Global convergence of model function based Bregman proximal minimization algorithms - Mahesh Chandra Mukkamala; Jalal Fadili; Peter Ochs (Journal of Global Optimization, 2021)
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 |x4-1|[12pt]minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt documentx^4-1 document 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 structured 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 and Poisson linear inverse problems, when compared to a state of the art optimization method that is valid for generic nonconvex nonsmooth optimization problems.
Keywords: 510; Composite minimization; Bregman proximal minimization algorithms; Model function framework; Bregman distance; Global convergence; Kurdyka–Łojasiewicz property