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.

References

Most of this work is tailored to the convex setting, and it is unclear and nontrivial to generalize the results to a nonconvex setting.

— Accelerated Gradient Descent Escapes Saddle Points Faster than Gradient Descent  (1711.10456 - Jin et al., 2017) in Subsection: Related Work (Acceleration)

The second is whether a different Lyapunov function would allow recovering part (or the whole) of this expanded spectral domain for non-quadratic objective functions.

— A Refined Parameter Condition in the Lyapunov Analysis of IGAHD  (2608.28088 - Adly, 28 Aug 2026) in Section 6, Conclusion

The $O (\varepsilon{-2})$ complexity bound is established for the nonconvex setting. In the convex case, however, one can generally expect a more favorable complexity guarantee. For instance, the accelerated techniques developed in could potentially be incorporated to further improve the convergence rate. A systematic investigation along this direction is left for future work.

— An Adaptive Projected-Gradient Algorithm for Sample-Average Approximations of Stochastic Multi-Objective Optimization  (2609.02722 - Li et al., 2 Sep 2026) in Section 4, immediately after Theorem 4.2 (complexity analysis)