# Direct search based on probabilistic descent in reduced spaces

arXiv: https://arxiv.org/abs/2204.01275 · HTML: https://mlsanity.com/paper/2204.01275/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2022.

## Built on by (most cited first, 7 of 7)

- [Zeroth-Order Fine-Tuning of LLMs in Random Subspaces](https://mlsanity.com/paper/2410.08989/) (2024)
- [Leveraging Coordinate Momentum in SignSGD and Muon: Memory-Optimized Zero-Order](https://mlsanity.com/paper/2506.04430/) (2025)
- [Stochastic Zeroth order Descent with Structured Directions](https://mlsanity.com/paper/2206.05124/) (2022)
- [A Sequential Quadratic Programming Method with High Probability Complexity Bounds for Nonlinear Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2301.00477/) (2023)
- [Tuning without Peeking: Provable Generalization Bounds and Robust LLM Post-Training](https://mlsanity.com/paper/2507.01752/) (2025)
- [Stochastic Dimension Zeroth-Order Estimator: Stable and Memory-Efficient Training of PINNs](https://mlsanity.com/paper/2603.24002/) (2026)
- [Randomized Subspace Nesterov Accelerated Gradient](https://mlsanity.com/paper/2605.00740/) (2026)

## Abstract (arXiv)

In this paper, we study a generic direct-search algorithm in which the polling directions are defined using random subspaces. Complexity guarantees for such an approach are derived thanks to probabilistic properties related to both the subspaces and the directions used within these subspaces. Our analysis crucially extends previous deterministic and probabilistic arguments by relaxing the need for directions to be deterministically bounded in norm. As a result, our approach encompasses a wide range of new optimal polling strategies that can be characterized using our subspace and direction properties. By leveraging results on random subspace embeddings and sketching matrices, we show that better complexity bounds are obtained for randomized instances of our framework. A numerical investigation confirms the benefit of randomization, particularly when done in subspaces, when solving problems of moderately large dimension.