An Inertial Block Majorization Minimization Framework for Nonsmooth Nonconvex Optimization
This work addresses optimization challenges in machine learning, such as sparse matrix problems, but appears incremental as it builds on existing block-coordinate and inertial methods.
The authors tackled the problem of nonsmooth nonconvex optimization by introducing TITAN, a novel inertial block majorization-minimization framework, which demonstrated effectiveness on sparse non-negative matrix factorization and matrix completion tasks, though no concrete numerical results were provided in the abstract.
In this paper, we introduce TITAN, a novel inerTIal block majorizaTion minimizAtioN framework for non-smooth non-convex optimization problems. To the best of our knowledge, TITAN is the first framework of block-coordinate update method that relies on the majorization-minimization framework while embedding inertial force to each step of the block updates. The inertial force is obtained via an extrapolation operator that subsumes heavy-ball and Nesterov-type accelerations for block proximal gradient methods as special cases. By choosing various surrogate functions, such as proximal, Lipschitz gradient, Bregman, quadratic, and composite surrogate functions, and by varying the extrapolation operator, TITAN produces a rich set of inertial block-coordinate update methods. We study sub-sequential convergence as well as global convergence for the generated sequence of TITAN. We illustrate the effectiveness of TITAN on two important machine learning problems, namely sparse non-negative matrix factorization and matrix completion.