Briefly explain alternating Least squares methods.

6 a] Briefly explain alternating Least squares methods.

Alternating Least Squares (ALS) is an optimization technique often used for solving matrix factorization problems in recommendation engines, such as predicting user preferences for items.

The ALS Algorithm Explained:

• Objective: The goal is to find two matrices, ( U ) and ( V ), such that their product approximates the original user-item matrix ( X ). You aim to minimize the difference between the actual values in ( X ) and the predicted values from ( U \cdot V^T ), while also keeping the values of ( U ) and ( V ) small.
• Iterative Process: ALS solves the optimization problem by alternating between fixing one matrix (either ( U ) or ( V )) and solving for the other in a series of iterations:

1. Fix ( V ) and solve for ( U ).
2. Fix ( U ) and solve for ( V ).
3. Repeat these steps until the changes in ( U ) and ( V ) become insignificant, indicating that the algorithm has converged.

• Closed-form solution: While the overall optimization problem does not have a simple closed-form solution, each step (optimizing ( U ) or ( V ) while the other is fixed) reduces to a least squares problem, which does have a closed-form solution.
• Parallelization: Because each user or item can be updated independently based on their specific ratings, ALS can be easily parallelized, making it computationally efficient for large datasets.
• Regularization: The algorithm can include regularization terms to ensure that the problem remains convex and to avoid overfitting.

Example Steps:

1. Start with a random matrix ( V ).
2. Optimize ( U ) while keeping ( V ) fixed: For each user, solve a least squares problem to find the optimal user vector ( u_i ).
3. Optimize ( V ) while keeping ( U ) fixed: For each item, solve a least squares problem to find the optimal item vector ( v_j ).
4. Repeat until convergence: The process is repeated until the changes in ( U ) and ( V ) are minimal.

ALS is widely used in recommendation systems due to its simplicity, scalability, and ease of parallelization, making it suitable for large-scale applications.