Two-level preconditioning for Ridge Regression
For practitioners solving regularized least squares problems in machine learning, this offers a faster iterative solver, though the gains are not quantified with specific numbers.
The paper develops a two-level preconditioner for ridge regression linear systems, using clustering to preserve principal components and accelerate Conjugate Gradient solvers, achieving speed-ups on artificial and real data.
Solving linear systems is often the computational bottleneck in real-life problems. Iterative solvers are the only option due to the complexity of direct algorithms or because the system matrix is not explicitly known. Here, we develop a two-level preconditioner for regularized least squares linear systems involving a feature or data matrix. Variants of this linear system may appear in machine learning applications, such as ridge regression, logistic regression, support vector machines and Bayesian regression. We use clustering algorithms to create a coarser level that preserves the principal components of the covariance or Gram matrix. This coarser level approximates the dominant eigenvectors and is used to build a subspace preconditioner accelerating the Conjugate Gradient method. We observed speed-ups for artificial and real-life data.