Cubic Regularized ADMM with Convergence to a Local Minimum in Non-convex Optimization

How to escape saddle points is a critical issue in non-convex optimization. Previous methods on this issue mainly assume that the objective function is Hessian-Lipschitz, which leave a gap for applications using non-Hessian-Lipschitz functions. In this paper, we propose Cubic Regularized Alternating Direction Method of Multipliers (CR-ADMM) to escape saddle points of separable non-convex functions containing a non-Hessian-Lipschitz component. By carefully choosing a parameter, we prove that CR-ADMM converges to a local minimum of the original function with a rate of $O(1 /T^{1/3})$ in time horizon T, which is faster than gradient-based methods. We also show that when one or more steps of CR-ADMM are not solved exactly, CRADMM can converge to a neighborhood of the local minimum. Through the experiments of matrix factorization problems, CRADMM is shown to have a faster rate and a lower optimality gap compared with other gradient-based methods. Our approach can also find applications in other scenarios where regularized non-convex cost minimization is performed, such as parameter optimization of deep neural networks.