Online Optimization for Large-Scale Max-Norm Regularization

TL;DR

This paper introduces an online algorithm for large-scale max-norm regularized matrix decomposition, enabling scalable low-rank estimation suitable for big data applications, with proven convergence and competitive performance.

Contribution

It reformulates max-norm regularization as a matrix factorization problem and develops an online method that maintains scalability and convergence guarantees.

Findings

Algorithm converges to a stationary point asymptotically.

Numerical results show improved efficiency and robustness over nuclear norm methods.

Scalable to large datasets due to fixed memory footprint of basis component.

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.