Privacy Amplification in Differentially Private Zeroth-Order Optimization with Hidden States

Zeroth-order optimization has emerged as a promising approach for fine-tuning large language models under differential privacy (DP) and memory constraints. While privacy amplification by iteration (PABI) provides convergent DP bounds for first-order methods, establishing similar guarantees for zeroth-order methods remains an open problem. First-order PABI analysis relies on the fact that gradients are perturbed with isotropic noise, allowing privacy bounds to be iteratively tracked via shifted R'enyi divergence. In contrast, DP zeroth-order methods inject scalar noise along random update directions to maintain utility. This anisotropic update fails standard shifted divergence frameworks, as the global Lipschitz property no longer holds almost surely. We provide the first convergent hidden-state DP bound for zeroth-order optimization by proposing a hybrid noise mechanism and a novel coupling analysis. We bypass the purely shifted-divergence approach by constructing a coupled auxiliary process, which circumvents the global Lipschitz barrier and yields a convergent privacy bound. Furthermore, our results induce better DP zeroth-order algorithmic designs that are previously unknown to the literature.