SPriFed-OMP: A Differentially Private Federated Learning Algorithm for Sparse Basis Recovery
This addresses a critical gap for privacy-sensitive applications in high-dimensional data analysis, though it is incremental as it adapts OMP to federated settings with privacy enhancements.
The paper tackles the problem of sparse basis recovery in federated learning with differential privacy, where existing methods degrade when dimensions exceed samples. It introduces SPriFed-OMP and an enhanced version, achieving efficient recovery with only O(√p) samples and outperforming prior DP-FL solutions in accuracy-privacy trade-off.
Sparse basis recovery is a classical and important statistical learning problem when the number of model dimensions $p$ is much larger than the number of samples $n$. However, there has been little work that studies sparse basis recovery in the Federated Learning (FL) setting, where the client data's differential privacy (DP) must also be simultaneously protected. In particular, the performance guarantees of existing DP-FL algorithms (such as DP-SGD) will degrade significantly when $p \gg n$, and thus, they will fail to learn the true underlying sparse model accurately. In this work, we develop a new differentially private sparse basis recovery algorithm for the FL setting, called SPriFed-OMP. SPriFed-OMP converts OMP (Orthogonal Matching Pursuit) to the FL setting. Further, it combines SMPC (secure multi-party computation) and DP to ensure that only a small amount of noise needs to be added in order to achieve differential privacy. As a result, SPriFed-OMP can efficiently recover the true sparse basis for a linear model with only $n = O(\sqrt{p})$ samples. We further present an enhanced version of our approach, SPriFed-OMP-GRAD based on gradient privatization, that improves the performance of SPriFed-OMP. Our theoretical analysis and empirical results demonstrate that both SPriFed-OMP and SPriFed-OMP-GRAD terminate in a small number of steps, and they significantly outperform the previous state-of-the-art DP-FL solutions in terms of the accuracy-privacy trade-off.