Bayesian Subspace Gradient Estimation for Zeroth-Order Optimization of Large Language Models

Abstract: Fine-tuning LLMs with zeroth-order (ZO) optimization reduces memory by approximating gradients through function evaluations, but existing methods rely on one-step gradient estimates from random perturbations. We introduce Bayesian Subspace Zeroth-Order optimization (BSZO), a ZO optimizer that applies Kalman filtering to combine finite-difference information across multiple perturbation directions. By treating each finite-difference measurement as a noisy observation, BSZO builds a posterior distribution over the projected gradient and updates it through Bayesian inference, with a residual-based adaptive mechanism to adjust perturbation scales. Theoretical analysis shows that BSZO improves the convergence rate by a factor of $k/γ$ compared to standard ZO methods. Experiments on RoBERTa, Mistral, and OPT models show that BSZO outperforms MeZO, MeZO-Adam, and HiZOO across various tasks, achieving up to 6.67\% absolute average improvement on OPT-13B while keeping memory usage close to inference-only baselines (1.00$\times$--1.08$\times$ of MeZO).