Generalizing acceleration analyses from convex to nonconvex settings

Develop rigorous generalizations of analytical frameworks for accelerated gradient methods from convex objectives to nonconvex objectives, establishing conditions and guarantees under which accelerated methods retain convergence properties in nonconvex settings.

Background

A substantial literature interprets accelerated gradient descent through Lyapunov functions, differential equations, and geometric viewpoints, but these treatments predominantly rely on convexity or knowledge of a global minimizer, limiting applicability to nonconvex problems.

The authors explicitly note uncertainty about extending these acceleration analyses to nonconvex settings, motivating future work to develop nonconvex counterparts that provide computable progress measures and convergence guarantees.