Inexact Coordinate Descent: Complexity and Preconditioning
This work addresses optimization efficiency for machine learning and numerical computing, presenting an incremental improvement over existing exact methods.
The paper tackles the problem of minimizing convex functions by relaxing the requirement for exact subproblem solutions in randomized block coordinate descent, allowing inexact updates and incorporating preconditioning for acceleration.
In this paper we consider the problem of minimizing a convex function using a randomized block coordinate descent method. One of the key steps at each iteration of the algorithm is determining the update to a block of variables. Existing algorithms assume that in order to compute the update, a particular subproblem is solved exactly. In his work we relax this requirement, and allow for the subproblem to be solved inexactly, leading to an inexact block coordinate descent method. Our approach incorporates the best known results for exact updates as a special case. Moreover, these theoretical guarantees are complemented by practical considerations: the use of iterative techniques to determine the update as well as the use of preconditioning for further acceleration.