Parallel Majorization Minimization with Dynamically Restricted Domains for Nonconvex Optimization

We propose an optimization framework for nonconvex problems based on majorization-minimization that is particularity well-suited for parallel computing. It reduces the optimization of a high dimensional nonconvex objective function to successive optimizations of locally tight and convex upper bounds which are additively separable into low dimensional objectives. The original problem is then broken into simpler and parallel tasks, while guaranteeing the monotonic reduction of the original objective function and convergence to a local minimum. This framework also allows one to restrict the upper bound to a local dynamic convex domain, so that the bound is better matched to the local curvature of the objective function, resulting in accelerated convergence. We test the proposed framework on a nonconvex support vector machine based on a sigmoid loss function and on nonconvex penalized logistic regression.