Stochastic global optimization of continuous functions via random walks on Grassmannians

• URL: http://arxiv.org/abs/2605.14151v1
• Date: Wed, 13 May 2026 22:06:50 GMT
• Title: Stochastic global optimization of continuous functions via random walks on Grassmannians
• Abstract summary: We introduce a global optimization method based on random walks on Grassmannian manifold.<n>The method repeatedly samples random $k$-dimensional linear subspaces.<n>We identify a gap parameter that controls the rate at which the iterates approach the global minimum value.
• Score: 19.659410865201384
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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