Differential Privacy in Distributed Learning: Beyond Uniformly Bounded Stochastic Gradients

This paper explores locally differentially private distributed algorithms that solve non-convex empirical risk minimization problems. Traditional approaches often assume uniformly bounded stochastic gradients, which may not hold in practice. To address this issue, we propose differentially **Pri**vate **S**tochastic recursive **M**omentum with gr**A**dient clipping (PriSMA) that judiciously integrates clipping and momentum to enhance utility while guaranteeing privacy. Without assuming uniformly bounded stochastic gradients, given privacy requirement $(\epsilon,\delta)$, PriSMA achieves a learning error of $\tilde{\mathcal{O}}\big((\frac{\sqrt{d}}{\sqrt{M}N\epsilon})^\frac{2}{5}\big)$, where $M$ is the number of clients, $N$ is the number of data samples on each client and $d$ is the model dimension. This learning error bound is better than the state-of-the-art $\tilde{\mathcal{O}}\big((\frac{\sqrt{d}}{{\sqrt{M}N\epsilon}})^\frac{1}{3}\big)$ in terms of the dependence on $M$ and $N$.