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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.14151 |
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| _version_ | 1866914601681027072 |
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| author | Gupta, Kartik Miller, Stephen D. Ravikumar, Pradeep Venkatesan, Ramarathnam |
| author_facet | Gupta, Kartik Miller, Stephen D. Ravikumar, Pradeep Venkatesan, Ramarathnam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14151 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic global optimization of continuous functions via random walks on Grassmannians Gupta, Kartik Miller, Stephen D. Ravikumar, Pradeep Venkatesan, Ramarathnam Optimization and Control Machine Learning 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. |
| title | Stochastic global optimization of continuous functions via random walks on Grassmannians |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2605.14151 |