Zeroth-Order Riemannian Optimization on Fixed-Rank Manifolds for LLM Fine-Tuning

Zeroth-order (ZO) optimization estimates descent directions from function values alone, which removes the activation storage required by backpropagation and makes it attractive for memory-efficient fine-tuning of large language models (LLMs). Existing low-rank ZO methods either parameterize the additive update through Euclidean factors with a non-unique representation, or restrict each finite-difference direction to be low-rank. In this work, we propose to constrain each additive update block to a fixed-rank matrix manifold while keeping the pretrained weights frozen, so the learned update remains rank-constrained throughout fine-tuning. On this geometry, we build a Riemannian ZO gradient estimator that projects a random matrix onto the tangent space, normalizes it to correct the scale mismatch induced by projection, and further propose Zeroth-order Riemannian Gradient Descent (ZO-RGD) and its smoothed variant, ZO-RGD-S. We show that the ambient finite difference matches its retracted counterpart, the normalized estimator recovers the Riemannian gradient in expectation, and ZO-RGD converges under trajectory-regularity conditions. Experiments on LLM fine-tuning tasks show that the proposed algorithms are competitive with strong forward-only baselines across model families and tasks, with gains in several settings.