Paper
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
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.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2602.17155v1</id>\n <title>Powering Up Zeroth-Order Training via Subspace Gradient Orthogonalization</title>\n <updated>2026-02-19T08:08:33Z</updated>\n <link href='https://arxiv.org/abs/2602.17155v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2602.17155v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.LG'/>\n <published>2026-02-19T08:08:33Z</published>\n <arxiv:primary_category term='cs.LG'/>\n <author>\n <name>Yicheng Lang</name>\n </author>\n <author>\n <name>Changsheng Wang</name>\n </author>\n <author>\n <name>Yihua Zhang</name>\n </author>\n <author>\n <name>Mingyi Hong</name>\n </author>\n <author>\n <name>Zheng Zhang</name>\n </author>\n <author>\n <name>Wotao Yin</name>\n </author>\n <author>\n <name>Sijia Liu</name>\n </author>\n </entry>"
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