Active strict saddles in nonsmooth optimization

Nonconvex optimization techniques are increasingly playing a major role in modern signal processing, high dimensional statistics, and machine learning. A driving theme, fully supported by empirical evidence, is that simple algorithms often work well in highly non-convex and even nonsmooth settings. Gradient descent, for example, often finds points with small objective value, despite existence of many highly suboptimal critical points. A growing body of literature provides one compelling explanation for this phenomenon. Namely, typical smooth objective functions provably satisfy the strict saddle property, meaning each critical point is either a local minimizer or has a direction of strictly negative curvature (e.g., [6, 29, 30, 62, 63]).