Welcome back! In our previous lesson, you explored the foundation of user-item explicit rating matrices used in recommendation systems. Today, we'll expand on that knowledge by diving into one of the powerful techniques for collaborative filtering known as the Alternating Least Squares (ALS) algorithm.

Recommendation systems have become essential in offering personalized experiences, with collaborative filtering being a primary method. Collaborative filtering works by understanding user preferences through their past interactions and leveraging similar users or items to provide recommendations. The ALS algorithm is a matrix factorization approach that enables us to predict missing ratings effectively, making it a valuable tool in recommendation systems.

Before we proceed with implementing the ALS algorithm, let's quickly recap the fundamental steps we covered in the previous lesson for setting up our environment. You may remember how we:

1. Read data from a file to create a user-item interaction matrix.
2. Marked some entries with -1 to simulate missing data for testing purposes while saving actual ratings for future evaluation.

Here's a concise code snippet capturing the setup:

Python
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1import numpy as np
2import random
3
4# Initialize user-item interaction matrix from file
5R = []
6with open('explicit_ratings.txt', 'r') as file:
7    users = file.readlines()
8    for user in users:
9        ratings = list(map(int, user.split(' ')))
10        R.append(ratings)
11R = np.array(R)
12
13# Mark some entries as missing (-1) for testing
14missing_ratio = 0.1  # Density of missing entries
15num_entries = np.count_nonzero(R != -1)
16missing_indices = [(random_point // R.shape[1], random_point % R.shape[1]) for random_point in random.sample(range(num_entries), int(missing_ratio * num_entries))]
17for (u, i) in missing_indices:
18    R[u, i] = -1
19
20# Save the original matrix for evaluation
21original_R = np.copy(R)

This setup is crucial as it establishes the data landscape we will work with throughout the ALS implementation.

To predict missing ratings using ALS, we need to decompose the interaction matrix into two matrices — user and item factors. These factors capture latent characteristics that influence user preferences and item popularity. Initially, these factors are filled with random values, which will then be optimized through ALS iterations.

Python
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1num_users, num_items = R.shape
2num_factors = 3
3
4# Random initialization of user and item factors
5U = np.random.rand(num_users, num_factors) * 0.01
6V = np.random.rand(num_items, num_factors) * 0.01

Here, U represents user factors and V represents item factors, with num_factors indicating the dimensionality of these latent features.

The ALS algorithm is the heart of our lesson. Before diving into the steps involved in ALS, it's essential to understand the optimization problem we're tackling and how ratings are predicted.

The ALS algorithm addresses the problem of predicting missing ratings in a user-item interaction matrix by factorizing it into two matrices: user factors (U) and item factors (V). We aim to approximate the matrix $R$ (user-item ratings) by minimizing the difference between the actual and predicted ratings through the following optimization problem:

$\min_{\underset{U,V}{}} \sum_{u,i \in \text{observed}} (R_{ui} - U_u \cdot V_i^T)^2 + \lambda \left(\| U_u \|^2 + \| V_i \|^2 \right)$

Here,

• $R_{ui}$ is the actual rating given by user $u$ to item $i$.
• $U_u$ and $V_i$ are the user and item factor vectors, respectively.
• $\lambda$ is the regularization parameter that penalizes large values of the factors to prevent overfitting.

The square of the norm $\| U_u \|^2$ represents the sum of the squares of the components of the user factor vector $U_u$. This acts as a regularization term to prevent overfitting by discouraging large values in the factor vectors.

Importantly, during training, we only work with user-item pairs where the ratings are known. Unknown (missing) ratings are excluded from the optimization process.

Once we have the factorized matrices, the predicted rating $\hat{R}_{ui}$ for user $u$ and item $i$ is calculated as:

$\hat{R}_{ui} = U_u \cdot V_i^T$

The ALS algorithm solves this optimization problem iteratively by alternating between updating user factors and item factors. Here's how ALS specifically tackles this:

1. Fix Item Factors and Optimize User Factors:

   • For each user $u$, it minimizes the error for the observed ratings by updating $U_u$, while keeping V fixed.
   • The update rule for user factors is derived from setting the derivative of the loss function with respect to $U_u$ to zero, resulting in:

   $U_u = (V_u^T V_u + \lambda I)^{-1} V_u^T R_u$

   Here, $V_u$ consists of item factors for items rated by user $u$, and $R_u$ are the actual ratings by user $u$.

2. Fix User Factors and Optimize Item Factors:

   • For each item $i$, it minimizes the error for the observed ratings by updating $V_i$, while keeping U fixed.
   • The update rule for item factors is similarly derived and is given by:

   $V_i = (U_i^T U_i + \lambda I)^{-1} U_i^T R_i$

   Here, $U_i$ consists of user factors for users who rated item $i$, and $R_i$ are the actual ratings for item $i$.

Python
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1lambda_reg = 0.1
2num_iterations = 20
3
4def train_als():
5    global U, V
6    for iteration in range(num_iterations):
7        # Update user factors
8        for u in range(num_users):
9            V_u = V[R[u, :] != -1, :]
10            R_u = R[u, R[u, :] != -1]
11            if V_u.shape[0] > 0:
12                U[u, :] = np.linalg.solve(
13                    np.dot(V_u.T, V_u) + lambda_reg * np.eye(num_factors),
14                    np.dot(V_u.T, R_u)
15                )
16
17        # Update item factors
18        for i in range(num_items):
19            U_i = U[R[:, i] != -1, :]
20            R_i = R[R[:, i] != -1, i]
21            if U_i.shape[0] > 0:
22                V[i, :] = np.linalg.solve(
23                    np.dot(U_i.T, U_i) + lambda_reg * np.eye(num_factors),
24                    np.dot(U_i.T, R_i)
25                )
26
27train_als()

The algorithm iterates over these two steps, alternating between updating user and item factors until convergence or a predetermined number of iterations is reached. This alternating optimization procedure ensures that each step is solving a least-squares problem, making the factor updates computationally efficient.

Once user and item factors are optimized, we can predict the missing ratings by matrix multiplication of the two factors. To evaluate the model's accuracy, we calculate the Root Mean Square Error (RMSE) for excluded items:

Python
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1# Predict the ratings
2predicted_R = np.dot(U, V.T)
3
4# Calculate RMSE for excluded items
5def calculate_rmse(original_R, predicted_R, missing_indices):
6    mse = np.sum([(original_R[u, i] - predicted_R[u, i]) ** 2 for (u, i) in missing_indices]) / len(missing_indices)
7    return np.sqrt(mse)
8
9rmse = calculate_rmse(original_R, predicted_R, missing_indices)
10print(f"RMSE for the excluded items: {rmse:.4f}")

The RMSE offers insight into the prediction error for missing values. A lower RMSE indicates better predictive performance.

In this lesson, you've successfully implemented the ALS algorithm to tackle collaborative filtering challenges within recommendation systems. You've learned to construct user-item matrices, initialize factors, and update them to predict missing ratings. This understanding equips you with a robust technique for building recommendation models.

Now, it's time to consolidate this theoretical understanding with hands-on exercises in the CodeSignal IDE. These exercises are designed to reinforce the concepts learned, allowing you to apply ALS in varied scenarios. You've made significant progress, so keep up the great work as you continue to explore the exciting world of recommendation systems!