We develop a new algorithm for non-convex stochastic optimization that finds an $\epsilon$-critical point in the optimal $O(\epsilon^{-3})$ stochastic gradient and Hessian-vector product computations. Our algorithm uses Hessian-vector products to correct'' a bias term in the momentum of SGD with momentum. This leads to better gradient estimates in a manner analogous to variance reduction methods. In contrast to prior work, we do not require excessively large batch sizes and are able to provide an adaptive algorithm whose convergence rate automatically improves with decreasing variance in the gradient estimates.

Zeroth-order optimization (ZO) has demonstrated remarkable promise in efficient fine-tuning tasks for Large Language Models (LLMs). In particular, recent advances incorporate the low-rankness of gradients, introducing low-rank ZO estimators to further reduce GPU memory consumption. However, most existing works focus solely on the low-rankness of each individual gradient, overlooking a broader property shared by all gradients throughout the training, i.e., all gradients approximately reside within a similar subspace. In this paper, we consider two factors together and propose a novel low-rank ZO estimator, TeZO, which captures the low-rankness across both the model and temporal dimension. Specifically, we represent ZO perturbations along the temporal dimension as a 3D tensor and employ Canonical Polyadic Decomposition (CPD) to extract each low-rank 2D matrix, significantly reducing the training cost. TeZO can also be easily extended to the Adam variant while consuming less memory than MeZO-SGD, and requiring about only 35% memory of MeZO-Adam. Both comprehensive theoretical analysis and extensive experimental research have validated its efficiency, achieving SOTA-comparable results with lower overhead of time and memory.

We develop a new algorithm for non-convex stochastic optimization that finds an \epsilon -critical point in the optimal O(\epsilon {-3}) stochastic gradient and Hessian-vector product computations. Our algorithm uses Hessian-vector products to "correct'' a bias term in the momentum of SGD with momentum. This leads to better gradient estimates in a manner analogous to variance reduction methods. In contrast to prior work, we do not require excessively large batch sizes and are able to provide an adaptive algorithm whose convergence rate automatically improves with decreasing variance in the gradient estimates.