What are some recent advances in non-convex optimization research?

What are some recent advances in non-convex optimization research? Non-convex optimization is now ubiquitous in machine learning. While previously, the focus was on convex relaxation methods, now the emphasis is on being able to solve non-convex problems directly. It is not possible to find the global optimum of every non-convex problem due to NP-hardness barrier. An alternate approach is: when can it be solved efficiently (preferably in low order polynomial time).