Leveraging Coordinate Momentum in SignSGD and Muon: Memory-Optimized Zero-Order

TL;DR

This paper introduces memory-efficient zero-order optimization algorithms, JAGUAR SignSGD and JAGUAR Muon, for fine-tuning large language models, providing theoretical guarantees and competitive empirical results with reduced resource requirements.

Contribution

It presents the first convergence guarantees for SignSGD in stochastic zero-order optimization and introduces novel ZO algorithms leveraging model structure for LLM fine-tuning.

Findings

Achieves comparable or better convergence than first-order methods.

Significantly reduces memory usage during LLM fine-tuning.

Provides theoretical convergence rates for the proposed ZO algorithms.

Abstract

Fine-tuning Large Language Models (LLMs) is essential for adapting pre-trained models to downstream tasks. Yet traditional first-order optimizers such as Stochastic Gradient Descent (SGD) and Adam incur prohibitive memory and computational costs that scale poorly with model size. In this paper, we investigate zero-order (ZO) optimization methods as a memory- and compute-efficient alternative, particularly in the context of parameter-efficient fine-tuning techniques like LoRA. We propose $\texttt{JAGUAR SignSGD}$, a ZO momentum-based algorithm that extends ZO SignSGD, requiring the same number of parameters as the standard ZO SGD and only $\mathcal{O}(1)$ function evaluations per iteration. To the best of our knowledge, this is the first study to establish rigorous convergence guarantees for SignSGD in the stochastic ZO case. We further propose $\texttt{JAGUAR Muon}$, a novel ZO extension of the Muon optimizer that leverages the matrix structure of model parameters, and we provide its convergence rate under arbitrary stochastic noise. Through extensive experiments on challenging LLM fine-tuning benchmarks, we demonstrate that the proposed algorithms meet or exceed the convergence quality of standard first-order methods, achieving significant memory reduction. Our theoretical and empirical results establish new ZO optimization methods as a practical and theoretically grounded approach for resource-constrained LLM adaptation. Our code is available at https://github.com/brain-mmo-lab/ZO_LLM