Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions
This resolves an open theoretical problem in optimization, ensuring convergence to minimizers rather than saddle points in non-convex settings.
The paper proves that gradient descent almost always avoids strict saddle points for non-convex functions, even with non-isolated critical points, and provides an upper bound on step-size.
Given a non-convex twice differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.