Gradient descent avoids strict saddle points — critical points with at least one direction of negative curvature. For constant step sizes, the proof is clean: the set of initial conditions that converge to a saddle has measure zero. The saddle's stable manifold is a thin set that random initialization almost surely misses.
Muşat and Boumal extend this to non-autonomous systems: gradient descent with varying step sizes, including vanishing schedules where the step size approaches zero. The extension is not trivial because the standard Center-Stable Manifold Theorem assumes the dynamics are the same at every iteration. With changing step sizes, the map applied at each step is different, and the stable manifold is no longer a fixed geometric object — it evolves.
Their Center-Stable Set Theorem proves that even with non-constant dynamics, the set of initial conditions converging to strict saddle points remains measure-zero. The result applies to Euclidean and Riemannian gradient descent and to the proximal point method, without requiring Lipschitz continuity or isolated saddle points.
The through-claim is about what robustness means for optimization guarantees. The constant-step-size result was reassuring but fragile — real implementations use learning rate schedules, warm-up phases, decay. If the guarantee only held for constant steps, it was a theorem about algorithms nobody runs. Proving saddle avoidance for the non-autonomous case closes the gap between theory and practice. The guarantee now covers the algorithms as deployed, not just as idealized. The changing step is what makes gradient descent practical, and now the saddle avoidance guarantee follows the step wherever it goes.