The densest columns: products with more than 50 reviews
pids = df.groupby('ProductId').count()['Id']
hi_pids = pids[pids > 50].index

# reviews that are for these products
hi_pid_rec = [r in hi_pids for r in df['ProductId']]

# the densest rows: users with more than 50 reviews
uids = df.groupby('UserId').count()['Id']
hi_uids = uids[uids > 50].index

# reviews that are from these users
hi_uid_rec = [r in hi_uids for r in df['UserId']]

# The result is a list of booleans equal to the number of rewviews
# that are from those dense users and movies
goodrec = [a and b for a, b in zip(hi_uid_rec, hi_pid_rec)]

Now we create a \(\textnormal{UserID} \times \textnormal{ProductID}\) matrix from these reviews.

Missing entries will be filled with NaNs.

Code 
dense_df = df.loc[goodrec]
good_df = dense_df.loc[~df['Score'].isnull()]
R = good_df.pivot_table(columns = 'ProductId', index = 'UserId', values = 'Score')
print(f"Review matrix shape: {R.shape}")
Review matrix shape: (3677, 7244)

And we can look at a small part of the matrix:

Code 
R.iloc[900:905, 1000:1004]
ProductId 1417024917 1417030321 1417030976 1417054069
UserId
A1WLZYEOIL1HLT NaN NaN NaN NaN
A1WNJVA59HLMO5 NaN NaN NaN NaN
A1WR12AC35R3K6 NaN NaN NaN NaN
A1WSFHRBY2ZD1R NaN NaN NaN NaN
A1WUMTJOASEL5F NaN NaN NaN NaN

We’ll use code from the Antidote Data Framework to do the matrix factorization and ALS. We have local copies recommender_MF.py, recommender_als.py and recommender_lmafit.py in our repository.

# Import local python package MF.py
import recommender_MF as MF

# Instantiate the model
# We are pulling these hyperparameters out of the air -- that's not the right way to do it!
RS = MF.als_MF(rank = 20, lambda_ = 1)


%time pred, error = RS.fit_model(R)
CPU times: user 27 s, sys: 306 ms, total: 27.3 s
Wall time: 27.1 s


print(f'RMSE on visible entries (training data): {np.sqrt(error/R.count().sum()):0.3f}')
RMSE on visible entries (training data): 0.343

And we can look at a small part of the predicted ratings matrix and see that it is a dense matrix:

Code 
print(f'Shape of predicted ratings matrix: {pred.shape}')
pred.iloc[900:905, 1000:1004]
Shape of predicted ratings matrix: (3677, 7244)
ProductId 1417024917 1417030321 1417030976 1417054069
UserId
A1WLZYEOIL1HLT 2.828816 4.067808 3.950319 3.949854
A1WNJVA59HLMO5 4.043491 4.930218 4.928916 4.785983
A1WR12AC35R3K6 5.356613 2.955982 4.108460 5.506213
A1WSFHRBY2ZD1R 2.724232 3.618981 3.932077 4.570697
A1WUMTJOASEL5F 4.712673 4.985779 3.714126 4.616174
Code 
## todo: hold out test data, compute oos error

# We create a mask of the known entries, then calculate the indices of the known
# entries, then split that data into training and test sets.

# Create a mask for the known entries
RN = ~R.isnull()

# Get the indices of the known entries
visible = np.where(RN)

# Split the data into training and test sets
import sklearn.model_selection as model_selection
X_train, X_test, Y_train, Y_test = model_selection.train_test_split(visible[0], visible[1], test_size = 0.1)

Just for comparison’s sake, let’s check the performance of \(k\)-NN on this dataset.

Again, this is only on the training data – so overly optimistic for sure.

And note that this is a subset of the full dataset – the subset that is “easiest” to predict due to density.

Code 
from sklearn.neighbors import KNeighborsRegressor
from sklearn.metrics import mean_squared_error

# Drop the columns that are not features
X_train = good_df.drop(columns=['Id', 'ProductId', 'UserId', 'Text', 'Summary'])

# The target is the score
y_train = good_df['Score']

# Using k-NN on features HelpfulnessNumerator, HelpfulnessDenominator, Score, Time
model = KNeighborsRegressor(n_neighbors=3).fit(X_train, y_train)
%time y_hat = model.predict(X_train)
CPU times: user 276 ms, sys: 4.97 ms, total: 281 ms
Wall time: 281 ms
Code 
print(f'RMSE on visible entries (test set): {np.sqrt(mean_squared_error(y_train, y_hat)):.3f}')
RMSE on visible entries (test set): 0.494

Assessing Matrix Factorization

Matrix Factorization per se is a good idea.
However, many of the improvements we’ve discussed for CF apply to MF as well.

To illustrate, we’ll look at some of the successive improvements used by the team that won the Netflix prize (“BellKor’s Pragmatic Chaos”).

When the prize was announced, the Netflix supplied solution achieved an RMSE of 0.951.

By the end of the competition (about 3 years), the winning team’s solution achieved RMSE of 0.856.

Let’s restate our MF objective in a way that will make things clearer:

\[ \min_{U, V} \sum_{(u, i)\in S}(r_{ui} - u_u^Tv_i)^2 + \lambda(\Vert U\Vert^2 + \Vert V\Vert^2) \]

where we have written out the vector \(\ell_2\) norm as the summation.

1. Adding Biases

If we add biases: \[ \min_{U, V} \sum_{(u, i)\in S}(r_{ui} - (\mu + \alpha_u + \beta_i + u_u^Tv_i)^2 + \lambda(\Vert U\Vert^2 + \Vert V\Vert^2 + \Vert \alpha\Vert^2 + \Vert \beta \Vert^2) \]

we see improvements in accuracy:

Figure 2: Matrix factorization models’ accuracy. The plots show the root-mean-square error of each of four individual factor models (lower is better). Accuracy improves when the factor model’s dimensionality (denoted by numbers on the charts) increases. In addition, the more refined factor models, whose descriptions involve more distinct sets of parameters, are more accurate.

2. Who Rated What?

In reality, ratings are not provided at random.

Take note of which users rated the same movies (ala CF) and use this information.

3. Ratings Change Over Time

Older movies tend to get higher ratings!

If we add time-varying biases:

\[ \min_{U, V} \sum_{(u, i)\in S}(r_{ui} - (\mu + \alpha_u(t) + \beta_i(t) + u_u^Tv_i(t))^2 + \lambda(\Vert U\Vert^2 + \Vert V\Vert^2 + \Vert \alpha\Vert^2 + \Vert \beta \Vert^2) \]

we see further improvements in accuracy:

To estimate these billions of parameters, we cannot use alternating least squares or any linear algebraic method.

We need to use gradient descent (which we covered previously).

Recap

  • Introduction to recommender systems and their importance in modern society.
  • Explanation of collaborative filtering (CF) and its two main approaches: user-user similarity and item-item similarity.
  • Discussion on the challenges of recommender systems, including scalability and data sparsity.
  • Introduction to matrix factorization (MF) as an improvement over CF, using latent vectors and alternating least squares (ALS) for optimization.
  • Practical implementation of ALS for matrix factorization on a subset of Amazon movie reviews.

References

Koren, Yehuda. 2009. “Collaborative Filtering with Temporal Dynamics.” Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 42: 447–56. https://dl.acm.org/doi/10.1145/1557019.1557072.
Koren, Yehuda, Robert Bell, and Chris Volinsky. 2009. “Matrix Factorization Techniques for Recommender Systems.” Computer 42 (8): 30–37. https://ieeexplore.ieee.org/document/5197422.
Naumov, Maxim et al. 2019. “Deep Learning Recommendation Model for Personalization and Recommendation Systems.” arXiv Preprint arXiv:1906.00091, May. http://arxiv.org/abs/1906.00091.
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