OASIS: Online Activation Subspace Learning for Memory-Efficient Training

Activations have become the primary memory bottleneck in large-batch LLM training. However, existing compression methods fail to exploit the spectral structure of activations, resulting in slow convergence or limited compression. To address this, we bridge the relationship between the algorithm's fast convergence and the requirements for subspace projection, and show that an effective compression should yield an unbiased estimate of the original activation with low variance. We propose Principal-Random Subspace for LLM Activation Compression (PRAC), which novelly decomposes activations into two components: a principal subspace captured via SVD to retain dominant information, and a random subspace sampled from the orthogonal complement to approximate the tail. By introducing a precise scaling factor, we prove that PRAC yields an unbiased gradient estimator with minimum variance under certain conditions. Extensive experiments on pre-training and fine-tuning tasks demonstrate that PRAC achieves up to 36% total memory reduction with negligible performance degradation and minimal computational cost.

Zeroth-order (ZO) optimization offers a memory-efficient alternative for LLM fine-tuning by estimating updates only from forward evaluations of perturbed parameters, without backpropagation or activation storage. However, in billion-parameter LLMs, isotropic perturbations often waste many forward evaluations on weakly informative directions. To make these evaluations more informative, existing ZO methods restrict perturbations to low-dimensional subspaces. Yet the quality of these subspaces is critical: overly compressed or poorly maintained spaces can miss useful update directions. To obtain a high-quality subspace for ZO updates, this paper proposes AIM-ZO, a ZO fine-tuning method based on Activation-Informed Subspace Maintenance. AIM-ZO uses forward activations as local directional information and continuously integrates them into a broad, evolving subspace over training. To access broader gradient-relevant structure while keeping individual perturbations low-dimensional, AIM-ZO activates only a smaller set of shared and sampled directions, decoupling the maintained width from the active width. We evaluate AIM-ZO across 5 LLMs and 11 downstream tasks under matched forward-evaluation budgets; its six-task average exceeds the strongest fully evaluated ZO baseline by 1.26 percentage points on OPT-2.7B and MeZO by 2.85 percentage points on OPT-30B. Our code is available at https://github.com/EkkoXy/AIM-ZO

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.