Powering Up Zeroth-Order Training via Subspace Gradient Orthogonalization

• URL: http://arxiv.org/abs/2602.17155v2
• Date: Sat, 21 Feb 2026 09:07:11 GMT
• Title: Powering Up Zeroth-Order Training via Subspace Gradient Orthogonalization
• Authors: Yicheng Lang, Changsheng Wang, Yihua Zhang, Mingyi Hong, Zheng Zhang, Wotao Yin, Sijia Liu,
• Abstract summary: We show that ZO optimization can be substantially improved by unifying two complementary principles.<n>We instantiate in a new method, ZO-Muon, admitting a natural interpretation as a low-rank Muon in the ZO setting.
• Score: 40.95701844244596
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Zeroth-order (ZO) optimization provides a gradient-free alternative to first-order (FO) methods by estimating gradients via finite differences of function evaluations, and has recently emerged as a memory-efficient paradigm for fine-tuning large-scale models by avoiding backpropagation. However, ZO optimization has a fundamental tension between accuracy and query efficiency. In this work, we show that ZO optimization can be substantially improved by unifying two complementary principles: (i) a projection-based subspace view that reduces gradient estimation variance by exploiting the intrinsic low-rank structure of model updates, and (ii) Muon-style spectral optimization that applies gradient orthogonalization to extract informative spectral structure from noisy ZO gradients. These findings form a unified framework of subspace gradient orthogonalization, which we instantiate in a new method, ZO-Muon, admitting a natural interpretation as a low-rank Muon optimizer in the ZO setting. Extensive experiments on large language models (LLMs) and vision transformers (ViTs) demonstrate that ZO-Muon significantly accelerates convergence and achieves a win-win improvement in accuracy and query/runtime efficiency. Notably, compared to the popular MeZO baseline, ZO-Muon requires only 24.7% of the queries to reach the same SST-2 performance for LLM fine-tuning, and improves accuracy by 25.1% on ViT-B fine-tuning on CIFAR-100.

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