On the Global Convergence of Randomized Coordinate Gradient Descent for Non-Convex Optimization

The paper proves the main claim rigorously: under Assumptions 1–3 and 0 < α_min < α_max < 1/M, randomized coordinate gradient descent almost surely avoids strict saddles, via a careful finite-block amplification near strict saddles, a positive Lyapunov exponent for the linearized cocycle (Proposition 3.1), and a law-of-large-numbers–style argument (Theorems 4.4 and 4.9 leading to Theorem 1) . The model’s solution, while thematically aligned (RDS viewpoint, Oseledets splitting, nontrivial unstable projection), misapplies off-the-shelf random stable manifold theorems in a setting the paper explicitly cautions against without additional quantitative control , and crucially asserts that x_t (or a linear functional of it) is an affine function of a past step size α_s. That “affine-in-α_s” claim is incorrect for nonlinear f: the dependence of later iterates on α_s is nonlinear via ∇f(x_k). The paper avoids this pitfall by deriving finite-time, high-probability amplification bounds instead of relying on such an affine argument.