A Flexible Coordinate Descent Method

TL;DR

The paper introduces Flexible Coordinate Descent (FCD), a randomized block coordinate method that leverages partial second-order information to improve robustness and convergence in convex optimization, especially for complex problems.

Contribution

It proposes a novel FCD algorithm that incorporates approximate curvature information and provides theoretical convergence guarantees for large-scale convex problems.

Findings

FCD converges with high probability under various conditions.

Numerical experiments show FCD's superior performance on large-scale problems.

The method effectively handles highly nonseparable and ill-conditioned problems.

Abstract

We present a novel randomized block coordinate descent method for the minimization of a convex composite objective function. The method uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems. We call the method Flexible Coordinate Descent (FCD). At each iteration of FCD, a block of coordinates is sampled randomly, a quadratic model is formed about that block and the model is minimized \emph{approximately/inexactly} to determine the search direction. An inexpensive line search is then employed to ensure a monotonic decrease in the objective function and acceptance of large step sizes. We present several high probability iteration complexity results to show that convergence of FCD is guaranteed theoretically. Finally, we present numerical results on large-scale problems to demonstrate the practical performance of the method.