解决了零阶优化与高秩LoRA结合时的信号坍缩问题，做大模型微调且受限于显存的开发者可以直接用AR1-ZO方法提升效果。
零阶优化（ZO）无需存储反向传播激活即可微调大模型，而LoRA提供紧凑的可训练适配器。但两者结合存在秩悖论：增加LoRA秩会提升适配器容量，但标准两点ZO要么扰动依赖秩的坐标数，要么在原子更新下使有限差分信号不可观测。本文证明瓶颈是测量拓扑问题，而非需要外部子空间。LoRA已分解为匹配的秩1原子，每个原子是一个完整因子坐标块。AR1-ZO通过交替秩1原子查询和拓扑感知缩放，恢复秩不变的有效信号，无需辅助基、激活钩子、曲率估计或额外前向查询。实验表明，在标准两前向查询预算下，AR1-ZO使高秩LoRA在匹配预算的ZO方法中有效。
AR1-ZO: Topology-Aware Rank-1 Zeroth-Order Queries for High-Rank LoRA Fine-Tuning
Zeroth-order (ZO) optimization enables large-language-model fine-tuning without storing backpropagation activations, while LoRA supplies compact trainable adapters. Combining them creates a rank paradox: increasing LoRA rank improves adapter capacity, but standard two-point ZO either perturbs a rank-dependent number of coordinates or, under atomwise updates, can make the finite-difference signal unobservable. This paper shows that the bottleneck is a measurement-topology problem rather than a need for an external subspace. LoRA already decomposes into matched rank-$1$ atoms, each a complete factor-coordinate block of dimension $d_\text{out}+d_\text{in}$. Querying one atom per step keeps the stored adapter rank $r$ while removing $r$ from the single-query perturbation dimension. The naive atomwise query is still miscalibrated: if it inherits canonical LoRA scaling $α/r$, the active finite-difference signal shrinks as $1/r$ and the active finite-difference signal-to-noise ratio (FD-SNR) as $1/r^2$, producing directional collapse under a fixed residual evaluation-noise floor. AR1-ZO pairs alternating rank-$1$ atom queries with topology-aware scaling $γ=αr$, restoring rank-invariant active signal without auxiliary bases, activation hooks, curvature estimates, or extra forward queries. Theory proves atom minimality, rank-independent active query dimension, directional collapse and restoration, and the remaining rank dependence as an amortized coverage cost. Experiments on OPT and Qwen3 models validate the signal mechanism and show that AR1-ZO makes high-rank LoRA effective among matched-budget ZO methods under the standard two-forward-pass query budget.