[Paper Review] Online Optimization for Max-Norm Regularization

Jie Shen, Huan Xu+1 more|arXiv (Cornell University)|Jun 12, 2014

Sparse and Compressive Sensing Techniques25 citations

TL;DR

This paper proposes an online algorithm for max-norm regularized matrix decomposition that reformulates the max-norm via matrix factorization into basis and coefficient components, enabling scalable, memory-efficient processing of large-scale data by maintaining only the basis in memory. The method provably converges to a stationary point of the expected loss and outperforms nuclear norm solvers in efficacy and robustness.

ABSTRACT

Max-norm regularizer has been extensively studied in the last decade as it promotes an effective low-rank estimation for the underlying data. However, such max-norm regularized problems are typically formulated and solved in a batch manner, which prevents it from processing big data due to possible memory budget. In this paper, hence, we propose an online algorithm that is scalable to large-scale setting. Particularly, we consider the matrix decomposition problem as an example, although a simple variant of the algorithm and analysis can be adapted to other important problems such as matrix completion. The crucial technique in our implementation is to reformulating the max-norm to an equivalent matrix factorization form, where the factors consist of a (possibly overcomplete) basis component and a coefficients one. In this way, we may maintain the basis component in the memory and optimize over it and the coefficients for each sample alternatively. Since the memory footprint of the basis component is independent of the sample size, our algorithm is appealing when manipulating a large collection of samples. We prove that the sequence of the solutions (i.e., the basis component) produced by our algorithm converges to a stationary point of the expected loss function asymptotically. Numerical study demonstrates encouraging results for the efficacy and robustness of our algorithm compared to the widely used nuclear norm solvers.

Motivation & Objective

• Address the scalability limitations of batch max-norm regularization in large-scale data settings due to high memory requirements.
• Enable online processing of streaming or large-scale data by reformulating max-norm regularization into an online optimization framework.
• Develop a memory-efficient algorithm where only the basis component is stored, independent of sample size.
• Ensure theoretical convergence of the solution sequence to a stationary point of the expected loss function.
• Demonstrate superior performance and robustness compared to widely used nuclear norm solvers in practical settings.

Proposed method

• Reformulate the max-norm as an equivalent matrix factorization involving a basis matrix and coefficient matrix.
• Maintain the basis component in memory and alternate optimization between the basis and coefficients for each incoming sample.
• Use an online update strategy that processes data sequentially, updating the basis and coefficients incrementally.
• Apply alternating optimization to the factorized components, ensuring the basis remains fixed during coefficient updates and vice versa.
• Leverage the structure of the factorization to decouple memory usage from the number of samples, enabling scalability.
• Prove asymptotic convergence of the basis component sequence to a stationary point of the expected loss function.

Experimental results

Research questions

• RQ1 Can max-norm regularization be effectively adapted to an online, scalable learning setting for large-scale data?
• RQ2 Does the proposed factorization-based reformulation of the max-norm enable memory-efficient online optimization?
• RQ3 Can the online algorithm achieve convergence to a stationary point of the expected loss function?
• RQ4 How does the performance of the proposed method compare to nuclear norm-based solvers in terms of efficacy and robustness?
• RQ5 Is the algorithm robust and scalable when processing large collections of samples in practice?

Key findings

• The proposed online algorithm achieves scalable learning by storing only the basis component, which is independent of the number of samples.
• The sequence of basis components generated by the algorithm converges asymptotically to a stationary point of the expected loss function.
• Numerical experiments show the method outperforms widely used nuclear norm solvers in terms of efficacy and robustness.
• The algorithm maintains strong performance across large-scale data settings due to its low memory footprint and incremental update mechanism.
• The factorization-based reformulation enables effective online optimization without sacrificing the theoretical benefits of max-norm regularization.