Global Convergence of Model Function Based Bregman Proximal Minimization Algorithms

Lipschitz continuity of the gradient mapping of a continuously differentiable function plays a crucial role in designing various optimization algorithms. However, many functions arising in practical applications such as low rank matrix factorization or deep neural network problems do not have a Lipschitz continuous gradient. This led to the development of a generalized notion known as the L-smad property, which is based on generalized proximity measures called Bregman distances. However, the L-smad property cannot handle nonsmooth functions, for example, simple nonsmooth functions like |x^4-1| and also many practical composite problems are out of scope. We fix this issue by proposing the MAP property, which generalizes the L-smad property and is also valid for a large class of nonconvex nonsmooth composite problems. Based on the proposed MAP property, we propose a globally convergent algorithm called Model BPG, that unifies several existing algorithms. The convergence analysis is based on a new Lyapunov function. We also numerically illustrate the superior performance of Model BPG on standard phase retrieval problems, robust phase retrieval problems, and Poisson linear inverse problems, when compared to a state of the art optimization method that is valid for generic nonconvex nonsmooth optimization problems.

@article{MFO22,
  title        = {Global Convergence of Model Function Based Bregman Proximal Minimization Algorithms},
  author       = {M.C. Mukkamala and J. Fadili and P. Ochs},
  year         = {2022},
  journal      = {Journal of Global Optimization},
  volume       = {83},
  pages        = {753--781}
}