Enhancing Zeroth-order Fine-tuning for Language Models with Low-rank Structures

Parameter-efficient fine-tuning (PEFT) significantly reduces memory costs when adapting large language models (LLMs) for downstream applications. However, traditional first-order (FO) fine-tuning algorithms incur substantial memory overhead due to the need to store activation values for back-propagation during gradient computation, particularly in long-context fine-tuning tasks. Zeroth-order (ZO) algorithms offer a promising alternative by approximating gradients using finite differences of function values, thus eliminating the need for activation storage. Nevertheless, existing ZO methods struggle to capture the low-rank gradient structure common in LLM fine-tuning, leading to suboptimal performance. This paper proposes a low-rank ZO gradient estimator and introduces a novel low-rank ZO algorithm (LOZO) that effectively captures this structure in LLMs. We provide convergence guarantees for LOZO by framing it as a subspace optimization method. Additionally, its low-rank nature enables LOZO to integrate with momentum techniques while incurring negligible extra memory costs. Extensive experiments across various model sizes and downstream tasks demonstrate that LOZO and its momentum-based variant outperform existing ZO methods and closely approach the performance of FO algorithms.

Linear interpolation gives better gradients than Gaussian smoothing in derivative-free optimization

In this paper, we consider derivative free optimization problems, where the objective function is smooth but is computed with some amount of noise, the function evaluations are expensive and no derivative information is available. We are motivated by policy optimization problems in reinforcement learning that have recently become popular [Choromaski et al. 2018; Fazel et al. 2018; Salimans et al. 2016], and that can be formulated as derivative free optimization problems with the aforementioned characteristics. In each of these works some approximation of the gradient is constructed and a (stochastic) gradient method is applied. In [Salimans et al. 2016] the gradient information is aggregated along Gaussian directions, while in [Choromaski et al. 2018] it is computed along orthogonal direction. We provide a convergence rate analysis for a first-order line search method, similar to the ones used in the literature, and derive the conditions on the gradient approximations that ensure this convergence. We then demonstrate via rigorous analysis of the variance and by numerical comparisons on reinforcement learning tasks that the Gaussian sampling method used in [Salimans et al. 2016] is significantly inferior to the orthogonal sampling used in [Choromaski et al. 2018] as well as more general interpolation methods.