We introduce a stochastic global optimization method based on random walks on Grassmannian manifolds. To minimize a continuous objective \\(\\ell:\\mathbb{R}^d \\rightarrow \\mathbb{R}\\), the method repeatedly samples random \\(k\\)-dimensional linear subspaces (with \\(k \\ll d\\)), solves the resulting low-dimensional restrictions of these problems to these subspaces using an arbitrary black-box optimizer, and updates the iterate (which monotonically improves upon the previous iterate). Unlike classical optimization analyses that rely on convexity, smoothness, Lipschitz bounds, or Polyak-Lojasiewicz-type conditions, our convergence guarantees depend only on the geometric distribution of restricted minima across the \\(k\\)-dimensional subspaces passing through a given point in \\(\\mathbb{R}^d\\). We identify a gap parameter — an analogue of a spectral gap for random walks — that controls the rate at which the iterates approach the global minimum value. Finally, we argue that the same analysis yields a blind-spot robustness property: sufficiently narrow, deep dips of the loss function (small-measure regions where \\(\\ell\\) spikes downward) have limited influence on the algorithm’s trajectory, since they are unlikely to be encountered by random subspace sampling.<\/p>\n","protected":false},"excerpt":{"rendered":"
We introduce a stochastic global optimization method based on random walks on Grassmannian manifolds. 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