# Zeroth-Order Fine-Tuning of LLMs in Learnable Low-Rank Subspaces

arXiv: https://arxiv.org/abs/2610.15225 · HTML: https://mlsanity.com/paper/2610.15225/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 17 references that are in the map.
- First posted on arXiv 10 Oct 2026.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [SuperGLUE: A Stickier Benchmark for General-Purpose Language Understanding Systems](https://mlsanity.com/paper/1905.00537/) (2019)
- [Fine-Tuning Language Models with Just Forward Passes](https://mlsanity.com/paper/2305.17333/) (2023)
- [Optimal rates for zero-order convex optimization: the power of two function evaluations](https://mlsanity.com/paper/1312.2139/) (2013)
- [Variance-reduced Zeroth-Order Methods for Fine-Tuning Language Models](https://mlsanity.com/paper/2404.08080/) (2024)
- [ZO-AdaMU Optimizer: Adapting Perturbation by the Momentum and Uncertainty in Zeroth-order Optimization](https://mlsanity.com/paper/2312.15184/) (2023)

## Abstract (arXiv)

Zeroth-order (ZO) optimization enables memory-efficient fine-tuning of large language models (LLMs). However, ZO optimization suffers from high variance in high-dimensional parameter spaces. While recent methods restrict perturbations to random low-rank subspaces, these subspaces often misalign with task-specific descent directions. We propose AdaSub-ZO, a novel forward-only fine-tuning framework in learnable low-rank subspaces. Instead of relying on fixed random matrices, AdaSub-ZO iteratively learns layer-wise projection matrices directly from finite-difference feedback to track task-aware descent directions. Furthermore, AdaSub-ZO incorporates a subspace momentum acceleration mechanism to smooth noisy gradient estimates across evolving subspace bases. Extensive experiments on natural language understanding tasks demonstrate that AdaSub-ZO outperforms existing ZO baselines, substantially reducing gradient estimation variance and accelerating convergence.