Xiangda Yan

Ziye Chen, Hongbin Lin, Chenyu Zhang et al.

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.

Junbin Qiu, Zhengpeng Xie, Xiangda Yan et al.

Zeroth-Order Optimization (ZOO) provides powerful tools for optimizing functions where explicit gradients are unavailable or expensive to compute. However, the underlying mechanisms of popular ZOO methods, particularly those employing randomized finite differences, and their connection to other optimization paradigms like Reinforcement Learning (RL) are not fully elucidated. This paper establishes a fundamental and previously unrecognized connection: ZOO with finite differences is equivalent to a specific instance of single-step Policy Optimization (PO). We formally unveil that the implicitly smoothed objective function optimized by common ZOO algorithms is identical to a single-step PO objective. Furthermore, we show that widely used ZOO gradient estimators, are mathematically equivalent to the REINFORCE gradient estimator with a specific baseline function, revealing the variance-reducing mechanism in ZOO from a PO perspective.Built on this unified framework, we propose ZoAR (Zeroth-Order Optimization with Averaged Baseline and Query Reuse), a novel ZOO algorithm incorporating PO-inspired variance reduction techniques: an averaged baseline from recent evaluations and query reuse analogous to experience replay. Our theoretical analysis further substantiates these techniques reduce variance and enhance convergence. Extensive empirical studies validate our theory and demonstrate that ZoAR significantly outperforms other methods in terms of convergence speed and final performance. Overall, our work provides a new theoretical lens for understanding ZOO and offers practical algorithmic improvements derived from its connection to PO.