# Stochastic global optimization of continuous functions via random walks on Grassmannians

arXiv: https://arxiv.org/abs/2605.14151 · HTML: https://mlsanity.com/paper/2605.14151/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 26 references that are in the map.
- First posted on arXiv 13 May 2026.

## Builds on (most cited first, 5 of 5)

- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Wild Patterns Reloaded: A Survey of Machine Learning Security against Training Data Poisoning](https://mlsanity.com/paper/2205.01992/) (2022)
- [Learning from Untrusted Data](https://mlsanity.com/paper/1611.02315/) (2016)
- [Robust Estimation via Robust Gradient Estimation](https://mlsanity.com/paper/1802.06485/) (2018)
- [SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation](https://mlsanity.com/paper/2006.10311/) (2020)

## Abstract (arXiv)

We introduce a stochastic global optimization method based on random walks on Grassmannian manifolds. To minimize a continuous objective ℓ:ℝdrightarrowℝ, the method repeatedly samples random k-dimensional linear subspaces (with kll 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 ℝd. 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 ℓ spikes downward) have limited influence on the algorithm's trajectory, since they are unlikely to be encountered by random subspace sampling.