CurvZO: Adaptive Curvature-Guided Sparse Zeroth-Order Optimization for Efficient LLM Fine-Tuning

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 with SGD in random subspaces enables memory-efficient fine-tuning of large language models without backpropagation. However, high gradient estimation noise fundamentally undermines adaptive optimizers like Adam. We propose SubZero+, which achieves practical adaptive ZO optimization through a carefully designed dual low-dimensionality strategy: (i) multi-query forward-difference gradient estimation in periodically refreshed random subspaces to mitigate noise amplification in moment buffers, and (ii) Adam updates with periodic restarts performed directly in low-dimensional space rather than full-parameter space. In experiments, this dual design retains memory overhead comparable to momentum-free ZO methods while achieving stronger optimization performance than the evaluated ZO baselines. Theoretically, in the exact-directional limit, $K$-query averaging preserves conditional unbiasedness, while the coefficient estimator's covariance and mean-squared error, as well as query-induced second-moment inflation, scale exactly as $1/K$. Extensive experiments across SuperGLUE with models from 1.3B to 32B parameters under both full fine-tuning and LoRA schemes demonstrate consistent improvements over competing ZO methods. SubZero+ significantly narrows the performance gap with first-order optimization while preserving ZO's inference-time memory efficiency.

Fine-tuning large language models at scale is increasingly limited by the memory cost of backpropagation. Zeroth-Order (ZO) optimizers reduce this burden by relying only on forward function evaluations, but they still exhibit a substantial performance gap to first-order (FO) optimizers such as Adam. We argue that this gap does not arise solely from the absence of explicit gradients: FO fine-tuning also benefits from informative low-rank optimization geometry, large-scale signal aggregation, and well-conditioned update directions. Motivated by this, we introduce StructZO, a structured Zeroth-Order optimization framework for memory-efficient LLM fine-tuning. StructZO performs large-query ZO estimation within an activation-induced informative subspace and applies spectral control to the resulting reduced-space update before mapping it back to the original parameters. This lets ZO aggregate useful directional signals in a compact, geometry-aware space while avoiding the activation and optimizer-state memory costs of backpropagation. We show theoretically that, when the induced subspace captures concentrated optimization-relevant directions, StructZO reduces the effective ZO estimation dimension from $mn$ to $mr$, with the remaining projection error governed by the spectral tail of the activation Gram matrix. Empirically, across OPT-series and Llama3.1-8B on diverse downstreamtasks, StructZO improves over the strongest existing ZO baselines by up to $8.7$ relative on average and up to $26.9$ relative on individual tasks, while remaining within $1.5$ of FO-Adam on average. These results suggest that structured and spectrally controlled ZO updates can substantially narrow the gap between ZO and FO optimization.