# Powering Up Zeroth-Order Training via Subspace Gradient Orthogonalization

arXiv: https://arxiv.org/abs/2602.17155 · HTML: https://mlsanity.com/paper/2602.17155/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 64 of its 80 references that are in the map.
- First posted on arXiv 19 Feb 2026.

## Builds on (most cited first, 30 of 64)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [SQuAD: 100,000+ Questions for Machine Comprehension of Text](https://mlsanity.com/paper/1606.05250/) (2016)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [SuperGLUE: A Stickier Benchmark for General-Purpose Language Understanding Systems](https://mlsanity.com/paper/1905.00537/) (2019)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [ZOO: Zeroth Order Optimization based Black-box Attacks to Deep Neural Networks without Training Substitute Models](https://mlsanity.com/paper/1708.03999/) (2017)
- [Black-box Adversarial Attacks with Limited Queries and Information](https://mlsanity.com/paper/1804.08598/) (2018)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [WiC: the Word-in-Context Dataset for Evaluating Context-Sensitive Meaning Representations](https://mlsanity.com/paper/1808.09121/) (2018)
- [Fine-Tuning Language Models with Just Forward Passes](https://mlsanity.com/paper/2305.17333/) (2023)
- [Black-Box Tuning for Language-Model-as-a-Service](https://mlsanity.com/paper/2201.03514/) (2022)
- [Optimal rates for zero-order convex optimization: the power of two function evaluations](https://mlsanity.com/paper/1312.2139/) (2013)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [A Primer on Zeroth-Order Optimization in Signal Processing and Machine Learning](https://mlsanity.com/paper/2006.06224/) (2020)
- [AutoZOOM: Autoencoder-based Zeroth Order Optimization Method for Attacking Black-box Neural Networks](https://mlsanity.com/paper/1805.11770/) (2018)
- [Query-Efficient Hard-label Black-box Attack:An Optimization-based Approach](https://mlsanity.com/paper/1807.04457/) (2018)
- [Revisiting Zeroth-Order Optimization for Memory-Efficient LLM Fine-Tuning: A Benchmark](https://mlsanity.com/paper/2402.11592/) (2024)
- [Zeroth-Order Stochastic Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1805.10367/) (2018)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [Flora: Low-Rank Adapters Are Secretly Gradient Compressors](https://mlsanity.com/paper/2402.03293/) (2024)

## Built on by (most cited first, 12 of 12)

- [Rethinking Muon Beyond Pretraining: Spectral Failures and High-Pass Remedies for VLA and RLVR](https://mlsanity.com/paper/2605.19282/) (2026)
- [MiMuon: Mixed Muon Optimizer with Improved Generalization for Large Models](https://mlsanity.com/paper/2605.19619/) (2026)
- [Position: Zeroth-Order Optimization in Deep Learning Is Underexplored, Not Underpowered](https://mlsanity.com/paper/2605.15622/) (2026)
- [ZO-Act: Efficient Zeroth-Order Fine-Tuning via One-Shot Activation-Informed Low-Rank Subspaces](https://mlsanity.com/paper/2607.01125/) (2026)
- [Accelerating Zeroth-Order Spectral Optimization with Partial Orthogonalization from Power Iteration](https://mlsanity.com/paper/2605.09034/) (2026)
- [Inference-Native Zeroth-Order Optimization for LLMs](https://mlsanity.com/paper/2605.28760/) (2026)
- [GRZO: Group-Relative Zeroth-Order Optimization for Large Language Model Fine-Tuning](https://mlsanity.com/paper/2606.02857/) (2026)
- [SubZero+: Memory-Efficient Adaptive Zeroth-Order LLM Fine-Tuning in Random Subspaces](https://mlsanity.com/paper/2608.15665/) (2026)
- [AIM-ZO: Activation-Informed Subspace Maintenance for Zeroth-Order LLM Fine-Tuning](https://mlsanity.com/paper/2609.35257/) (2026)
- [Scaling Zero-Order Pretraining through Model Sharding](https://mlsanity.com/paper/2609.37899/) (2026)
- [TACO: Ternary Absolute-max Column-wise One-sparse Optimizer for LLM Fine-Tuning](https://mlsanity.com/paper/2610.02199/) (2026)
- [StructZO: Structured Zeroth-Order Optimization for Memory-Efficient LLM Fine-Tuning](https://mlsanity.com/paper/2610.16451/) (2026)

## Abstract (arXiv)

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.