Provable Efficient Online Matrix Completion via Non-convex Stochastic Gradient Descent (1605.08370v1)

Abstract: Matrix completion, where we wish to recover a low rank matrix by observing a few entries from it, is a widely studied problem in both theory and practice with wide applications. Most of the provable algorithms so far on this problem have been restricted to the offline setting where they provide an estimate of the unknown matrix using all observations simultaneously. However, in many applications, the online version, where we observe one entry at a time and dynamically update our estimate, is more appealing. While existing algorithms are efficient for the offline setting, they could be highly inefficient for the online setting. In this paper, we propose the first provable, efficient online algorithm for matrix completion. Our algorithm starts from an initial estimate of the matrix and then performs non-convex stochastic gradient descent (SGD). After every observation, it performs a fast update involving only one row of two tall matrices, giving near linear total runtime. Our algorithm can be naturally used in the offline setting as well, where it gives competitive sample complexity and runtime to state of the art algorithms. Our proofs introduce a general framework to show that SGD updates tend to stay away from saddle surfaces and could be of broader interests for other non-convex problems to prove tight rates.

• The paper introduces a provably efficient online matrix completion algorithm that updates dynamically with non-convex SGD.
• It achieves near-linear total runtime and competitive sample complexity compared to offline methods.
• The approach avoids saddle points, ensuring robust, fast convergence and scalability for real-time systems.

Provable Efficient Online Matrix Completion via Non-convex Stochastic Gradient Descent

The paper "Provable Efficient Online Matrix Completion via Non-convex Stochastic Gradient Descent" addresses a significant problem in the domain of matrix completion, which is the recovery of a low-rank matrix using a minimal set of observed entries. Traditional algorithms focus predominantly on offline settings, processing all available observations at once, which is not feasible for real-time applications. This paper introduces a novel approach for online matrix completion using non-convex stochastic gradient descent (SGD), providing theoretical guarantees on its efficiency.

Contributions and Results

The core contribution of the paper is the development of the first efficient algorithm for online matrix completion, backed by proofs of performance and convergence. The algorithm dynamically updates its estimates as each new entry is observed, optimizing computational efficiency and sample complexity. Key highlights of the paper include:

• Algorithm Performance: The proposed algorithm executes quick updates per observation in $O(k^3)$ time complexity, with $O(\mu d k^4 (k + \log(\sigma/\epsilon)) \log d)$ observations needed to achieve $\epsilon$ accuracy, where $\mu$ is the incoherence parameter, $d$ is the matrix dimension, $k$ is the rank, and $\sigma$ is the condition number of the matrix.
• Competitive with Offline Algorithms: The algorithm boasts a sample complexity and total runtime linear with respect to the matrix dimension, making it comparable to leading offline algorithms. This efficiency is crucial for applications requiring real-time data processing like recommendation systems.
• Saddle Point Avoidance: The framework introduced by the authors ensures that the SGD updates avoid saddle surfaces, enhancing the overall stability and speed of convergence. This feature implies broader applicability to other non-convex optimization problems.

Theoretical Implications

The proofs presented leverage a novel framework that demonstrates how SGD updates self-regulate away from saddle points, maintaining geometric rates of convergence. In addition to expanding the horizon of matrix completion algorithms to online settings, the paper provides valuable insights into non-convex optimization, contributing techniques that could be applicable to other domains.

Practical Implications

The practical ramifications are substantial. With the algorithm’s efficiency in processing observations as they arrive, it can be integrated into systems that benefit from timely updates, such as live recommendations and adaptive filtering. The computation requirement per step (near-linear complexity in d) remains feasible even for large-scale matrices, ensuring scalability.

Future Directions

This research opens pathways to refine online matrix completion further, improving upon initialization techniques and exploring extensions to more complex data structures and settings. Additionally, it may incite exploration into its applicability to other real-time data streams and recommendations beyond user-item matrices. Future work could delve into distributed implementations or adaptations that tackle even broader classes of non-convex problems, cementing SGD’s utility in real-world applications.

In conclusion, the paper makes significant advancements in efficient online matrix completion, ensuring robustness against saddle points and providing competitive performance metrics relative to offline algorithms. As real-time systems gain prominence across industries, such enumeration will likely become a cornerstone of adaptive data science applications.