Learning Dynamics of Zeroth-Order Optimization: A Kernel Perspective

We investigate the effectiveness of adaptive zeroth-order (ZO) optimization for memory-constrained fine-tuning of large language models (LLMs). Contrary to prior claims, we show that adaptive ZO methods such as ZO-Adam offer no convergence advantage over well-tuned ZO-SGD, while incurring significant memory overhead. Our analysis reveals that in high dimensions, ZO gradients lack coordinate-wise heterogeneity, rendering adaptive mechanisms memory inefficient. Leveraging this insight, we propose MEAZO, a memory-efficient adaptive ZO optimizer that tracks only a single scalar for global step size adaptation. We support our method with theoretical convergence guarantees under standard assumptions. Experiments across multiple LLM families and tasks demonstrate that MEAZO matches ZO-Adam's performance with the memory footprint of ZO-SGD. Additional experiments on synthetic quadratic problems and LLM fine-tuning further demonstrate MEAZO's enhanced robustness to step size choices, particularly in grouped or block-structured optimization settings.

Zeroth-order (ZO) optimization enables fine-tuning large language models when backpropagation is unavailable or memory-prohibitive, but existing methods often perturb full model weights or randomly constructed low-dimensional subspaces, yielding high-variance estimates and limited performance. We propose ZO-Act, an activation-informed ZO fine-tuning method that restricts perturbations to a fixed low-rank subspace derived from input activations. For each linear layer, ZO-Act computes a small activation basis once at initialization and optimizes only lightweight coefficient matrices using forward-only loss evaluations. This reduces the effective perturbation dimension, exposes explicit trainable variables compatible with momentum-based optimizers such as Adam, and naturally supports quantized LLM fine-tuning by keeping low-bit weights frozen. We analyze ZO-Act as zeroth-order optimization over a restricted coefficient space and show that perturbing the low-dimensional coefficients reduces both the variance-dependent convergence term and the finite-difference error of the ZO estimator, at the cost of a controlled subspace approximation bias that is mitigated by the low-rank structure of LLM activations and gradients. Experiments on Llama-3-8B, OPT-13B, and INT4 Llama-3-8B show consistent gains over strong ZO fine-tuning baselines across language understanding, question answering, and commonsense reasoning.

Zeroth-order (ZO) optimization enables memory-efficient fine-tuning of large language models (LLMs) using only forward passes, but it remains unclear how useful adaptation is distributed across layers. In this work, we reveal a surprising phenomenon: ZO fine-tuning is sharply dominated by a single decoding layer. Across multiple LLM families and downstream tasks, fine-tuning this dominant layer alone consistently matches or even exceeds full-model ZO fine-tuning. We further show that the dominant layer is task-agnostic but model-specific, and can be identified before training through a simple inference-only analysis of activation outliers. Specifically, the dominant layer consistently aligns with the first activation-outlier layer in the pre-trained model. To explain this phenomenon, we analyze how perturbation effects propagate under ZO optimization. We find that the dominant layer combines two key properties: high perturbation sensitivity and early placement in the residual stream, allowing perturbation-induced effects to propagate and accumulate through remaining subsequent decoding layers. As a result, this layer produces disproportionately strong and stable optimization signals under forward-only updates. Extensive experiments on LLaMA2-7B and Qwen3-8B across nine benchmarks show that dominant-layer ZO fine-tuning improves average performance over full-model MeZO and LoRA-based ZO fine-tuning while achieving up to 4.52$\times$ training speedup.